3rd Prep Maths 2nd Term by Ahmed Kamal

Solving Two Equations of 1st Degree in two Variables Algebraically and Graphically Example 1 – One Solution 2x + y = −2 Solve the two simultaneous equations  .  x − y = −7

Graphing Method Graph each equation to find the point of intersection.

Substitution Method Solve the first equation for y.

y = −2x − 2 Substitute this into the second equation. x – (−2x − 2) = −7 Solve for x to get x = −3. The lines intersect at the coordinate (−3, 4), so that is the solution to the system.

Substitute x = −3 into the first equation to get y. y = −2(−3) − 2 = 4 The solution is (−3, 4).

Elimination Method Add down to eliminate y and solve for x.

2x + y = −2   x − y = −7 3x = −9 x = −3 Multiply the second equation by −2, then add down to eliminate x and solve for y.  2x + y = −2  −2x + 2y = 14 3y = 12 y=4 The solution is (−3, 4).

S.S. = { ( - 3 , 4 ) } Number of solution = 1

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Example 2 – Infinitely Many Solutions 2x − 4y = 2 Solve the two simultaneous equations  . −x + 2y = −1

Graphing Method Graph each equation to find the point of intersection.

Substitution Method Solve the second equation for x.

Elimination Method Multiply the second equation by 2, then add down to eliminate x.

x = 2y + 1 Substitute this into the first equation.

 2x − 4 y = 2  −2x + 4 y = −2 0=0

2(2y + 1) – 4y = 2 Solve to get 2 = 2, a true statement (or an identity). The lines are coinciding (they are the same line when graphed), so there are infinitely many solutions to the system.

Since the variable dropped out and resulted in a true statement, there are infinitely many solutions to the system.

Since both variables were eliminated and a true statement resulted, there are infinitely many solutions to the system.

S.S. = { ( x , y ) : x , y ∈ R and 2 x – 4 y = 2 } Number of solution = infinte number

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Example 3 – No Solution

 3x − y = 2 Solve the two simultaneous equations  . − 3x + y = 0 

Graphing Method Graph each equation to find the point of intersection.

Substitution Method Solve the second equation for y. y = 3x Substitute this into the first equation. 3x – (3x) = 2 Solve to get 0 = 2, a false statement.

The lines are parallel and never intersect, so there are no solutions to the system.

Elimination Method Add down to eliminate x.

 3x − y = 2  −3x + y = 0 0=2 Since both variables were eliminated and a false statement resulted, there are no solutions to the system.

Since the variable dropped out and resulted in a false statement, there are no solutions to the system.

S.S. = ϕ Number of solution = 0

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Exercises

1) Find graphically the S.S. of each two simultaneous equations:

a) y = x and y = -x

c) y + 2 x = 1 and 4 x = 10 − 2 y

……………………………………

b) y + x = 4 and 2 x – y = 5

d) x + 2 y = 3 and 2 x − 6 = 4 y

……………………………………

2) Find algebraically the S.S. of the two simultaneous equations

a) x – y = 5 and x + y = 1

d) 2 x – 3 y = 3 and 4 x + 5 y = 0

……………………………………

b) x – y = 0 and 2 x + y = 6

e) x – y = 5 and x + y = 1

……………………………………

c) 2 x + y = 7 and x + 3 y = 6

f) 2 x = 1 − 2 y and x + y = 5

……………………………………

…………………………………… 6

3) Complete each of the following

1) The S.S. of the system; x = - 3 and 2 x + y = 0 is …… ……………………………………

……………………………………

2) The S.S. of the system; y = 1 and 2 x - y = 3 is …… ……………………………………

……………………………………

3) The S.S. of the system; x = - 3 and y = 3 is …… ……………………………………………………………………………………… 4) The system 2 x + y = - 1 and 4 x +2 y = 3 have …… solution ……………………………………

……………………………………

5) ( 1 , - 1 ) is the common solution for the system a x + y = 1 and x – by = 2, then a = …… and b = …… ……………………………………

……………………………………

4) Choose the correct answer ;

(1) The system x – 3 y – 1 = 0 and 3 x = 9 y + 3 have …… common solution(s)

[ one , infinite , no , two ]

……………………………………

(2) The system x – 3 y – 1 = 0 and 3 x = 9 y + 2 have …… common solution(s)

[ one , infinite , no , two ]

……………………………………

…………………………………… 7

(3) The system x – 3 y – 1 = 0 and 3 x = y + 3 have …… common solution(s)

[ one , infinite , no , two ]

………………………………………

(4) One of the solutions of the equation 2 x – y = 1 is …… [(0,1),(1,0),(1,1),(1,2)] ………………………………………

………………………………………

(5) The intersection point of the 2 lines x = 7 and y + 1 = 0 is … [ ( 7 , 1 ) , ( 7 , - 1 ) , ( -7 , 1 ) , ( -7, -1 ) ] ………………………………………

………………………………………

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Solving Equation of 2nd Degree in one Variables Algebraically and Graphically

a x² + b x + c = 0 is a quadratic equation, where a ≠ 0 There formula to solve the quadratic equation x =

−b ± b2 − 4ac 2a

Example 1: Solve the equation x 2 − 2 x − 7 = 0 , to the nearest 2 decimal places

a = 1 , b = −2 , c = −7 x=

−(−2) ± (−2)2 − 4 × 1× (−7) 2 ± 4 + 28 = 2 ×1 2

2 + 32 2

2 − 32 2

S.S. = { 3.83 , - 1 .83 }

Example 2: Solve the equation 3x 2 − x + 3 = 0 a = 3 , b = −1 , c = 3

−( −8) ± ( −1)2 − 4 × 3 × 3 x= = undefined 2×3 S.S. = ϕ

Example 3: Solve the equation x 2 − 2x + 1 = 0

a = 1 , b = −2 , c = 1

−( −2) ± ( −2)2 − 4 × 1× 1 x= =1 2 ×1 S.S. = { 1 } 9

Exercises 1) Find the solution set of each of the following ( using the formula ):

a) x² = 6 x – 7

where

2 ≈ 1.41

………………………………………

b) x² = 2 ( x + 6 )

approximate to one decimal place

………………………………………

2 2 = 2 where x x ………………………………………

3 ≈ 1.73

………………………………………

c) 1 -

d) ( x – 2 )² = 6

………………………………………

approximate to two decimals places

………………………………………

2) Graph the function f ( x ) = - 2 + x² + x where x ∈ [ - 3 , 2 ] , then from the

graph find the solution set of - 2 + x² + x = 0 …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… …………………………………………………… ……………………………………………………

3) Determine the solution set of each of the following:

S.S. = ………………

4) An aircraft hangar at Bush intercontinental airport is parabolic in shape.

