Grade 10 workbook paper 2

Compiled by Chesley Nell

Grade 10 Core Mathematics

2

GOMATH WORKBOOKS

Forward: Welcome to “ GO MATH WORKBOOKS”. This workbook is designed to be text book and class work book in one.

There are sufficient exercises to ensure that learners get the required practice.

A detailed memorandum booklet is available for each workbook.

The statement “ You get out what you put in.” is very apt where mathematics is concerned.

To succeed in mathematics one must be prepared to invest the time and effort to achieve that success.

The partnership that you as a learner and this GO MATH WORKBOOK develop will be profitable if you allow it to be.

Chesley Nell: Mathematics Educator.  Chesley Nell 2011

Grade 10 Core Mathematics

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GRADE 10 CORE MATHEMATICS. PAPER TWO:

Topic:

Pages:

1.

Analytical Geometry

(4 – 21)

2.

Triangles & Quadrilaterals

(22 – 40)

3.

Trigonometry

(41 - 67)

4.

Data Handling

(68 – 100)

5.

Volumes & Surface Area

(101 - 115)

Grade 10 Core Mathematics

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PAPER TWO: 1.

ANALYTICAL GEOMETRY

Analytical geometry - Studies the properties of geometric figures Algebraically. This is pursued by the means of examining significant points (co-ordinates) of these figures in a Cartesian Plane. Hence also referred to as Co-ordinate Geometry.

Formulae: 1.

Length of a line:

A(2 ; 5)

B(-4 ; -3)

Length of AB = (x1  x 2 ) 2  (y1  y 2 ) 2 = (6) 2  (8) 2 = 100 = 10 2.

Mid – Point of a line  (x  x 2 ) (y 1  y 2 )  ; Mid – point =  1  2 2    A(2 ; 5) C (x ; y )

B(-4 ; -3)

Mid – Point AB = C (-1; 1) 3.

Gradient of Straight Line: Gradient is represented using the symbol ‘m’ [from y= mx+c] M=

y [ i.e the difference in y divided by the difference in x] x

A( 2;4)

m

B(3:6)

y y1  y 2 6  (4) 10 2     x x1  x2 3  (2) 5 1

Parallel Lines have the same gradients: m1  m2 Perpendicular lines have inverse gradients: m1  m2  1

Grade 10 Core Mathematics

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Exercise 1.1: Distance between points: 1.

Find the distance between the given pairs of points: 1.1 (3;7 ) and (5 ; -2)

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 1.2 (4 ; 5) and ( -4 ;3) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.3 (2 ; -3) and (-2 ; 7) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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1.4 (-7 ; 1) and (0 ; 0) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.5 (-3 ; 2) and (-6 ; -7) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.6 (-5 ;-4) and (3 ; -8) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

2.

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Given the coordinates of the vertices of ď „ABC , in each case ( 2.1 to 2.5) Determine: A. the perimeter of the triangle.

B. Whether the triangle is equilateral, isosceles or scalene.

C. Whether or not the triangle has a right angle.

2.1

A(2 ; -5) ; B(5 ; 5); C(-2 ; 4)

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2.2

A(5 ;1) ; B(1 ; 3) ; C(1 ; -2)

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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2.3 A(2 ; 1) ; B(2 ; -2) ; C(7 ; -2) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2.4

A(0 ; 0) B( 3 ; 1) ; C( 3 ; -1)

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

2.5

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A(-4 ; 1) B(-1 ;-3) ;C(0; 4)

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 3.

Show that: 3.1

A(-3 ; 2) , B(3 ;6), C(9 ;-2) and D(3 ; -6) are vertices of a parallelogram.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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3.2 (6 ;-4) , (5 ;3) (-2 ; 2) and (-1 ; -5) are vertices of a square. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ Mid-points of lines: 4.

Calculate the coordinates of the midpoints of the line joining the following points: 4.1 (-3 ;1) and (1 ; 5)

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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4.2 (-2 ; 3) and (6 ; 3) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 4.3 (4 ; -1) and (-1 ; 3) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 4.4 (0 ;0 ) and (3 ; -8) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 4.5 ( 3;1) and (3 3;1) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

5.

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Determine the values of x and y if: 5.1 (-3 ; 2) is the mid-point of the line joining (-1 ; 5) and (x ; y).

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 5.2

(-1 ; y) is the mid-point of the line joining (0 ; -2) and x ; 8)

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

5.3

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(x ; y) is the centre of a circle on diameter AB where A(-2 ; -1) and B(-1 ; 9).

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 5.4 (x ; 3) is the centre of a circle with diameter MN. M (5 ; -2) and N(-7 ; y) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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Exercise 1.2: Formulae:

AB  ( x1  x2 ) 2  ( y1  y2 ) 2 Gradient = m =

 x  x y  y2  Mid-point =  1 2 ; 1  2   2

y1  y2 x1  x2

1.

AB is a straight line on a Cartesian plane where A(-3; -4) and B( 2 ; 6) Calculate the following:

1.1

the length of AB in units.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

1.2

the co-ordinates of the mid – point ( C ) of line AB.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.3

the gradient of line AB.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

1.4

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show that points A;B and C are collinear.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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2. y

D

A C 0

x

B

The Points A(-4 ;3) ; B(-4 ; -4) ; C(6 ; 1) and D(6 ; 8) lie on a cartesian plane.

Determine: 2.1

the length of AD.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

2.2

the mid-point of DC

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

2.3

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the gradient of BC

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

2.4

show that ABCD is a parallelogram.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

2.5

the co-ordinates of the point of intersection of the diagonals AC & BD

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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3.

4

A(-3;3) C(8;2)

2

-5

5

-2

B(3;-3) -4

The figure above represents a triangle on a Cartesian plane.

A( -3 ;3) ; B(3 ; -3) and C(8 ; 2) 3.1 Calculate the perimeter (distance around) of ď „ABC .

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

10

Grade 10 Core Mathematics

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GOï‚·MATH WORKBOOKS

3.2 Prove that triangle ABC is right angled at B.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

3.3 Give the coordinates of the mid -point of AC, AB & BC ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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4.

Determine whether the following triangles are Isosceles, Equilateral or Scalene.

4.1

A(1;2) , B (6;3) and C (6;1)

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

P (4;1) , Q (3;0) and R (1;3) 4.2 ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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GOMATH WORKBOOKS

U (5;2) , V (1;1) and W (13;1) 4.3 ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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1. Triangles: Theorems: 1.

The sum of the angles of a triangle equals 180ď‚° A

E

2 1

3

D

C

B

Given: ∆ABC. RTP: đ??´Ě‚ + đ??ľĚ‚ + đ??śĚ‚ = 180ď‚° Proof: Produce BC to D and draw CE parallel to AB. đ??śĚ‚2 = đ??´Ě‚ (đ?‘Žđ?‘™đ?‘Ą ďƒ?, đ?‘ đ??´đ??ľ ď źď ź đ??śđ??¸) đ??śĚ‚3 = đ??ľĚ‚ (đ?‘?đ?‘œđ?‘&#x;đ?‘&#x; ďƒ?, đ?‘ đ??´đ??ľď źď ź đ??śđ??¸) But đ??śĚ‚1 + đ??śĚ‚2 + đ??śĚ‚3 = 180ď‚° ( ďƒ?, đ?‘ đ?‘œđ?‘› đ?‘Ž đ?‘ đ?‘Ą đ?‘™đ?‘–đ?‘›đ?‘’) ď œđ??´Ě‚ + đ??ľĚ‚ + đ??śĚ‚ = 180ď‚° 2.

The exterior angle of a triangle is equal to the sum of the interior opposite angles A E

2 1

B

Given: ∆ABC with BC produced to D. RTP: đ??´đ??śĚ‚ đ??ˇ = đ??´Ě‚ + đ??ľĚ‚ Proof: Draw CE parallel to AB. đ??śĚ‚1 = đ??´Ě‚ (đ?‘Žđ?‘™đ?‘Ą ďƒ?, đ?‘ đ??´đ??ľ đ??źđ??ź đ??śđ??¸ đ??śĚ‚2 = đ??ľĚ‚ (đ?‘?đ?‘œđ?‘&#x;đ?‘&#x; ďƒ?, đ?‘ đ??´đ??ľ đ??źđ??ź đ??śđ??¸

C

3

D

Grade 10 Core Mathematics

3.

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In an isosceles triangle , the sides opposite the equal angles are equal.