The arch of the building can be modeled using the equation

y = -0.007x2 +1.7x. How wide is the hangar at the base? ………………………………………………….. ………………………………………………….. ………………………………………………….. …………………………………………………..

x Width = ?

Revision on Factorization 1) Factorization by taking out H. C. F.

1) 5 a + 5 b = ………

2) 10 a + 5 b + 15 c = ………

3) a3 b4 – a3 b3 + a5 b2 = ………

4) a ( a + 1 ) + b ( a + 1 ) = ………

2) Factorization of a trinomial

1) x2 + 5 x + 6

4) x 2 – 5 x – 6

7) 2 x2 – 5 x + 3

………………………………………………………………………………………… 2) x2 – 5 x + 6

5) 22 x – 75 + x2

8) 2 x2 + 5 x – 12

………………………………………………………………………………………… 3) x2 + 5 x – 6

6) 2 x2 + 5 x + 3

9) 6 a2 + 5 a b – b2

………………………………………………………………………………………… 3) Factorization of a perfect square

1) x2 + 2 x + 1 = ……… 2) 4 x2 – 4 x + 1 = ……… 3) 9 x2 – 12 x y + 4 y2 = ……… 4) Factorization of the difference between two squares

1) x2 – 4 = ………

2) x2 – y2 = ………

3) 4 x2 – 9 y2 = ………

4) 1 – 25 x2 = ………

5) Factorization of sum and difference of two cubes

1) x3 – 27 = ……… 2 ) 8 x3 + 64 = ……… 3 ) x6 + y6 = ……… 6) Factorizing by grouping

1) a x + b y + b x + a y = ……… 2) a b + a e – b d – d e = ……… 3 ) x3 + 2 + x + 2 x2 = ……… 4 ) x3 + x2 + 2 x + 2 = ……… 12

Solving two equations in two variables, one of the 1st degree and the other of the 2nd degree x = 1 – y → (1)

Example: Find the S.S. of

x2 + y2 = 13 → (2)

By substituting (1) in (2) ( 1 – y )2 + y2 = 13 1 – 2 y + y2 + y2 = 13 2 y2 – 2 y + 1 – 13 = 0 (÷2)

2 y2 – 2 y – 12 = 0 y2–y–6=0 Factorization:

(y+2)(y–3)=0 y+2=0

y–3=0

y=-2

y=3

By substituting in ( 1 ) x = 1 – ( -2 ) = 3

x=1–3=-2

S.S. = { ( 3, - 2 ) , ( - 2 , 3 ) }

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Exercises

1) Find the solution set in R × R for each of the following equations:

(1)

y=3x

and x2 – y2 = - 32

(3)

y = 2 x + 1 and 4 x2 + y2 = 13

…………………………………………

x – 5 y = 0 and x2 + y2 = 26

(4) 2x + y = 10 and x2 + y2 = 25 …………………………………………

(2)

…………………………………………

2) Choose the correct answer from the brackets:

1) The S.S of the system y = x and x y = 1 in R × R is ……………. (a) {(1,1)} ( b ) { ( 0 , 0) } (c) {(1,1),(-1,-1)}

( d ) { ( 1 , 1 ) , ( 0, 0 ) }

2) The S.S of the equations x = 1 and x2 – y2 = 10 in R × R is ……………. (a) {(1,3)} (b){(1,-3)} (c) {(1,3),(1,-3)}

(d){

}

3) The ordered pair which satisfies each of the two equations x y = 2 and x – y = 1 is…. (a) (1,1) ( b ) ( 2 , 1) (c) (1,2) ( d ) (0.5 , 1 ) 4) If x2 + x y = 15 and x + y = 5, then x = …….. (a) 3 (c) 5

(b)4 (d)6

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Set of Polynomial Function Zeros

f( x ) is a polynomial function, The set of all the values which makes f (x) = 0 is called the set of zeros of the Function f (x) and is written as z ( f )

Exercises 1) Choose the correct answer from the brackets:

(i)

The set of zeros of the function F(x) = x2 + 1 is ……………….

(a) { - 1 , 1 }

(b) { - 1 }

(d) Ф

(ii) The set of zeros of the function F(x) = x2 – 2 x + 1 is ………………. (a) { 0 }

(b) { 0 , - 1 }

(d) { 1 }

(iii) The set of zeros of the function F(x) = x2 + x + 3 is ………………. (a) { 0 }

(b) { 3 }

(d) { 1 }

(iv) The set of zeros of the function F(x) = x ( x2 – 4 ) is ………………. (a) { 0 }

(b) { 0 , - 2 }

(d) { 0 , 2 }

(v) If the set of zeros of the function F(x) = x2 + b is Ф , then b = ……… (a) 9

(b) - 9

(d) - 1

2) Complete each of the following

a. The set of zeros of the function f(x) = 0 is ………………. b. The set of zeros of the function g(x) = − 3 is ………………. c. The set of zeros of the function k(x) = ( x – 3 )2 ( x +3 ) is ………… d. If the set of zeros of the function m(x) = a x – 8 is { - 4 } , then a = ………. e. If f(x) = a x2 + b x + c does not intersect the X- axis, then Z ( f ) = ………