A 1 2





B

C

D

Given: ∆đ??´đ??ľđ??ś đ?‘¤đ?‘–đ?‘Ąâ„Ž đ??ľĚ‚ = đ??śĚ‚ RTP: AB = AC. Proof: Construct AD to bisect đ??´Ě‚ , cutting BC at D. In ∆đ??´đ??ľđ??ˇ đ?‘Žđ?‘›đ?‘‘ ∆đ??´đ??śđ??ˇ Ě‚1 = đ??´ Ě‚2 (đ?‘?đ?‘œđ?‘›đ?‘ đ?‘Ąđ?‘&#x;đ?‘˘đ?‘?đ?‘Ąđ?‘–đ?‘œđ?‘›) đ??´ Ě‚ đ??ľĚ‚ = đ??ś ( đ?‘”đ?‘–đ?‘Łđ?‘’đ?‘›) AD is common ∆đ??´đ??ľđ??ˇ ≥ ∆đ??´đ??śđ??ˇ (đ??´đ??´đ?‘†) ∴ đ??´đ??ľ = đ??´đ??ś Conversely: The angles opposite equal sides of a triangle are equal.

4.

The Mid – Point Theorem.

The line joining the mid-points of two sides of a triangle is parallel to the third side and equal to half its length.

1

RTP: DE ‖ đ??ľđ??ś and đ??ˇđ??¸ = 2 đ??ľđ??ś A

D

1

E

F

2

B

C

Proof: Construct EF = DE and join FC. đ??źđ?‘› ∆đ??´đ??ˇđ??¸ đ?‘Žđ?‘›đ?‘‘ ∆đ??śđ??šđ??¸ AE = CE ( Given) DE = EF ( by construction) đ??¸Ě‚1 = đ??¸Ě‚2 (đ?‘‰đ?‘’đ?‘&#x;đ?‘Ąđ?‘–đ?‘?đ?‘Žđ?‘™đ?‘™đ?‘Ś đ?‘œđ?‘?đ?‘?đ?‘œđ?‘ đ?‘–đ?‘Ąđ?‘’ đ?‘Žđ?‘›đ?‘”đ?‘™đ?‘’đ?‘ ) ∆đ??´đ??ˇđ??¸ ≥ ∆đ??śđ??šđ??¸ ( đ?‘†đ??´đ?‘†) Ě‚ đ??¸ = đ??śđ??šĚ‚ đ??¸ ∴ đ??´đ??ˇ ∴ đ??´đ??ľâ€–đ??šđ??ś ( đ??´đ?‘™đ?‘Ąđ?‘’đ?‘&#x;đ?‘›đ?‘Žđ?‘Ąđ?‘’ đ?‘Žđ?‘›đ?‘”đ?‘™đ?‘’đ?‘ đ?‘Žđ?‘&#x;đ?‘’ đ?‘’đ?‘žđ?‘˘đ?‘Žđ?‘™) BD = DA (given) BD= FC ( proved above) ∴ đ??ˇđ??śđ??šđ??ľ đ?‘–đ?‘ đ?‘Ž đ?‘?đ?‘Žđ?‘&#x;đ?‘Žđ?‘™đ?‘™đ?‘’đ?‘™đ?‘œđ?‘”đ?‘&#x;đ?‘Žđ?‘š ( đ?‘œđ?‘›đ?‘’ đ?‘?đ?‘Žđ?‘–đ?‘&#x; đ?‘œđ?‘?đ?‘?đ?‘œđ?‘ đ?‘–đ?‘Ąđ?‘’đ?‘ đ?‘ đ?‘–đ?‘‘đ?‘’đ?‘ đ?‘’đ?‘žđ?‘˘đ?‘Žđ?‘™ đ?‘Žđ?‘›đ?‘‘ đ?‘?đ?‘Žđ?‘&#x;đ?‘Žđ?‘™đ?‘™đ?‘’đ?‘™) ∴ đ??ˇđ??š = đ??ľđ??ś ( đ?‘œđ?‘?đ?‘?đ?‘œđ?‘ đ?‘–đ?‘Ąđ?‘’ đ?‘ đ?‘–đ?‘‘đ?‘’đ?‘ đ?‘œđ?‘“ đ?‘Ž đ?‘?đ?‘Žđ?‘&#x;đ?‘Žđ?‘™đ?‘™đ?‘’đ?‘™đ?‘œđ?‘”đ?‘&#x;đ?‘Žđ?‘š) 1 đ??ˇđ??¸ = 2 đ??ľđ??ś and đ??ˇđ??¸â€–đ??ľđ??ś

Grade 10 Core Mathematics

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Congruence: Definition: Two triangles are said to be congruent if they are equal in all respects.

There are four cases of congruence: i.e. methods of proving triangles are congruent. â&#x;šTwo angles and one corresponding side are equal. ( AAS). â&#x;šTwo sides and the angle between them are equal. (SAS). â&#x;šThree sides are equal. (SSS) â&#x;šRight angle, hypotenuse and one further side are equal. (RHS) NB: If two parallel lines are cut by a transversal the angles formed are: Alternate angles, which are equal.  

Corresponding angles, which are equal. 



Co-interior angles, which are supplementary. x x + y = 180ď‚° y

L1

1 4

2 3

L2

If two lines intersect then two types of angles are formed. ( đ?‘łđ?&#x;? & đ?‘łđ?&#x;? ) 1.

Adjacent supplementary angles ( angles on a straight line) i.e. 1 + 2 = 180° & 3 + 4 = 180°.

2.

Vertically opposite angles. đ?‘–. đ?‘’ 1 = 3 & 2 = 4

Grade 10 Core Mathematics

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Exercise 3.1: 1.

In the sketch below, AB = CD and đ??´đ??ľĚ‚ đ??ś = đ??ˇđ??śĚ‚ đ??ľ .

A

D E

B

C

Prove that: 1.1 ∆đ??´đ??ľđ??ś ≥ ∆đ??ˇđ??śđ??ľ _________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 1.2 ∆đ??´đ??ľđ??¸ ≥ ∆đ??ˇđ??śđ??¸ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

2.

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In the sketch below đ?‘„đ?‘… = đ?‘†đ?‘‡ ; đ?‘ƒđ?‘… = đ?‘ƒđ?‘† đ?‘Žđ?‘›đ?‘‘ đ?‘ƒđ?‘…Ě‚ đ?‘† = đ?‘ƒđ?‘†Ě‚đ?‘….

P

Q

2.1

R

S

T

Prove ∆đ?‘ƒđ?‘…đ?‘„ ≥ ∆đ?‘ƒđ?‘†đ?‘‡.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 2.2

Deduce that PQ = PT

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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Prove that ∆đ??´đ??¸đ??ˇ ≥ ∆đ??ľđ??śđ??ˇ in the diagram below: Given: AD = BD ; AE = BC; AE ď ž BD and BC ď ž AD. A C

F D E

B

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ __________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

4.

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Prove that ∆đ??´đ??ľđ??ˇ ≥ ∆đ??´đ??ľđ??ś in the diagram below: Given: AD = BC ; đ??ľđ??´Ě‚đ??ˇ = đ??´đ??ľĚ‚ đ??ś D

A

C

B

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

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Similarity: Two triangles are said to be similar if they are equiangular or conversely if the ratios of their corresponding sides are in proportion.

Examples: A

A r

D x

r

D

E 6cm

8cm

3cm

2cm

E 4cm

y x

B

F

F

y

C

B

­Юљ╝­ЮЉЏ РѕєРђ▓ ­ЮЉа­Юљ┤­Юљх­ЮљХ ­ЮЉј­ЮЉЏ­ЮЉЉ ­ЮљИ­Юљи­Юљ╣: ­Юљ┤╠ѓ = ­ЮљИ╠ѓ (­ЮЉћ­ЮЉќ­ЮЉБ­ЮЉњ­ЮЉЏ) ╠ѓ (­ЮЉћ­ЮЉќ­ЮЉБ­ЮЉњ­ЮЉЏ) ­Юљх╠ѓ = ­Юљи ­ЮљХ╠ѓ = ­Юљ╣╠ѓ (­ЮЉћ­ЮЉќ­ЮЉБ­ЮЉњ­ЮЉЏ) Рѕ┤ Рѕє­Юљ┤­Юљх­ЮљХ ле Рѕє­ЮљИ­Юљи­Юљ╣ ( ­Юљ┤­Юљ┤­Юљ┤) ­Юљ┤­Юљх ­Юљх­ЮљХ ­Юљ┤­ЮљХ Рѕ┤ ­ЮљИ­Юљи = ­Юљи­Юљ╣ = ­ЮљИ­Юљ╣

4cm

C

­Юљ╝­ЮЉЏ РѕєРђ▓ ­ЮЉа­Юљ┤­Юљх­ЮљХ ­ЮЉј­ЮЉЏ­ЮЉЉ ­ЮљИ­Юљи­Юљ╣: ­Юљ┤­Юљх 6 =3=2 ­ЮљИ­Юљи ­Юљх­ЮљХ

­Юљи­Юљ╣ ­Юљ┤­ЮљХ

4

=2=2 8

=4=2 Рѕ┤ Рѕє­Юљ┤­Юљх­ЮљХ ле Рѕє­ЮљИ­Юљи­Юљ╣ (­ЮЉа­ЮЉќ­ЮЉЉ­ЮЉњ­ЮЉа ­ЮЉќ­ЮЉЏ ­ЮЉЮ­ЮЉЪ­ЮЉю­ЮЉЮ­ЮЉю­ЮЉЪ­ЮЉА­ЮЉќ­ЮЉю­ЮЉЏ) ­ЮљИ­Юљ╣

Triangles: Properties: 1. The sum of the interior angles is 180№ѓ░.

2. The exterior angle of a triangle is equal to sum of the interior opposite angles.

3. Isosceles triangles have two sides equal and the angles opposite these sides are also equal.

4. Equilateral triangles have all three sides and all three angles equal.

5. Scalene triangles have no sides or angles equal.

Grade 10 Core Mathematics

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Quadrilaterals:

Number None

opposites sides parallel

Both pairs

Both pairs

One pair Both pairs



One pair

  













 











 



 

Square

Rectangle

Rhombus



 



 

 

 



 

Both pairs None



 

All four sides None

Opposite angles equal



One pair

None

Adjacent sides equal



One pair

None opposites sides equal

Parallelogram

Property

Trapezium

Kite

Properties of Quadrilaterals





  







 

 









 



One pair Consecutive angles equal

Both pairs All four angles none only one

Line of symmetry











 





 



 

only two





 

four

Diagonals are equal Diagonals bisect each other



Diagonals are perpendicular



 







 

Grade 10 Core Mathematics

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31

Relationship between sides and angles:

Number of sides of a polygon 3 4 5 6 7 8 9 10 n Sum of interior angles 180º 360º 540º 720º 900º 1080º 1260º 1440º (n-2)180º Size of one interior angle of regular polygons 60º

90º

108º 120º 128,570º

135º

140º

144º

(n-2)180º n

The pattern is (number of sides-2)180º Sum of interior Angles = (n-2)180º Number of sides of a polygon 3 4 5 6 7 8 9 Sum of exterior angles 360º 360º 360º 360º 360º 360º 360º

Number of sides of a polygon 4 5 6 7 8 9 Number of Triangles in a Regular Polygon 2 3 4 5 6 7

3 1

10 360º

10 8

Exercise 3.2: 1

Find the values of the variables in the following (i.e. x ; y & z) 1.1

AB//DC and AB=DC A

B



x

z

62 D



y C

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

1.2

GOMATH WORKBOOKS

32

PQRS is a rectangle: P

Q



36

y x



S

R

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 1.3

ABCD is a rhombus: A

B

y

35

x D

C

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

1.3

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33

LMNO is a trapezium with LM // PO and LM = LP ; đ??żĚ‚ = 105° Ě‚ đ?‘‚ = 65° and đ?‘ƒđ?‘€ L

105°

M

ď‚Ž

x

65°

y z

ď‚Ž

P

O

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

1.5

ABCD is a kite. AB = AC & BD = CD. đ??´đ??śĚ‚ đ??ľ = 60ď‚°, đ??ˇđ??ľĚ‚ đ??ś = 70° A x

B

60ď‚°

y 70ď‚°

C

z

D

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

1.6

34

GOï‚·MATH WORKBOOKS

ABCDEF is a regular hexagon.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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35

Paralellograms: Theorem A Given: Parallelogram ABCD Ě‚ RTP: AB = CD ; AD = BC; đ??´Ě‚ = đ??śĚ‚ đ?‘Žđ?‘›đ?‘‘ đ??ľĚ‚ = đ??ˇ Proof: Construct diagonal BD Ě‚2 đ??ľĚ‚1 = đ??ˇ (đ?‘Žđ?‘™đ?‘Ąđ?‘’đ?‘&#x;đ?‘›đ?‘Žđ?‘Ąđ?‘’ đ?‘Žđ?‘›đ?‘”đ?‘™đ?‘’đ?‘ đ??´đ??ľâ€–đ??śđ??ˇ) Ě‚ Ě‚ đ??ˇ1 = đ??ľ2 (đ?‘Žđ?‘™đ?‘’đ?‘Ąđ?‘&#x;đ?‘’đ?‘›đ?‘Žđ?‘Ąđ?‘’ đ?‘Žđ?‘›đ?‘”đ?‘™đ?‘’đ?‘ đ??´đ??ˇâ€–đ??ľđ??ś) đ??ľđ??ˇ = đ??ľđ??ˇ (đ?‘?đ?‘œđ?‘šđ?‘šđ?‘œđ?‘›) ∴ ∆đ??´đ??ľđ??ˇ ≥ ∆đ??śđ??ˇđ??ľ (đ??´đ??´đ?‘†) ∴ đ??´đ??ˇ = đ??ľđ??ś ∴ đ??´đ??ľ = đ??śđ??ˇ Ě‚2 đ?‘Žđ?‘›đ?‘‘ đ??ˇ Ě‚1 = đ??ľĚ‚2 đ??ľĚ‚1 = đ??ˇ Ě‚ đ??ľ đ?‘Žđ?‘›đ?‘‘ đ??´đ??ˇ Ě‚ đ??ľ = đ??śđ??ľĚ‚ đ??ˇ đ??´đ??ľĚ‚ đ??ˇ = đ??śđ??ˇ Theorem B: Converse to A: States that if the opposite sides of a quadrilateral are equal then the quadrilateral is a parallelogram.

Theorem C: Converse to A: States that if the opposite angles of a quadrilateral are equal then the quadrilateral ios a parallelogram.

To prove that a quadrilateral is a parallelogram one must be able to prove at least two of the properties of this particular shape, Properties concerned are: 1.

both pairs of opposite sides are parallel.

2.

both pairs of opposite sides are equal.

3.

opposite angles are equal.

4.

the diagonals bisect each other.

Hint: a good method is to prove that one pair of opposite sides are equal and parallel.

Examples: 1. Prove that ABCD is a parallelogram.

4

B (0 ; 3) 2

A (-4 ; 0) -10

C (1 ; 0)

-5

-2

D (-3 ; -3) -4

5

10

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36

Proof: đ??´đ??ľ = √(0 + 4)2 + (3 − 0)2 = 5 đ??ˇđ??ś = √(1 + 3)2 + (0 + 3)2 = 5

3

đ?‘šđ??´đ??ľ = 4 3

đ?‘šđ??ˇđ??ś = 4

ABCD is a parallelogram ( one pair of opposite sides are equal and parallel) 2. ABCD is a quadrilateral with AB = CD & AD = BC A 2

B 1

1

D

2.1

2.2

2 1 C

RTP: ∆đ?‘¨đ?‘Ťđ?‘Ş â‰Ą ∆đ?‘¨đ?‘Šđ?‘Ş Proof: đ??źđ?‘› ∆đ??´đ??ˇđ??ś đ?‘Žđ?‘›đ?‘‘ ∆đ??´đ??ľđ??ś đ??´đ??ˇ = đ??ľđ??ś ( đ?‘”đ?‘–đ?‘Łđ?‘’đ?‘›) đ??śđ??ˇ = đ??´đ??ľ ( đ?‘”đ?‘–đ?‘Łđ?‘’đ?‘›) đ??´đ??ś = đ??´đ??ś (đ?‘?đ?‘œđ?‘šđ?‘šđ?‘œđ?‘› đ?‘ đ?‘–đ?‘‘đ?‘’) ∆đ?‘¨đ?‘Ťđ?‘Ş â‰Ą ∆đ?‘¨đ?‘Šđ?‘Ş (đ?‘şđ?‘şđ?‘ş) RTP: đ?‘¨đ?‘Šâ€–đ?‘Ťđ?‘Ş đ?’‚đ?’?đ?’… đ?‘¨đ?‘Ťâ€–đ?‘Šđ?‘Ş Proof: đ??´Ě‚2 = đ??śĚ‚2 (đ?‘?đ?‘&#x;đ?‘œđ?‘Łđ?‘’đ?‘‘ đ?‘–đ?‘› 2.1 ∆đ??´đ??ˇđ??ś ≥ ∆đ??´đ??ľđ??ś đ??´đ??ˇâ€–đ??ľđ??ś ( alternate angles equal) đ??´Ě‚1 = đ??śĚ‚1 (đ?‘?đ?‘&#x;đ?‘œđ?‘Łđ?‘’đ?‘‘ đ?‘–đ?‘› 2.1 ∆đ??´đ??ˇđ??ś ≥ ∆đ??´đ??ľđ??ś đ??´đ??ľâ€–đ??ˇđ??ś ( alternate angles equal)

2.3

)

)

RTP: ABCD is a parallelogram Proof: đ??´đ??ľ = đ??śđ??ˇ & đ??´đ??ľâ€–đ??śđ??ˇ (đ?‘?đ?‘&#x;đ?‘œđ?‘Łđ?‘’đ?‘‘ đ?‘Žđ?‘?đ?‘œđ?‘Łđ?‘’) đ??´đ??ˇ = đ??ľđ??ś & đ??´đ??ˇâ€–đ??ľđ??ś (đ?‘?đ?‘&#x;đ?‘œđ?‘Łđ?‘’đ?‘‘ đ?‘Žđ?‘?đ?‘œđ?‘Łđ?‘’) ABCD is a parallelogram ( both pairs opposite sides equal and parallel)

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37

Exercise 3.3: 1.