3) Find the set of zeros for each of ; 1. F(x ) = x3 – 4 x2 + 4 x

4. F(x) = x ( x2 + 1 ) – 5 ( x2 + 1 )

…………………………………………

2. F(x) = x3 + x2 + x + 1

…………………………………………

5. F(x) = ( x + 3 ) ( x2 - 3 x + 9 )

…………………………………………

3. F(x) = ( x2 + 6 x + 9 ) ( x + 2 )

…………………………………………

6. F(x) = x4 – 8 x2 – 9

…………………………………………

4) If Z ( f ) = { - 2 , 2 } where f ( x ) = x2 + a . Find the value of a ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

The domain of the function

The domain of the Polynomial function: The domain of a polynomial function is all real numbers

Example: The domain of g ( x ) = x² + 3 x – 5 is R

Algebraic fractional function: If m ( x ) and n ( x ) are two polynomial function, Then f ( x ) =

m( x ) is called algebraic fractional function n( x )

Example: f ( x ) =

3x + 5 5x + 2

The domain of the fractional function: The domain of the fractional function is all real numbers except the zeros of denominator

Example 1: a)

5 ∈ R if x ≠ 0 x

2+x ∈ R if x ≠ 4 x−4

x² − 5 x + 6 ∈ R if x ≠ ± 3 x² − 9

x 3 − 27 Example 2: Find the domain of f ( x ) = x ² + 7 x + 10 Let x² + 7 x + 10 = 0 (x+2)(x+5)=0 The domain of f ( x ) is R ∼ { -2 , -5 }

x = -2 or x = - 5

Example 3: Find the domain of g ( x ) =

x ² − 3x − 4 x² + x − 2

Let x² + x – 2 = 0 (x+2)(x–1)=0 x = -2 or x = 1

The domain of g ( x ) is R ∼ { - 2 , 1 }

The set of zeros of fractional function: f(x)=

m( x ) is a fractional functions n( x )

Then The set of zeros of f ( x ) = z ( m ) ∼ z ( n )

Example 1: Find the set of zeros of f ( x ) =

x² − 5 x + 6 x² − 9

The set of zeros of x² – 5 x + 6 is { 3 , 2 } The set of zeros of x² – 9 is { 3 , - 3 } Then the set of zeros of f ( x ) is { 2 }

Example 2: Find the set of zeros of f ( x ) =

x −5 x² + 4 x − 5

The set of zeros of x – 5 is { 5 } The set of zeros of x² + 4 x – 5 is { - 1 , 5 } Then the set of zeros of f ( x ) is ϕ

Example 3: Find the set of zeros of f ( x ) = The set of zeros of x – 1 is { 1 } The set of zeros of x + 1 is { - 1 } Then the set of zeros of f ( x ) is { 1 }

x −1 x +1

Exercises 1) Find the domain and the set of zeros for each of 1)

N(x) =

x+1 x−2

……………………………………… ………………………………………

………………………………………

……………………………………… 5)

………………………………………

2x 2 + 5x − 12 N(x) = 3 3x − 14x 2 − 24x

………………………………………

N(x) =

3 x−4

……………………………………… ………………………………………

………………………………………

……………………………………… 6)

……………………………………… ………………………………………

N(x) =

x 2 + 5x + 6 x 4 − 81

………………………………………

N(x) =

5x x − 2x

………………………………………

……………………………………… 7)

……………………………………… ………………………………………

N(x) =

− x 3 + 3x 2 − 2x 13x 2 − x 4 − 36

………………………………………

x +8 x 2 − 5x − 6 3

N(x) =

……………………………………… ………………………………………

………………………………………

…………………………………………… 20

2) Choose the correct answer from the brackets: x+1 1) The domain of the algebraic fraction equals the domain of the 3 algebraic fraction …………………. x+1 3 x+1 x +1 (a) 2 (b) (c) (d) x+1 x +3 x−3 x +4

2) The domain of the algebraic fraction (a) R ~ { - 1 , 1 }

x ( x + 1)

(b) R ~ {1 }

is ………………. x2 − 1 (c) {1 , - 1 } (d) R

3) The set of zeros of the algebraic fraction

(a) { - 1 , 0 }

(b) { - 1 }

4) The set of zeros of the algebraic fraction (a) { - 4 , 0, 4 }

(b) { - 4, 4 }

5) If the domain of the algebraic fraction (a) 4

(b) 2

3) Complete each of the following a) The set of zeros of the function F(x) = b) The function F(x) =

x ( x + 1) x2 − 1 (c) { 0 }

is ………………. (d) {1 , - 1 }

x−4 is ………………. x 3 − 16x (c) { 4 } (d) Ф

3x is R ~ {2 } ,then a = … x − 4x + a (c) - 2 (d) - 4 2

( x − 3 ) ( x + 2) is ………… x 2 − 2x − 3

3x is undefine if x ∈ ………………. x − 4x + 4 2

x2 − a c) If the set of zeros of the function F(x) = 3 is { - 4, 4 } .Then a …. x − 27 x3 + 8 d) If N(x) = 2 is defined for every x ∈ ………………. x − 5x − 6

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Common Domain of algebraic fractions

If D1 is the domain of N1(x) and D2 is the domain of N2(x) , then the common domain of N1(x) , N2(x) is D1 ∩ D2 .

Note: m( x ) h( x ) and g (x) = are fractional functions n( x ) k( x) The domain of f ( x ) is R – z ( n ) f(x)=

The domain of g ( x ) is R – z ( k ) Then the common domain of f ( x ) and g ( x ) is R – [ z ( n ) ∪ z ( k ) ]

Example: 5x −1 3x g(x)= and k ( x ) = x+4 x−5 The domain of g ( x ) is R ∼ { 5 } The domain of k ( x ) is R ∼ { -4 } The common domain of g ( x ) and k ( x ) is R ∼ { 5, - 4 }