A (0 ; 4)

B( 5 ; 4)

C ( 2 ; -3)

D ( -3 ; -3)

Prove with reasons that the shape above is a parallelogram. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 2

PRSQ is a parallelogram. RS = ST & TR is a straight line.

Prove: 2.1 QA = AT. 2.2 PT // QS. T

A

P

Q



S

 R

2.1 ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

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38

2.2 ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 3.

FBCD is a parallelogram. AF = FB. Prove that FE = ED

A

F

1 2

1 2

E

D

3

2

B

1

C

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

4.

F

A 2

1

B 2

1

1

D

GOMATH WORKBOOKS

39

2

E

2

1

C

In the figure above, ABCD is a parallelogram. đ??´đ??ˇ = đ??´đ??¸ đ?‘Žđ?‘›đ?‘‘ đ??šđ??ś = đ??ľđ??ś.

Prove that đ??´đ??¸đ??śđ??š is a parallelogram. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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40

5.

P 2

Q 1

2

S

1

R

In the figure above ∆đ?‘ƒđ?‘„đ?‘… đ?‘Žđ?‘›đ?‘‘ ∆đ?‘ƒđ?‘†đ?‘… are isosceles triangles with đ?‘ƒđ?‘† = đ?‘ƒđ?‘… = đ?‘„đ?‘… đ?‘Žđ?‘›đ?‘‘ đ?‘ƒđ?‘„‖đ?‘†đ?‘….

Prove that PQRS is a parallelogram. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

4.1

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41

Trigonometry:

Use the sketches that follow and complete the tables.

The angle sizes at A are 30º ; 45º and 60º. Measure the sides accurately and write the ratios down in decimal form.

The task is to compare the ratios of the different sides of the 3 triangles formed.

Decimal answers to 3 places.

WHAT CONCLUSIONS CAN BE MADE? Task 1: 30º E

F G

30  A

B

C

D

Measure the sides as indicated in the table below and represent the answers first as a fraction and then as a decimal. Compare your answers and come to a conclusion. NB: it is important to be very accurate.

AGB

AFC

BG @30º= _____= AB

FC @30º=______= AC

AED ED @30º=_____= AD

AB @30º=______= AG

AC @30º=______= 0, AF

AD @30º=_____= AE

BG @30º=______= AG

FC @30º=______= AF

ED @30º=_____= AE

AGB

AFC

AB BG @30º= _____=

AC FC @30º=______=

AED AD ED @30º=_____=

AG AB @30º=______=

AF AC @30º=______= 0,

AE AD @30º=_____=

AG BG @30º=______=

AF FC @30º=______=

AE ED @30º=_____=

Grade 10 Core Mathematics

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42

Task 2: 45º

E

F

G

45  A

B

C

D

Measure the sides as indicated in the table below and represent the answers first as a fraction and then as a decimal. Compare your answers and come to a conclusion. NB: it is important to be very accurate.

AGB

AFC

BG @45º= _____= AB

FC @45º=______= AC

AED ED @45º=_____= AD

AB @45º=______= AG

AC @45º=______= AF

AD @45º=_____= AE

BG @45º=______= AG

FC @45º=______= AF

ED @45º=_____= AE

AGB

AFC

AB BG @45º= _____=

AC FC @45º=______=

AED AD ED @45º=_____=

AG AB @45º=______=

AF AC @45º=______=

AE AD @45º=_____=

AG BG @45º=______=

AF FC @45º=______=

AE ED @45º=_____=

Grade 10 Core Mathematics

43

GOMATH WORKBOOKS

Task 3: 60º E

F

G

60 A

B

C

D

Measure the sides as indicated in the table below and represent the answers first as a fraction and then as a decimal. Compare your answers and come to a conclusion. NB: it is important to be very accurate.

AGB

AFC

BG @60º= _____= AB AB @60º=______= AG BG @60º=______= AG

FC @60º=______= AC AC @60º=______= AF FC @60º=______= AF

AED ED @60º=_____= AD AD @60º=_____= AE ED @60º=_____= AE

AGB

AFC

AB BG @60º= _____=

AC FC @60º=______=

AED AD ED @60º=_____=

AG AB @60º=______=

AF AC @60º=______=

AE AD @60º=_____=

AG BG @60º=______=

AF FC @60º=______=

AE ED @60º=_____=

Grade 10 Core Mathematics

GOMATH WORKBOOKS

44

Summary:

C

C h

r y

30Âş

A

x

B

O

30Âş

a

A

B

x  a  adjacent y  o  opposite r  h  hypotenuse In general format

ď „ACB

BC y @30Âş = tan30Âş = x AB AB x @30Âş = cos30Âş = AC r BC y @30Âş = sin30Âş= r AC

Correct Trigonometric format

ď „ACB

y o tan30Âş = OR x a x a cos30Âş = OR r h y o sin30Âş = OR r h

ď „ACB đ??´đ??ľ đ??ľđ??ś đ??´đ??ś đ??ľđ??ś đ??´đ??ś đ??ľđ??ś

@30Âş = cot30Âş = @30Âş = sec30Âş =

đ?‘Ľ đ?‘Ś đ?‘&#x;

@30Âş = cosec30Âş=

đ?‘Ľ đ?‘&#x; đ?‘Ś

cot30Âş = sec30Âş =

đ?‘Ľ đ?‘Ś đ?‘&#x;

ď „ACB OR

đ?‘Ž

đ?‘œ â„Ž

OR đ?‘Ľ đ?‘Ľ đ?‘&#x; â„Ž cosec30Âş = OR đ?‘Ś đ?‘œ

N B: In the above table , the right hand side is the correct way to represent the different trigonometric ratios. It important to remember that a trig ration must be followed by a specific angle size or a variable representing an angle. After the equal sign a ratio is written: e.g. ���60° = 1,732 ‌ ..A trig function cannot stand on its own and must be written with an angle or a variable (denoting an angle).

Grade 10 Core Mathematics

sin x 

y r

cos x 

x r

tan x 

y x

 cos ecx 

 sec x 

r x

 cot x 

x y

GOMATH WORKBOOKS

45

r y

1 cos ecx 1 cos x  sec x

sin x 

1 tan x  cot x

tan x 

sin x cos x

cot x 

cos x sin x

cos2 x  sin 2 x  1

Grade 10 Core Mathematics

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46

4.2 Pythagoras in trigonometry: Solving basic equations & solution of triangles.

Example: Using Pythagoras:

y

 (3 ;4)



x

0

Complete the triangle by joining the point (3;4) to the x – axis to form a right –angled triangle. Use Pythagoras theorem to calculate r (hypotenuse). You are now ready to find the values of the different ratios for .

As follows below:

y

 (3 ;4) 4

r 2  x 2  y 2 (Pythagoras theorem) 4 sin   5 r 2  32  4 2 3 cos  r 2  25 5 r5 4 Tan  3



0

3

x

Grade 10 Core Mathematics

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47

Exercise 4.1: In each of the cases below calculate the length of the radius and the find all the trig ratios for  y 15 e.g tan    x 8 3.1.1.1.1.1.1

2.

y

y

 (6 ;8)

 (8 ;15)



0



x

0

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

x

Grade 10 Core Mathematics

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48

2.

4.

y

y

 (7 ;24)

 (9 ;12)



0



x

0

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

5.

6.

x

Grade 10 Core Mathematics

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49

y

y  (44 ;33)

0

 (40;9)

x

0

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

x

Grade 10 Core Mathematics

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50

In each of the cases below calculate the length of the radius and use the values from the sketch to answer the questions that follow: 7.

8.

y

y

 (11;60) ;40)

 (20;21) ;16)

ď ą

ď ą

x

0 7.1 đ?‘?đ?‘œđ?‘ đ?œƒ =

0 8.1 đ?‘Ąđ?‘Žđ?‘›đ?œƒ =

___________________________________________________________________ ___________________________________________________________________ 7.2

đ?‘ đ?‘–đ?‘›đ?œƒ đ?‘?đ?‘œđ?‘ đ?œƒ

=

8.2 đ?‘ đ?‘–đ?‘›2 đ?œƒ =

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 7.3đ?‘ đ?‘–đ?‘›2 đ?œƒ + đ?‘?đ?‘œđ?‘ 2 đ?œƒ

8.3 1 − đ?‘?đ?‘œđ?‘ 2 đ?œƒ =

____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

x

Grade 10 Core Mathematics

51

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4.3 Equations in trigonometry: Solving for  given a ratio: i.e Solving basic equations: [NB:  is a variable and represents an angle.]

Example: Solve for  where   [0  ;90 ] 1.

sin   0,5

  30

Method: use you calculator as follows: enter {shift} {sin-1 } followed by the ratio 0,5 then equals (=) 2.

Tan x = 3,456 x = 73,86º Method: use you calculator as follows: enter {shift} {tan-1 } followed by the ratio 0,5 then equals (=) NB: The functions sin;cos;tan on the calculator convert trig functions to decimal ratios.

The functions sin-1;cos-1;tan-1 convert trig ratios back to angles.

Exercise 4.2: Solve for x where x  [0  ;90 ] sin x  0,336 .

1.

7. sin( x  10 )  0,800 ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

cos x  0,786 cos(x  15 )  0,642 2.

8. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

tan x  1,732 tan( x  25 )  5,482 3.

9. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

52

GOMATH WORKBOOKS

2 sin x  0,412 4. đ?‘?đ?‘œđ?‘ đ?‘’đ?‘?đ?‘Ľ = 1,1223 10. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

3 cos x  1,236 5. đ?‘ đ?‘’đ?‘?đ?‘Ľ = 1,743 11. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2 tan( x  30ď Ż )  11,3426 6. đ?‘?đ?‘œđ?‘Ąđ?‘Ľ = 2,1445 12. ____________________________________________________________________

___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

53

GOMATH WORKBOOKS

4.4 Solving Triangles:

B h y  A y r x cos  r y tan   x

C

x

sin  

BC AB AC cos  AB BC tan   AC

sin   or

Exercise 4.3 : 1.

A

C D

B

BC AC BD sin A  AB sin A 

Task complete all ratios for above triangle: ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

GOï‚·MATH WORKBOOKS

54 _S

_B _T 1.

2. _C

_A

_P

_R

_Q

_N _0 3.

_L

_M

In diagram 1 & 2: Find the ratios for: 2.1

sin A

2.7

sinP

___________________________________________________________________ ____________________________________________________________________ 2.2

cosA

2.8

cosP

___________________________________________________________________ ____________________________________________________________________ 2.3

tanA

2.9

tanP

___________________________________________________________________ ____________________________________________________________________ 2.4

sinB

2.10

sinT

___________________________________________________________________ ____________________________________________________________________ 2.5

cosB

2.11

sinS

___________________________________________________________________ ____________________________________________________________________ 2.6

tanB

2.12

cosS

___________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

55

GOï‚·MATH WORKBOOKS

In diagram 3: Write down two possible values for: 3.1 sinL __________________________________________________________ ____________________________________________________________________ 3.2 cosL ___________________________________________________________________ ____________________________________________________________________ 3.3 tanL ___________________________________________________________________ ____________________________________________________________________ 3.4 sinN ___________________________________________________________________ ____________________________________________________________________

3.5 cosN ___________________________________________________________________ ____________________________________________________________________

3.6 tanN ___________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

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56

Exercise 4.4 1.

2.

C

C

30cm 15 cm 35ď‚°

56ď‚°

B

A

B

A

Calculate the lengths of BC and AB

Calculate the lengths of AC & AB

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ 3.

S

B

4.

C 24 cm

14 cm

2

P

12 cm

Q

26ď‚° 32ď‚°

1

8 cm

R

A

D

In the diagram above: PQ = 12 cm; In the diagram above: AB = 24 cm BAˆ C  26ď Ż and CAˆ D  32ď Ż QR = 8 cm and SR = 14 cm. Ě‚ = 90° đ?‘…Ě‚ = 90° đ??ľĚ‚ & đ??ˇ Calculate: Calculate : ˆ ˆ 3.1. the size of Q1 and Q2 4.1.

AC ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

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57

3.2. the length of PS. 4.2. the length of CD ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ _____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 4.5

Solution of Triangles ( problem/reality questions)

A ladder rests 4 metres up a wall and has an angle at the foot of the ladder of 45.

Calculate the length of the ladder.

Ladder 4m 45º Answer: 4m 4  Ladder L  L sin 45  4

sin 45 

L sin 45 4   sin 45 sin 45 4 L sin 45

4 sin 45 Ladder  5,7metres NB: L

A x

Angle of depression

y B

Angle of elevation

Line 1 Lines 1 and 2 are parallel.

The angle of elevation from B to A is yº The angle of depression from A to B is xº

Line 2

Grade 10 Core Mathematics

58

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Exercise 4.5: 1.

A swimmer crosses a swiftly flowing river and is washed downstream by the current. He reaches the opposite bank and by swimming a distance of 100m.

100m

River Current flow

35º Task: If the angle he swam was 35º with the near side bank , calculate the actual width of the river. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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Calculate the height of the taller building if the distance between the buildings is 10m. The angle between a line from the top of the tall building (excluding the roof) to the bottom of the short one is 35º.

Task: Calculate the height of the taller building.

35º

Height of building 10m

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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Cliff Face

Height of cliff

Surface of Sea

50º

1200m 3.

In the diagram above , a yacht is anchored 1200m from the base of a cliff. The angle of elevation from the yacht to the top of the cliff is 50º. Calculate the height of the cliff. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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4.6 Graphs in Trigonometry : Sketching graphs of trigonometric functions by table method where x  [0  ;360 ] Method: Draw up a table using x –values of [ 0º ; 30º ; 60º ; 90º ;120º ; 150º ; 180º ;210º ; 240º ; 270º ;300º ; 330º ; 360º] Use your calculator to get the corresponding y – values Plot these points on a Cartesian plane and make an accurate sketch joining the points in a smooth curve shape.

Example 1: Sketch the graph of f ( x)  sin x where x  [0  ;360 ] X y = sinx

0º 0

90º 1

180º 0

270º -1

360º 0

y

0

90 

180

270

360

f x  = sin x 

x

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Exercise 4.: Copy and complete the following table and sketch the graph of y  sin x 0º 90º 180º 270º 360º 0 -1

1.

X y = sinx y 2

1

x 0 90

180

360

270

-1

-2

2.

Copy and complete the following table and sketch the graph of y  cos x Decimals to nearest 1 decimal place

X

0º 1

y  cos x

90º 0

180º

270º

360º 1

y 2

1

x 0 90

-1

-2

180

270

360

Grade 10 Core Mathematics

3.

X y = sinx +1 y  sin x  1

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Copy and complete the following table and sketch the graph of y  sin x  1 0º 90º 180º 270º 360º 0 1 0 -1 0 +1 +1 +1 +1 +1

y 2

1

x 0 90

180

360

270

-1

-2

4.

Copy and complete the following table and sketch the graph of y  cos x  1

X y = cos x +1

0º 1 +1

90º 0 +1

180º +1

270º 0 +1

360º +1

y  cos x  1 y 2

1

x 0 90

-1

-2

180

270

360

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Copy and complete the following table and sketch the graph of y  sin x  1

X y = sinx -1 y  sin x  1

0º 0 -1

90º -1

180º 0 -1

270º -1

360º 0 -1

y 2

1

x 0 90

180

270

360

-1

-2

6.

Copy and complete the following table and sketch the graph of y  cos x  1 0º 90º 180º 270º 360º

x y = cosx -1

y  cos x  1 y 2

1

x 0 90

-1

-2

180

270

360

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Copy and complete the following table and sketch the graph of y   cos x

X y = -cosx

0º -1

90º

180º 1

270º

360º -1

y 2

1

x 0 90

180

270

360

-1

-2

8.

X y = -sinx

Copy and complete the following table and sketch the graph of y   sin x 0º 90º 180º 270º 360º 0 -1 1 0

y 2

1

x 0 90

-1

-2

180

270

360

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Copy and complete the following table and sketch the graph of y   cos x  1

x y = -cosx +1 y   cos x  1

0º

90º

180º

270º

360º

y 2

1

x 0 90

180

360

270

-1

-2

10.

Copy and complete the following table and sketch the graph of y   sin x  1

x y = -sinx +1 y   sin x  1

0º

90º

180º

270º

360º

y 2

1

x 0 90

-1

-2

180

270

360

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X y = tanx

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Copy and complete the following table and sketch the graph of y  tan x 0º 0

90º ∞

180º

270º ∞

360º 0

y 2

1

x 0 90

-1

-2

180

270

360

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DATA HANDLING:

Statistics : A branch of mathematics concerned with collection, interpretation and analyses of data.

“The science of making decisions in face of uncertainty” Data: Name given to the collection of information usually expressed in numerical form.

Data Handling involves: 

Collecting data for a particular purpose.



Sorting data.



Representing data in tables, charts or graphs.



Analyzing the results.



Coming to conclusions.

Once collected data must be “interpreted” or analyzed . Only then can conclusions be made . Interpreting is Pictoral or Arithmetic. 

Pictoral Methods: Graphs: Bar; Histograms; Frequency Polygons; Pie Charts; Line and Broken Line graphs.



Arithmetic:



Measures of Central Tendency: Mean; Median Mode.



Measures of Dispersion: Range; deciles; percentiles; quartiles and interquartile range.

5.1

Displaying Data: 

A Bar Graph is a diagram consisting of a series of columns(bars) that are parallel(vertical or horizontal) that the length show frequency.

Example: 80 households surveyed & number of children in each household.