Exercises 1) Find the common domain for each of the following fractions: a)

x+2 x 2 − 3x

x x −1

x−3 2x , 3 x − 5x + 6 x − 4x 2 2

………………………………………

x 2 − 8x + 15 c) 3

x 3 − 27 , 2 x − 3x − 18

2x + 1 2x 2 + x e) 3 , 3 , x + 4x x − 8 x 2 − 25 2 , x 4 − 5x 2 − 36 x 2 − 3x

……………………………………… ……………………………………… ………………………………………

………………………………………

x −2 2x 2 − 7x − 4

2x − 5 , 4x 2 + 8x + 3

……………………………………… ………………………………………

x 2 + 3x 2x 2 − 5x − 7

……………………………………… ………………………………………

………………………………………

2) Complete each of the following a) The common domain of the functions b) The common domain of N1(x) =

( x − 3 ) ( x + 2) x 2 − 2x − 3

2x + 1 is … x 3 + 4x

4 , and N2(x) = x2 – 9 is … x

x−2 1 and N2(x) = is 3x − a x² − b R ~ { 3, 2 , - 2 }, then a = … and b = …

c) If the common domain of N1(x) =

d) If the common domain of the functions N1(x) = N2(x) =

1 is R ~ { 3 }, then b = … x − 6x − b 2

x and x2 + 9

Reducing the algebraic fraction Reducing an algebraic fraction means putting it in its simplest from

Example:

2× 2× 5 20 4 = = 5×5 25 5

Note: We should determine the domain of the algebraic fraction before reducing it.

Exercises 1) Reduce each of the following showing its domain, then find n ( 1 ), n ( -2 ) and n ( 3 ) if it possible

x2 − 4 a) N ( x ) = x2 − 5 x + 6 …………………………………………

x 2 − 36 d) M ( x ) = 3 x − 216 …………………………………………

…………………………………………

x2 − 1 b) M ( x ) = 2 x −x …………………………………………

1 2 e) N ( x ) = 1 x− 3 …………………………………………

…………………………………………

x2 + x − 6 c) N ( x ) = 2 x − 2 x − 15 …………………………………………

f) M ( x ) =

2 x

…………………………………………

3 x² …………………………………………

…………………………………………

………………………………………………...

………………………………………… 24

g) N ( x ) =

x −1 1− x …………………………………………

c) N ( x ) =

…………………………………………

5 − 10x 3x − 1 − 2x 2 …………………………………………

…………………………………………

2x 2 − 14x + 20 7x − 2x 2 − 6 …………………………………………

h) M ( x ) =

x3 + 8 d) M ( x ) = − 16 + 2 x − x 2 …………………………………………

………………………………………… …………………………………………

…………………………………………

2) Complete the following: a)

a ........ = b+c 9b + 9c

2x + 4 2x b) x² − 4 = .........

c) If f(x) =

x−3 , then f( x ) = ………. ( in the simplest form ) 3−x

d) If N ( x ) =

x+3 , then N ( 2 ) does not exist because …… x−2

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Equality of two algebraic fractions

Two algebraic fractions f ( x ) and g ( x ) are equal if 1st : They have the same domain 2nd : f ( x ) = g ( x ) for every x ∈ the domain Note: Two algebraic fractions f ( x ) and g ( x ) have the same values in the common domain if f ( x ) = g ( x ) for every x ∈ the common domain

Example 1: If f ( x ) =

x 5x and g ( x ) = Prove that f = g x² + 1 5 x² + 5

Since the domain of f ( x ) is R and the domain of g ( x ) is R Since g ( x ) =

5x 5x x = = =f(x) 5 (x² + 1) 5 ( x² + 1) x² + 1

Therefore f = g

Example 2: If f1 ( x ) =

x −1 x−3 and f2 ( x ) = 2 . Is f1 = f2 ? 2 x −x x − 3x

If it is not, find the domain which makes f1 = f2. f1 (x) =

x −1 1 = x ( x − 1) x

D1 = R ∼ { 0 ,1 }

f2 (x) =

x−3 1 = x ( x − 3) x

D2 = R ∼ { 0 , 3 }

Then f1 ( x ) ≠ f2 ( x ) because D1 ≠ D2 but f1 ( x ) = f2 ( x ) for every x ∈ the common domain = R ∼ { 0 , 1 , 3 }

Exercises 1) If N1 ( x ) =

x² + 2 x − 3 x² − 3 x + 2 , N2 ( x ) = , does N1 equal N2 ? x² + 5 x + 6 x² − 4

Give reason ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

2) If F1 ( x ) =

x³ − x² − 6x x² − 4 and F2 ( x ) = . x³ − 9x x² + x − 6

Is F1 ( x ) = F2 ( x ) ? Why? ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… x3 + 1 x 3 + x 2 + 2x + 2 3) If F1 ( x ) = 3 and F2 ( x ) = . Prove that F1 = F2 x − x2 + x x 2 + 2x ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

4) N1 ( x ) =

x 2 + 3x + 2 x2 − 1 , N ( x ) = . Does N1 equal N2 ? 2 x2 − 4 x 2 − 3x + 2

Find the common domain which makes N1 and N2 equals ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

x 3 − x 2 − 6x x2 − 4 5) If F1 ( x ) = 2 and F2 ( x ) = . x3 − 9x x + x−6 Prove that F1 (x) = F2 (x) for every x ∈ the common domain ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… x2 − 9 x −3 and n ( x ) = . Prove that n1 ≠ n2 2 x +1 x2 + 4 x + 3 ………………………………………………………………………………………… 6) If n1 ( x ) =

………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

Addition and subtraction of algebraic fractions

n1 ( x ) =

f (x) g(x)

and

Then n1 ( x ) + n2 ( x ) =

n2 ( x ) =

t (x) k (x)

f (x) t (x) f ( x ) k ( x ) + t ( x ) g( x ) + = g(x) k (x) g(x ) k ( x )

The domain of the resulted algebraic fraction is R – [ z ( g ) ∪ z ( k ) ]

Example 1: If n1 ( x ) =

5 2x and n2 ( x ) = x−2 x+1

Then n1 ( x ) + n2 ( x ) =

5 ( x + 1) 2x ( x − 2 ) 5 2x = + + ( x − 2 )( x + 1) ( x + 1)( x − 2 ) x − 2 x +1 5 ( x + 1) + 2 ( x + 2 ) = ( x − 2 )( x + 1) 5x+5+2x+2 = ( x − 2 )( x + 1) 7x+7 = ( x − 2 )( x + 1) 7 ( x + 1) = ( x − 2 )( x + 1) 7 = x−2