Number of children Per family 0 1 2 3 4 5

Frequency 8 14 20 17 10 11

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Vertical Bar Graph depicts the data:

Number of children per family 80 households f r e q u e n c y

Number of children per family Exercise 5.1: An educator was trying to ascertain why certain learners were not doing set homework . She asked the learners to calculate the number of hours spent watching TV the previous night. The time was rounded up to the nearest hour. The following data was collected by an educator.

1 2 2 1 0 1 3 1 0 0 2 1 0 1 1 1 3 1 1 1 0 1 2 3 3 1 0 0 0 1 1 1 1 1 0 0 0 1 0 1 3 3 1 2 0 1 3 0 1 3 1.

How many learners in the class?

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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Draw up table to tally the numbers of learners who watched TV for 0 hrs, 1 hr, 2hrs and for 3 hrs.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

3.

Draw a bar graph to show the results.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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Was the educator correct in thinking that the learners were not doing the set homework because they were watching too much TV. Validate your answers.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 5.2

Compound Bar Graphs:

Dual bar graphs : Used when two different sets of information given for connected topics: Example: 20 people recorded which TV station they watched at 8:15 p.m. on two consecutive nights in July 2004.

First Night Second Night TV1 6 7 TV2 4 1 TV3 6 6 M-NET 3 4 E-TV 1 2

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Dual Bar Graph :

f r e q u e n c y

TV 1

TV 2

TV 3 TV Stations

MNET

E - TV

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Sectional Bar Graphs:

Airport survey: 100 passengers traveling overseas - “which countries were traveled to” Passengers USA Male 12 Female 6 Total 18

UK 15 17 32

GERMANY HOLLAND INDIA 8 5 4 3 11 5 11 16 9

AUSTRIA 9 5 14

f r e q u e n c y

USA

UK

GER HOLL COUNTRIES

IND

AUST

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Exercise 5.2:

Ave Temp in ÂşC

Jan July

The Average day-time temperature for 9 provincial capital cities were recorded.

Eastern Free State Gauteng KwaZulu- Mpuma- Northern Limpopo North west Cape BloemNatal langa Cape fontein JHB PMB KimberPoloMafikeng Bisho Nelspruit ley kwane 22 14

23 8

20 10

23 13

24 15

25 11

23 12

24 12

Task: 1.

Draw a dual bar graph to illustrate the above information.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2.

Which province has the greatest difference in temperature between January and July?

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Western Cape Cape Town 21 12

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Why would somebody need to know the average temp in ºC of the various cities in Jan and July?

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 5.4 Pie Charts: Example: 70% of the Earth’s structure is water; rest is land 11% of the Earths surface is habitable ; 6% is too cold.

4% is mountainous; 9% is too dry.

Water Land-Habit Land – cold Land – Mountainous Land – Dry

%

Calculation of % of 360º

Angle written correct to whole no

70 11 6 4 9 100

252º 39.6º 21.6º 14.4º 32.4º 360º

252º 40º 22º 14º 32º 360º

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Exercise 5.3: 240 learners were asked what they intended doing on leaving school.

The results were: 80 wanted to attend University 86 wanted to attend Technikon 64 wanted to get a job 10 did not know 1.

Copy and complete the table:

Go to University

No of learners 80

Go to Technikon

86

Get a job

64

Don’t Know

10

Total

240

Calculation 80  360 240

Angle 120º

64  360 240

15º ……

2.

Illustrate this information on a pie chart.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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Illustrate the same information as a bar graph.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

4.

Compare the two displays and identify:

One feature that the pie chart shows better.

One feature that the bar graph shows better. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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Broken Line Graphs.

Can Help:  Find Patterns or Trends  Spot relationships between data  Find & predict values not given in the data Example: Norma is in hospital. Every 3 hrs her temperature is taken and the points are plotted on a graph.

When data is collected in time intervals it is usual to plot them in this way.

40

39.5

39 38.5

38 37.5

37 0:00

3:00

6:00

9:00

12:00

15:00

18:00

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Misleading Statistics:

Statistics are notorious for being a way that people use to make false and misleading arguments.

1.

2.

Statistics can be made up.

Charts and graphs are drawn so that they also mislead.

CHECK: 1. Does it have a title.

2. Are both axes labeled.

3. Are the units included.

4. Are the scale units equal distances apart.

5.7

Measures of Central Tendency(M.O.C.T.)

Measures of central tendency of ungrouped data If one number can be compared with another ---- then this is called a measure of central tendency.

3 different Measures Of Central Tendency are: 1.

Mean The average of the scores.

2.

Mode The most repeated score.

3.

Median ( Midlemost) in a ordered set of data.

5.8

Frequency Tables:

NB Must use all the data when finding one of the M.O.C.T’s.

Example: Value Frequency

3 1

4 4

5 3

6 2

Mode = 4 4,5 Median = 4,5 [3 4 4 4 4 5 5 5 6 6] Mean = 4,6 5.9

Discrete and continuous data.

1.

2.

Discrete Data: Information collected by “counting”.

Number of children cannot include fractions.

But scores can include fractions.

Continuous Data: Collected by “measurement” Suitable degree of accuracy( not always exact.)

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Stem & Leaf Diagrams

Example: Heights in cms.

147; 145; 157; 159; 154; 164; 161; 164; 162; 166; 163; 162; 171; 172; 171 Stem 14 15 16 17

Leaf 57 579 1223446 112

Tally Table Hts in cms 140 – 149 150 – 159 160 – 179 180 – 189

// /// //// // ///

Measures of central tendency (MOCT) from stem and leaf diagrams.

Median & Mode.

Example: 33; 57; 46; 28; 68; 32; 60; 65; 54; 54; 40; 46; 45 ;26; 69.

Stem 2 3 4 5 6

Leaf 68 23 0566 447 0589

There are 2 modes (46) and (54) thus Bimodal Number: 15 data items.

Median is 46 Mean = 48,2

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Grouped Data:

If dealing with many values it is easier to group data into sub sets.

Groups of ‘5’ as an example.

1–5 6 – 10 11 – 15 16 – 20

Measures of Central Tendency in Grouped Data: Example: Width of leaf (mm) No of leaves

11 – 20

21 - 30

31 – 40

41 - 50

51 – 60

2

6

8

5

4

Mode: Modal class is the class (group) with the highest frequency e.g. 31 – 40 above.

Mean: To find the mean we must find a value to represent each class thus take the average of each class by adding 1st and last terms.

11  20 e.g  15,5 2 Width of leaf (mm) Mid point No of leaves

11 – 20

21 - 30

31 – 40

41 - 50

51 – 60

15.5 2

25.5 6

35.5 8

45.5 5

55.5 4

sum of midpo int s no of leaves 917.5 mean  25 mean  36.7 mean 

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HISTOGRAMS:

Bar graph used for Discrete Data.

Histogram used for Continuous Data.

F r e q u e n c y 1-5

6-10 11-15 16-20 21-25 26-30 31-35

Number of items bought

5.13

FREQUENCY POLYGONS:

Histograms and Frequency Polygons are frequency graphs.

Histograms-----Bar graph.

Frequency polygon-----Line graph.

A frequency Polygon is a 2 dimensional shape made of line segments joined up.

NB: We can use the midpoints of the bars of a histogram as the points for the frequency polygon.

Don’t start the histogram at zero BUT leave a space then start the bars. The Frequency polygon will then start and end on the bottom axis.

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MEASURES OF DISPERSION.

RANGE: This is a simple measure of the spread of data.

Range = Maximum Value minus Minimum Value.

Range cannot be used for grouped data as it ignores the distribution of values lying between the maximum and minimum values.

Example: 2 classes scores in a maths test are as follows: Class 1:

22; 48; 52 ; 54; 64; 66; 76; 86; 88; 92; 100.

Range = 100 – 22 = 78

Class 2:

56; 58; 60; 62; 66; 66; 78; 78; 78; 80; 82.

Range = 82 – 56 = 26

Class 1 :

2

Scattered between 22 and 100

8 4 2 6 4 6 8 6 2

1 2 3 4 5 6 7 8 9 10

Class 2: Bunched between 56 and 82.

6 0 8 0

8 2 6 6 8 8 2

Measures of Dispersion: A Median is sometimes a better measurement of Central Tendency than the mean.

Median divides the ordered data into 2 halves.

Further subdivisions are : Quartiles: Points that divide into quarters.

Deciles: points that divide into tenths

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Percentiles: points that divide into hundredths There are 3 quartiles.

Q1 – Lower quartile Q2 – Median (M) Q3 – Upper Quartile Percentiles are first and second.

75 for a test. To compare her marks with the rest of her class we can 90 calculate the percentile in which her mark falls. Arrange all marks in descending order.

Meg gets

100 learners 80% got better than 75  thus Megs mark is in the 80th percentile. 

If a mark is in the 99th percentile then 99% of the marks are below that value.



If a mark is in the 60th percentile then 60% of the learners scored lower than this mark.

The lower quartile is 25 percentile.

The upper quartile is 75 percentile.

Study of the values of quartiles, deciles and percentiles give us an idea of the spread of data.

5.15

CALCULATING QUARTILES:

22; 48; 52; 54; 64; 66; 76;86; 88; 92; 100 56; 58; 60; 62; 66; 66; 78; 78; 78; 80; 82 Q1

M

Q3

Dividing data into 4 quartiles: ± 25% data below Q1 ±

50% data below M.

± 75% data below Q3 Class 1: ± 25% marks less than 52 ± 50% marks less than 66 ± 75% marks below 78.

Q1

M

Q3

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Class 2: ± 25% marks less than 60 ± 50% marks less than 66 ± 75% marks below 78 INTER-QUARTILE RANGE: Lie between the two quartiles (lower and upper) The Inter- quartile range = Q3 – Q1 It measures a better dispersion than the range as it is not affected by extreme values.

It is based on middle half of the data. i.e 50% of the data lies here.

Example; Class 1: IQR : 88 -52 = 36 Class 2: IQR : 78 – 60 = 18 Class 2 has a smaller IQR which tells us that marks are not as spread out around the median as for class 1.

Semi Inter-quartile range: It is half the IQR. Q  Q1 SIQR = 3 2 Values for Quartiles are not necessarily part of the data given. e.g. Class 3:

20; 39; 40; 43; 43; 46; 53; 58; 63; 70; 75; 91; .

Q1 46  53 M   49,5 2 40  43 Q1   41,5 2 63  70 Q3   66,5 2 Range = 91 – 20 = 70

M

Q3

Range shows marks widely spread 25% lie below Q1 = 41,5 50% lie below Q2 = M = 49,5 75% lie below Q3 = 66,5 IQR = 66,5 – 41,5 = 25 25  12,5 SIQR = 2

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Exercise 5.4: 1.

A company wanted to evaluate the training program in its factory.

They gave the same task to trained and untrained employees and timed each one in seconds.

Trained: 121 137 120 118 Untrained: 135 142 134 139

131 125

135 134

130

128

130

126

132

127

129

126 140

147 142

145

156

152

153

149

145

144

Draw a back – to – back stem & leaf diagram to show the two sets of data. _______________________________________________________________ 1.1

___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 1.2

Find the medians and quartiles for both sets of data.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ _____________________________________________________________ 1.3

Find the Inter-quartile Range for both sets of data.

____________________________________________________________________ ___________________________________________________________________

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Comment on the results.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2..1

The heights, measured to nearest cm, of 75 girls picked at random at Glory High School, Are shown on the following frequency table: Height (h) in cm 135  h < 140 140  h < 145 145  h < 150 150  h < 155 155  h < 160 160  h < 165 165  h < 170 170  h < 175 175  h < 180

Frequency 2 5 10 17 19 15 4 2 1

Draw a histogram to illustrate this data: ___________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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2.2 The heights, measured to nearest cm, of 75 boys picked at random at Glory High School, Are shown on the following frequency table: Height (h) in cm Frequency 0 135  h < 140 1 140  h < 145 7 145  h < 150 11 150  h < 155 15 155  h < 160 18 160  h < 165 15 165  h < 170 6 170  h < 175 2 175  h < 180 On the same set of axes as the histogram draw a frequency polygon to illustrate this data. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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Use the frequency tables and the two graphs to help answer the following questions. For each group of learners state: 2.3.1 The modal class.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2.3.2

The median height.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2.3.3

The lower quartile

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2.3.4

The upper quartile

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 2.4 Are the girls in the sample taller than the boys or are the boys taller?

Use the statistical measures in 2.3 to back up your conclusions.

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Exercise 5.5: Example: 1.

The following marks were recorded for a maths class: 28 53 75 63

45 75 63

36 58 75 63

36 60 78 67

36 60 81 68

38 60 83 68

45 71 84 69

42 71 84 76

45 75 90 79

1.1 1.2 1.3 1.4

Do a stem and leaf diagram for the data Find the median, mode and mean for the data Find the lower and upper quartile Calculate: 1.4.1 the interquartile range 1.4.2 the semi-interquartile range 1.4.3 the range for the class 1.5 Write down the maximum and minimum scores.

1.6 Do a box and whisker diagram using the five-number summary

Answer: Stem 2 3 4 5 6 7 8 9

Leaf 8 6668 2555 38 0003337889 115555689 1344 0

Mode = 75 ; Mean = 62.9 ; Number = 35 Interquartile range = 30 ; Semi- interquartile ; range = 15 Range = 62 Standard Deviation = 16.6 Lowest = 28 ; Q1 = 45 ; Median = 67 ; Q3 = 75 ; Highest = 90

90

28 45 0

10

20

30

40

67 50

60

70

75 80

90

100

x  Q2  62.9  67  4.1  0 Data is negatively skewed i.e. skewed to the left. The marks are concentrated to the right of the median and spread out to the left of median.

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45 75 84

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The following marks were recorded for a maths class: 54 53 75 63

46 58 75 92

44 81 78 67

22 60 60 68

28 54 37 68

37 71 56 69

56 71 25 76

45 44 90 98

2.1 Do a stem and leaf diagram for the data.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

2.2 Find the median, mode and mean for the data.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

2.3 Find the lower and upper quartile.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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2.4 Calculate: 2.4.1

the interquartile range.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

2.4.2

the semi-interquartile range.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

2.4.3

the range for the class.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

2.4 Write down the maximum and minimum scores.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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Do a box and whisker diagram using the five-number summary (L;Q1;M;

Q3;H) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 3.

34 28 34 3.1

The following marks were recorded for a maths class: 12 15 37 80

15 12 42 65

34 45 23 28

22 65 50 19

56 33 54 39

23 24 25 32

22 9 8 40

20 18 20 31

Do a stem and leaf diagram for the data.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

3.2

Find the median, mode and mean for the data.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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Find the lower and upper quartile.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Calculate: 3.4.1

the interquartile range.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

3.4.2

the semi-interquartile range.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

3.4.3

the range for the class.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

3.4 Write down the maximum and minimum scores.

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

Grade 10 Core Mathematics

3.6

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95

Do a box and whisker diagram using the five-number summary

(L;Q1;M; Q3;H) ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

4.

Girls 34 Boys 75 4.1

The following marks for a class of Girls and Boys were recorded : 72 85 77 72

65 92 42 65

44 90 85 68

72 65 50 79

66 63 74 89

80 54 65 62

58 55 85 70

70 72 80 71

Do a back to back stem and leaf diagram for the data

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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4.2

96

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Find the median, mode and mean for both sets of data

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

4.3

Find the lower and upper quartile of each set of data

4.4 Calculate: 4.4.1

The inter-quartile ranges for:

4.4.1.1

girls

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

4.4.1.2

boys

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ 4.4.1.3

class

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

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4.4.2

97

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The semi-interquartile ranges for:

4.4.2.1 girls ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

4.4.2.2 boys ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

4.4.2.3

class

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________ ____________________________________________________________________

4.4.3

The ranges for:

4.4.3.1 girls ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 4.4.3.2

boys

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 4.4.3.3

class

Grade 10 Core Mathematics

4.5

98

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Write down the maximum and minimum scores of each set of data

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

4.6

Do separate box and whisker diagrams for the girls and the boys

____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

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5.

The following table represents the maths scores for the entire grade 11 maths group at Northwood School. The data is grouped due to the size of group.

Class 0 to 9 10 to 19 20 to 29 30 to 39 40 to 49 50 to 59 60 to 69 70 to 79 80 to 89 90 to 99 100 to 109 Totals

Frequency(f) 15 10 17 40 35 22 20 20 15 5 1 200

Mid-points(X) 4.5 14.5 24.5 34.5 44.5 54.5 64.5 74.5 84.5 94.5 104.5

fX 67.5

5.1 Complete the last column of the table i.e (fX) 5.2 Find the modal class ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 5.3 Find the median class ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 5.4 Find the interval where Q1 and Q3 lie. ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 5.5 Calculate the estimated mean. ď&#x192;Ľ fX NB estimated mean = n ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

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100

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5.6 Use the grouped data to display the data on a histogram ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ 5.7 Draw the relevant frequency polygon on the histogram.

Grade 10 Core Mathematics

6.

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101

Volume and Surface Area of 3-D Shapes Formulae: Volumes of Rectangular Prisms:

V  (area of base) height 1.

Square Base: V  (side of base)2  height (length) V  length breadth height(length 2.

Rectangular Base : 1 3.

Trapezium Base : V  sum of parallel sides  height (length) 2 1 4.