The domain of the resulted function is R – { 2 , - 1 }

Additive inverse of fractional function: 1) f ( x ) =

h (x) is a fractional function its additive inverse is written as - f ( x ) g(x)

2) - f ( x ) =

− h(x) g(x)

3) The domain of f ( x ) = The domain of - f ( x ) = R – z ( g )

Example 2: If n1 ( x ) =

5 2x and n2 ( x ) = x−2 x+1

Then n1 ( x ) − n2 ( x ) =

5 ( x + 1) 2x ( x − 2 ) 5 2x = − − x − 2 x + 1 ( x − 2 )( x + 1) ( x + 1)( x − 2 ) 5 ( x + 1) − 2( x + 2 ) = ( x − 2 )( x + 1) 5x+5−2x−2 ( x − 2 )( x + 1) 3x+3 = ( x − 2 )( x + 1) 3 ( x + 1) = ( x − 2 )( x + 1) 3 = x−2 =

The domain of the resulted function is R – { 2 , - 1 }

Exercises 1) Find N(x) for each of the following, and its domain a) N(x) =

x−3 x−2 + x2 − 4 x + 3 x2 − 5 x + 6

b) N(x) =

3x − 4 2x + 6 + x2 + 5 x + 6 x2 + x − 6

…………………………………………

3x − 15 x 2 − 3x − 18 d) N(x) = − x² − 8x + 15 9 − x2

2 x² − 7 x + 3 1 − 3 x − 4 x 2 + 2 x² − 2 x − 3 x −x −2 …………………………………………

…………………………………………

c) N(x) =

2) Reduce N ( x ) to its simplest form, then find n ( -1 ) and n ( -2 ) if possible N(x) =

3 x + 15 2x +1 − x² + 7 x + 10 x + 2

3) Complete the following: a) If n1 ( x ) =

x² − 1 x +1 , and n2 ( x ) = , then n1 ( x ) + n2 ( x ) = …… x² + x + 1 x² − 1

b) If N1 ( x ) =

12 3x , and N2 ( x ) = , then N1 ( x ) − N2 ( x ) = …… x² − 4 x² − 2 x

and its domain is …… .

Multiplying Algebraic Fraction

Let n1 ( x ) =

f (x) g(x)

and

Then n1 ( x ) × n2 ( x ) =

n2 ( x ) =

t (x) k (x)

f ( x ) × t( x ) g( x ) × k ( x )

The domain of the resulted algebraic fraction is R – [ z ( g ) ∪ z ( k ) ] Notes: 1) The multiplicative neutral element is 1 2) n1 ( x ) × n2 ( x ) = n2 ( x ) × n1 ( x )

( Commutative property )

3) n1 ( x ) × [ n2 ( x ) × n3 ( x ) ] = [ n1 ( x ) × n2 ( x ) ] × n3 ( x ) ( Associative property )

Example: If n1 ( x ) =

4 x² − 9 x−2 and n2 ( x ) = x² − x − 6 2 x² − 3 x

4x 2 − 9 x−2 Then n1 ( x ) × n2 ( x ) = 2 × x − x − 6 2x 2 − 3x =

( 2 x − 3 )( 2 x + 3 ) x+2 × ( x − 3 )( x + 2 ) x (2x − 3)

(2x +3) x ( x − 3)

2x +3 x² − 3 x

The domain of the resulted function is R – { 3 , 2 , 0 ,

3 } 2

To download the student handbook: www.nis-egypt.com 32

Exercises

1) Find each of the following, and its domain

x 2 − 3x − 4 x2 − x 1) × 2 x2 − 1 x +3 x

x3 − 8 x2 − 4 4) × x + 2 x2 + 2 x + 4

…………………………………………

2x − 6 5x + x 2 + 6 2) 2 × x +x−2 9 − x2

x 2 − 3x − 4 x2 − x 5) × 2 x2 − 1 x +3 x

…………………………………………

9x 2 + 6x − 8 4x 2 + 4 x − 3 × 3) 6x 2 + 5x − 4 6x 2 − x − 2

9x 2 − 4 x 2 − 5x + 4 6) × 3x 2 − 7x − 6 3x 2 − 5x + 2

………………………………………… …………………………………………

…………………………………………

…………………………………………………

4x 2 + 3x − 10 3x 2 − 6x + 12 2) N(x) = , then find N ( 4 ) , N ( -2 ) × 2 x3 + 8 8x − 22 x + 15 ……………………………………………………………………………………… ……………………………………………………………………………………… ……………………………………………………………………………………… ………………………………………………………………………………………

Bus Rules: 1- Go directly to your bus and find your seat. Do not get off the bus again after you are seated. 2- Stay in your seat until the bus comes to a complete stop at your home. 3- Talk in a quiet voice to the person next to you. 4- Sit with your bottom on the seat, your back against the seat, and your feet on the floor. 5- Keep your head, hands, arms, and all other objects to yourself and in the bus 6- Do not litter on the bus, damage or destroy property. 7- As you get on or off the bus, look both ways before you cross the street. 8- Always obey and respect the bus matron and the driver Student Handbook

Dividing Algebraic Fraction

Multiplyplicative inverse of fractional function: −1

1) f ( x ) is a fractional function, its multiplicative inverse is denoted by f ( x )

2) If f ( x ) =

−1 m( x ) n( x ) , then f ( x ) = n( x ) m( x )

3) If f ( x ) =

−1 m( x ) , then the domain of f ( x ) = R ∼ z ( n ) n( x )

Dividing fractional functions: t (x) k (x) f ( x) × k( x) n ( x ) ÷ m ( x ) = n ( x ) × m-1 ( x ) = g( x ) × t ( x )

Let n ( x ) = Then

f (x) g(x)

and

m(x)=

The domain of the resulted algebraic fraction is R∼ [ z ( g ) ∪ z ( t ) ∪ z ( k ) ]