Triangular Base : V  (base  height) length 2 Volume of cylinders: V  (area of base) height V   r 2h Volume of a Cone: 1 V   r 2h 3 Volume of Pyramid: 1 V  (area of base) height 3 Volume of Sphere: 4 V   r3 3 Surface Areas of Shapes:

Hint Draw a net diagram of the shape: Net Diagrams of 3 D shapes: Rectangular Prisms:

Volume  side1 side2  side3 Volume = area of base X height l Net: Rectangular Prisms: b

hxl

b

b h

hxl

h

hxb

b hxl

l

b

hxl Surface area = 2(h x l) + 2( h x b) + 2(l x b)

hxb

l

h

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102

Net : Square base prism:

s sxh

s sx s

s

s sxh

sxh

sxh

h

s sx s Surface area = 4(h x s) + 2( s x s) Net of a Cylinder

r

h 2r

Surface Area = area of circles plus area of rectangle Surface Area = r 2  r 2  2rh Surface Area = 2r 2  2rh

Sh  h 2  r 2

Cone:

Surface Area = r h 2  r 2 Circumference = 2  r OR Surface area =

1 circumference x slant height 2

NB Slant height =

h2  r 2

Slant height Net Diagram of a cone:

Arc length

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103

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Formulae for Surface Area of Shapes.

PRISMS: Rectangular bases:

S  2lb  4hb Triangular Bases: S  bh  bl  2 b 2  h 2

S  bh  3bl

 If triangle is isosceles.  If equilateral.

Cylinders: S  2 r 2  2 rh S  2( r 2   rh)

Square base pyramid:

1 SurfaceArea  ( side) 2  4( base  sh) 1 slant height  ( base) 2  (height) 2 2 2 SurfaceArea  ( side) 2  2(base  sh) (Pythagoras theorem) OTHER 3- D SHAPES Cone: CurvedSurface   r h 2  r 2 or 1 CurvedSurface  circumference  slant height 2 Sphere: S  4 r 2

Tetrahedron: 1 Vol = (Area of base x height) 3 4 3 1  (base) 2 Surface Area = 2 2

Grade 10 Core Mathematics

104

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Pictures of Different 3-Dimensional Shapes:

A:

Surface area and volume of cylinder with section removed: Calculate the surface area of the shape: Calculate the ratio of arc to circle



Arc Leave  as a symbol Circumference

Area of Arc(sector) = Use the ratio above and multiply  r2 ( area of full circle) Area of flat sections = l  b (length x breadth) they are rectangles.

Area of curved surface = Arc  height Total Surface area = 2( area of arc) + 2( area of rectangular flat sections + area of curved surface Find the volume of the shape.

Volume of shape = Area of arc x height of cylinder.

Grade 10 Core Mathematics

B:

105

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Surface area and volume of cone with section removed:

Calculate the ratio of arc to circle

Arc Leave  as a symbol Circumference

Area of Arc(sector) = Use the ratio above and multiply  r2 ( area of full circle) Calculate the Slant Height as follows ( Pythagoras) Slant height = h 2  r 2 1 Area of curved surface = ( area of curved sector) x slant height 2 1 Area of straight side =  base height 2 Total area = area of sector( arc) + area ofn curved side + 2( area of straight side) Volume of Cone =

1 (Area of Arc) x height 3

Volumes of prisms & the effect of the factor--k Exercise 6.1: 1. Calculate the volume and surface area of the following closed prisms: Prism P Q R S Length (mm) 52 47 43 39 Breadth (mm) 20 18 17 15 Height (mm) 85 77 70 64 Volume Surface Area Determine the following ratios correct to 2 decimals.

VolumeP VolumeQ 2.1 2.2 VolumeR VolumeQ ________________________________________________________________

T 36 14 58

___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ VolumeR VolumeS 2.4 VolumeS VolumeT ________________________________________________________________

2.3

___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________

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SurfaceAreaP SurfaceAreaQ 3.2 SurfaceAreaQ SurfaceAreaR ________________________________________________________________

3.1

___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ SurfaceAreaR SurfaceAreaS 3.4 SurfaceAreaS SurfaceAreaT ________________________________________________________________

3.3

___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________

4.

Are the volumes of the prisms approximately in proportion? Give reasons for your answers. ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 5.1 How much smaller in volume is prism T than prism P? Give the scale factor (not the change in volume). ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 5.2 Are the surface areas of the prisms in proportion? Give reasons for pour answers. ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

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107

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5.3 How much smaller in surface area is prism T than prism P? Give the scale factor (not the change in area).

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

6.

Determine the scale factor used : 6.1 to reduce the dimensions of the prisms.

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

6.2

To enlarge the dimensions of the prisms.

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ 7. What is “The Golden Ratio” ? ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 8.Determine which ratio of the faces comes closest to this ratio.

NB: You must choose a ratio greater than 1.

Grade 10 Core Mathematics

108

8.1 Reduce each of the dimensions of prism P by a factor of

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1 , then calculate 2

the volume and surface area of the new prism X. ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 8.2 How much smaller in volume and in surface area is this new prism X? Give the scale factor in each case. ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 9.1

Using the answers to question 8 , estimate the volume and surface area of prism Y, where each dimension of prism P has been enlarged by a factor of 2.

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 9.2

Calculate the volume and surface area of prism Y, and compare your answers to your answers to question 9.1

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

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109

Examples: h b l Dimensions of the prism is length; breadth and height. i.e. l ; b ; h Volume of prism = l  b  h k is a factor of 2] Prism l (cm) b (cm) H (cm)

A B C D E F G H

4 8 4 4 8 8 4 8

3 3 6 3 6 3 6 6

2 2 2 4 2 4 4 4

Volume (cm3) 24 48 48 48 96 96 96 192

Vxk

V Vx2 Vx2 Vx2 Vx4 Vx4 Vx4 Vx8

Factor No of Sides Doubled 0 k 1 k 1 k 1 2 K 2 K2 2 2 k 2 k3 3

It is noticed : When 1 dimension is doubled then the volume is doubled as well When 2 dimensions were doubled then the volume is 4 times the original.

When all 3 dimensions are doubled the volume is 8 times the original.

This holds for any factor value. i.e. k = 3 the volumes increase accordingly : 1 trebled thus volume trebled 2 trebled thus volume is 9 times original 3 trebled thus volume is 27 times the original.

Factors affecting the volumes are: k k2 k3 Volume of prism = l  b  h Prism L (cm) b (cm) h (cm) Volume Vxk (cm3) A 4 3 2 24 V B 4 3 6 72 Vx3 C 4 9 2 72 Vx3 D 12 3 2 72 Vx3 E 12 9 2 216 Vx9 F 8 3 4 216 Vx9 G 4 9 6 216 Vx9 H 12 9 6 648 V x 27

Factor k k k k2 k2 k2 k3

Grade 10 Core Mathematics

10.

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110

Copy and complete the following tables:

h b l Dimensions of the prism is length; breadth and height. i.e. l ; b ; h Volume of prism = l  b  h Factor k =______ Prism A B C D E F G H

L (cm) 6 12 6 6 12 12 4 12

b (cm) 4 4 8 4 8 4 8 8

h (cm) 3 3 3 6 3 6 6 6

Volume (cm3) 72

Vxk V Vx2

Factor k k

k2

Volume of prism = l  b  h Factor k =______ Prism A B C D E F G H

L (cm)

b (cm)

h (cm)

4 4 4 12 12 4 12 12

3 3 9 3 9 9 9 9

2 6 2 2 2 6 6 6

Volume (cm3) 24 72 216 216

Vxk Factor V Vx3 Vx Vx Vx 9 Vx Vx V x 27

K

k2

K3

Grade 10 Core Mathematics

111

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Key Ideas: The circumference of a circle is proportional to its area: arc length sector area sector  circumference of circle area circle

To find the area of a sector of a circle one must first find the ratio:

arc . circumference

Multiply this ratio by  r2 to get the area of the sector.

Area of a base that is a hexagon shape: It takes 6 equilateral triangle shapes to form 3 3.a a OR Area  the hexagon base : Area  ( where ‘a’ is the base of the 2 2 equilateral triangle) height  3.a    height Volume of Prism with Hexagon Base = 6  2  

a ( base of ) NB: The perpendicular height of any equilateral triangle with side ‘a’ is

3  a by 2

Pythagoras.

Exercise continued: 11.

A cold - drink can measures approximately 65 mm in diameter and 75mm in height.

11.1 Calculate the volume of the can ( in mm2 and cm2).

The writing on the can says that it contains 200 ml of liquid. How much air space is there in the can? ( 1ml  1cm 3 ) ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________

Grade 10 Core Mathematics

112

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11.2 What is the height of the liquid in the can? ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 12.1

Calculate the total surface area of the can (in cm3 ) , assuming that the can is a closed cylinder.

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

12.2

113

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If the metal to make the can costs 0,25 cents per square centimeter, calculate the cost of making each can.

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 13

The manufacturer of Lemon Twist wants to double the volume of the can , but keep the radius as it is. By which factor must the height be increased?

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

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114

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14.1 If the radius is increased by a scale factor of 2, but the height is kept the same by which factor will the volume increase?

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ 14.2 By which scale factor will the area of the top of the can increase?

________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________

Grade 10 Core Mathematics

115

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14.3 By which factor will the area of the lateral surface ( the area of the curved side) increase? ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ ___________________________________________________________________ ____________________________________________________________________ ___________________________________________________________________ __________________________________________________________________