Example: If n1 ( x ) =

2x + 6 x² + 2x − 3 and n2 ( x ) = x² + 2 x + 4 x³ − 8

Then n1 ( x ) ÷ n2 ( x ) =

2x + 6 x² + 2x − 3 ÷ x² + 2 x + 4 x³ − 8

2x + 6 x³ − 8 × x² + 2 x + 4 x² + 2x − 3

( x − 2 )( x² + 2 x + 4 ) 2( x +3) × =2 ( x − 1) ( x + 3 ) x² + 2 x + 4

The domain of the resulted function is R – { 2 , 3 }

Exercises

1) Find the domain in which each of the following algebraic fractions has a multiplicative inverse and then find this inverse in simplest form 1) N(x) = x2 + 5 x + 6

4) N(x) =

…………………………………………

3−x x ( x +3 )

…………………………………………

2) N(x) =

5 x

…………………………………………

8x 4 − 8 x 5) N(x) = x2 − x

…………………………………………

3) N(x) =

x+9 x2 − 9

………………………………………… …………………………………………

…………………………………………

2x 3 + 6 x 2 + x + 3 6) N(x) = x2 − 9

………………………………………… …………………………………………

…………………………………………

………………………………………… ………………………………………… ………………………………………… ………………………………………… ………………………………………...

2) Complete each of the following a) If N ( x ) =

( x − 3 ) ( x + 2) , then the domain of x − 2x − 3 2

N – 1 (x) is ……

b) The domain of the multiplicative inverse of N1 ( x ) = x2 – 9 is R ~ …… c) The common domain of N1 ( x ) =

x −2 1 and N2 ( x ) = is 3x − a x −3

R ~ { 3 , 2 }, then the domain of N – 1 ( x ) is …… d) The common domain of N1 ( x ) =

x 1 and N ( x ) = 2 x2 + 9 x2 − 6 x − b

is R ~ { 3 }, then b = …… and the domain of N – 1 (x) is ……

3) Find each of the following, and its domain

x −1 x2 − 5 x 1) 2 ÷ x − 1 x2 − 4 x − 5

5 1 ÷ x − 2x − 3 1 − x2 2

…………………………………………

x 2 − 3x + 2 3x − 15 2) ÷ 2 2 x − 1 x − 4x − 5

3x − 9 2x + 6 ÷ x 2 − 5x + 6 6 − x − x 2

…………………………………………

………………………………………… ………………………………………... 37

4) Choose the correct answer from the brackets: x 3 1) The domain in which + has a multiplicative inverse is …… x−5 x−5 (a) R ~ { 0 , 5 } (b) R ~ { 5 } (c) R ~ { 0 , 3 , 5 } (d) R ~ { 3 , 5 }

2) If f ( x ) = x + (a)

1 x+1

1 where x ≠ 0, Then f – 1 ( x ) = ………. x 1 x (b) 2 (c) +x x x +1

(d) − x −

1 x

x−4 , then the domain of N – 1 ( x ) is R ~ ………………. x + 16x (a) { - 4 , 0, 4 } (b) { - 4, 4 } (c) { 4 } (d) { 0, 4 }

3) If N (x) =

4) If the domain of the algebraic fraction

3x is R ~ { 2 } ,then the x − 4x + a 2

domain of its multiplicative inverse is R ~ ……… (a) { 4 }

5) If n ( x ) =

(b) {2 , 0 }

(d) { 0 }

x² − 4 x³ − 8 × x² − 4 x + 4 2 x³ + 4 x² + 8 x

i) Find n ( x ) to its simplest form and state its domain ii) If n ( x ) = 1, then the value of x ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... ........................................................................................................................... 38

Probabiltiy Notes (1): 1) The sample space is denoted by S 2) Any event of a sample space will be denoted by a capital letter A 3) For any event A of a sample space S, we deduce A ⊂ S 4) Probability of occurring event A is denoted by P ( A ) n( A ) 5) P ( A ) = n( S ) n ( A ) is the number of elements of the event A n ( S ) is the number of elements of the sample space 6) P ( S ) = 1 7) P ( A ) is written as a fraction, decimal or percentage. 8) If A is an impossible event, then P ( A ) = zero 9) If A is a certain event, then P ( A ) = 1 10) If A is an event, then P ( A ) ∈ [ 0 , 1 ]

Mutually exclusive events: A and B are 2 disjoint events ( mutually exclusive ) ( A ∩ B = ϕ ) in a sample space, then P ( A ∩ B ) = 0

Note (2): A and B are mutually exclusive events are represented by S the opposite Venn diagram A

Note (3): A and B are two events in a sample space of a random experiment P(A ∪ B)=P(A)+P(B)−P(A ∩B)

Complementary event: The complementary event to an event A is the event of not occurring A.

Note (4): 1) The complementary event to an event A is denoted by A’

S A

2) The complementary event to an event A is represented by the shaded part in opposite Venn diagram 3) If A is an event of the sample space S, then A ∪ A` = S 4) The event and its complementary are two mutually exclusive events ( A ∩ A’ = ϕ )

Difference between two events: A and B are two events in a sample space of a random experiment The event A − B is the event of the occurrence of A and the non-occurrence of B

Note (5):

S A

1) The event A − B is represented by the shaded part in opposite Venn diagram 2) The event B − A is represented by the shaded part in opposite Venn diagram

Example ( 1 ): A and B are two mutually exclusive events in the sample space where P(A)= Find:

1 3 and P ( B ) = 8 8 (i) P ( A ∪ B )

(ii) P ( A ~ B )

(iii) P ( A )

A and B are mutually exclusive events means P ( A ∩ B ) = 0 (i) P ( A ∪ B ) = P ( A ) + P ( B ) =

1 3 + = 0.5 8 8

(ii) P ( A ~ B ) = P ( A ) – P ( A ∩ B ) = (iii) P ( A' ) = 1 – P ( A ) = 1 –

1 1 –0= 8 8

1 7 = 8 8

Example ( 2 ): A and B are two events in the sample space where P ( A ) = P(B)= Find:

1 , 2

1 7 and P ( A ∩ B ) = 4 24 (i) P ( A ∪ B )

(ii) P ( B` )

(iii) P ( A ∩ B )`

(i) P ( A ∪ B ) = P ( A ) + P ( B ) – P ( A ∩ B ) = (ii) P ( B` ) = 1 – P ( B ) = 1 –

1 3 = 4 4

(iii) P ( A ∩ B )` = 1 – P ( A ∩ B ) = 1 –

7 17 = 24 24

1 1 7 11 + – = 2 4 24 24

Example ( 3 ): A and B are two events in the sample space where P ( A' ) =

3 5 1 , P ( B' ) = and P ( A ∪ B ) = . Find P ( A ∩ B ) 4 8 8

P ( A ) = 1 – P ( A' ) = 1 –

3 1 = 4 4

P ( B ) = 1 – P ( B' ) = 1 –

5 3 = 8 8

P(A ∩ B)=P(A)+P(B)–P(A ∪ B)=

1 3 1 1 + – = 4 8 8 2

Example ( 4 ): Let A and B be two events in the sample space where P ( A ) = P(B)= Find:

1 5 and P ( A ∪ B ) = 2 12 i) P ( A ~ B )

P(A ∩ B)=

(ii) P ( B ~ A )

5 1 5 11 + − = 6 2 6 12

(ii) P ( A ~ B ) = P ( A ) – P ( A ∩ B ) =

5 11 −1 – = 6 12 12

(iii) P ( B ~ A ) = P ( B ) – P ( A ∩ B ) =

1 11 − 5 – = 2 12 12

To download the student handbook: www.nis-egypt.com 42

5 6

REVISION SHEET

1) Choose the correct answer: 1) If A and B are two events in the sample space of a random experiment and p (A) = 0.3, p (B) = 0.6 , p (A ∩ B) = 0.2 then p (A

B) = ……………… ( 1 , 0.8 , 0.7 , 0.5 )

x −1 then the domain n-1 (x) is ……………… x−3

2) If n (x) =

(

- {3} ,

- {1} ,

- {1, 3} , {1, 3} )

3) If a regular circle die is rolled once and randomly and the upper face is observed, then the probability of getting an even prime number equals ………

(0,

4) If : p (A) = 0.5, p (B) = 0.2, p (A

1 1 , ,1) 6 3

B) = 0.7 then ………………

(B ⊂ A , A ⊂ B , p (A ∩ B) = 0.3 , A and B are mutually exclusive) 5) If: n (x) =

x 1 then the set of zeroes of n (x) = ……… x−3 x−3 ( {3} , {1} , {-1} , {-3} )

6) The solution of the two equations: x – y = 0 and x y = 25 is ……… ( { (5, 5) } , { (0, 0) } , { (-5, -5) } , { (5, 5) , (-5, -5) } ) 7) In experiment of thrown a die once the probability of appearance a number divisible by 7 = ……… 8) If n1(x) = n1, n2 is 9) If n(x) =

( zero ,

1 −1 , ,1) 7 7

x −1 x+4 , n2 (x) = and the common domain of the two functions x+2 x−k - {-2, 7} then k = ……………… 5 the domain of n is …… x−2

(

( 7 , -2 , -4 , 1 ) - {2} ,

5 -{ }, 2

- {5} ,

- {-2} )

10) The multiplicative inverse of the fraction

2 , x ≠ 3 is ………… x−3 (

−2 −2 x −3 3 + x , , , ) x−3 3−x 2 2

11) If the curve of the function f where f (x) = x2 – a passes through the point ( ± 1 , -1 , 1 , zero )

(1, 0) then a = ………………

12) If a regular coin is tossed once, then the probability of getting head or tail is ………….

( zero% , 25% , 50% , 100% )

13) The set of zeroes of f : where f(x) = x (x2 – 2x + 1) is …………. ( {0, 1} , {0, -1} , {0, 1, -1} , {1} ) 14) If AB, then p (A

( zero , p (A) , p (B) , p (A ∩ B) )

B) = ………….

15) If the algebraic fraction

x−a x−3 has a multiplicative inverse is then a = x−3 x+2

………….

( 2 , 3 , -3 , -2 )

16) The set of zeroes of f(x) = 3 x is …………. 17) The simplest form of f(x) =

( {0} , {3} , {0, 3} ,

)

4 x ÷ , x ≠ 1 or zero is …………. x −1 x −1 (

18) If the simplest form of the algebrical fraction n(x) = n(x) = x then a = ……

4x x −1 x ,4, , ) x −1 4x ( x − 1)² x(x − 2) , x ≠ 2 is x+a ( zero , 2, -2, -4 )

19) If x = 3 a root for the equation x2 + m x = 3 then m = …………. ( -1 , -2 , 2 , 1 ) 20) If n(x) =

x+2 then n-1 (2) = …… x+5

(

4 − 4 7 −7 , , , ) 7 7 4 4

To download the student handbook: www.nis-egypt.com 44

2) Complete: 1) If n1(x) =

3−a 5 , n2(x) = and n1(x) = n2(x) then a = ………… x +1 x +1

2) The common domain of n1 and n2 where n1(x) = 3) The simplest form of n(x) =

x−2 1 , n2 = is …………. x² − 4 x +1

2 x² + x , x ≠ 0 is …………. x

4) The zeroes of the function f in

: f (x) =

x +1 is …………. 4

5) It is said that A and B are mutually exclusive event if A ∩ B = …………. 6) The probability of impossible event is …………. 7) A is an event of a sample space S of a random experiment where n(A) = 3, n(S) = 5 then p(A) = …………. 8) The set of zeroes of f(X) = x² - 9 x + 8 is ………… 9) If x + y = 5 , x – y = 3 ,then x² – y² = ……….. 10)If A and B are two mutually exclusive events then P ( A - B ) = …… 11) If there are infinite number of solution of two equations x + 2 y = 5 , 3 x + 6 y = k,then k = …………. 12) The solution set of the two equations : x + y = 5 , y = - 1 is ……….. 13) The two events A and B are mutually exclusive if …………. 14) If A ⊂ B ,then P ( A - B ) =……………… 15 ) The solution set of the two equations : x + y = 0 , y - 5 = 0 is ……… 16) The common domain of the functions n2 (x) and n2 (x) is ………

3) Find n(x) in simplest form showing its domain n(x) =

4) Find in

3x−4 2x+ 6 + x² − 5 x + 6 x² + x − 6

the solution of the two equations

x – 5y = 0 and x2 + y2 = 26

5) If A and B are two events in the sample space of a random experiment and p (B) =

1 , p (A 12

B) =

1 then find p (A) in each of 3 2) B ⊂ A

1) A, B are mutually exclusive

6) A bag contains 12 balls number from 1 to 12 if a ball is drawn randomly and the event A is getting an odd number and event B is getting a prime number then find: p(A`) , p (A – B)

7) State the domain of the function n (x) =

x³ − 8 2 x² + 4 x + 8 ÷ x² − 4 x + 4 x² − 4

8) Find the solution set of the equation x2 – 2x – 4 = 0 by using the general rule rounding the result to nearest one decimal digit.

9) A card is drawn randomly from 100 identical cards numbered from 1 to 100. Find the probability that the drawn card 1) divisible by 10

2) divisible by 25

3) divisible by 10 and 25

10) Find the solution set of the following equations y – x = 2, x2 + x y – 4 = 0

11) Find the solution set of the equation 3x2 = 5x – 1 where 13 = 3.61

12) A class contains 30 students, 25 students of them succeeded in

mathematics, 24 students succeeded in Science and 20 succeeded in both of the two examinations if a student is chosen randomly. Find the probability that the chosen student 1) Succeeded in mathematics only 2) Succeeded in one of the two examinations at least. 46

Basic Definitions and concepts on circles

1) The circle is the set of points of the plane which are equidistance from a fixed point in the plane.

The fixed point is called the centre of the circle The constant distance is called the radius length

Example: A circle M means a circle of centre M

2) A chord of a circle is a line segment whose endpoints are points of the circle Example: AB and AC are chords of circle M

3) A diameter of a circle is a chord passing through the centre of the circle A diameter is the longest chord of the circle. Example: AC is a diameter of circle M

4) A raduis of the circle is a line segment whose endpoints are the centre of the circle and a point of the circle Example: 1) MA and MB are radii of circle M 2) AMB is an isoscelse triangle 3) m ∠ A = m ∠ B

5) The circle has an infinite number of axes of symmetry 6) The area of a circle = π r², The perimeter of the circle = 2 π r 48

Exercises 1) Complete:

a) In the opposite figure: AB is a diameter of circle M where m ∠ AMC = 110˚

m ∠ MAC = ……˚ m ∠ MCB = ……˚ m ∠ BCA = ……˚ b) π =

.......... The diameter

2) In the opposite figure:

AB is a diameter of circle M. AB // CD , m ∠ AMC = 30º Prove that:

i) m ∠ DBA = 75º

ii) AC = DB ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

3) In the given figure: DBC is a triangle inscribed in a circle M m ∠ BMD = 100º. m ∠ MCB = 40º Find the measures of the Interior angles of ▲ BCD ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… 49

Important Corollaries

Corollary 1: The straight line passing thorough the centre of the circle and the midpoint of any chord of it is perpendicular to this chord If AB is a chord of circle M, If MC is drawn where C is the midpoint of AB Then MC ⊥ AB

M C B

Corollary 2: The straight line passing thorough the centre of the circle and perpendicular to any chord of it bisects this chord If AB is a chord of circle M, If MC ⊥ AB M

Then AC = CB

C B

Corollary 3: The perpendicular bisector of any chord of a circle passes thorough the centre of the circle. If AB is a chord of circle M,

If L ⊥ AB at its midpoint C Then M ∈ straight line L

M C A

Exercises 1) Complete : a) In the opposite figure: AB is a diameter of circle M, AB = 10 cm, AC = 8 cm and MD ⊥ AC , MD = …… cm

BC = …… cm

b) In the opposite figure: AB is a chord of the circle M

C is the midpoint of AB and m ∠ CMB = 50˚ m ∠ B = ……˚

m ∠ A = ……˚

m ∠AMB = ……˚

c) In the opposite figure:

AB is a chord in circle M of radius 13 cm, MC ⊥ AB to cut circle M at D

B C M

If AB = 12 cm, then the area of ▲ ABD = …… cm²

2) In the given figure:

ABC is an equilateral ▲ inscribed in a circle M

D M

DA = DB and AE = EC Find m ∠ DME

3) In the given figure: AC and AB are 2 chords of circle M, E, D are midpoints of AC , AB m ∠ BAC = 45º, Draw DM to cut AC at F Prove that ME = EF ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

4) In the opposite figure:

AB is a chord of circle M D is a midpoint of AB, E is a midpoint of AM DE = 3.5 cm Find the circumference of circle M ( π =

22 ) 7

………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

5) In the given figure: Two concentric circles at M, AB is a chord of the great circle and cuts the small circle at C and D. ME ⊥ AB , ME = 12 cm, EC = 9 cm the radius of the small circle =

3 the radius of the great one 4

1) Prove that AC = BD ii) Find DB ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

Everyday Communication - Channel Book: 1- Students will use the Channel book to register their homework assignments. 2- The Channel Book will help keep students organized and serves as a communication tool between parents and teachers. 3- Parents are to check the Channel daily and sign it. 4- If the Channel book is lost, not kept orderly, or is damaged, students must purchase another one from the school. Student Handbook

Positions of a point with respect to a circle If X is any point in the plane of a circle M of radius = r Then we have 3 cases of the positions of point X with respect to the circle M

Positions

X is inside the circle

X is on the circle

MX , r

MX < r

MX = r

X is outside the circle MX > r

Exercises 1) Complete: a) A circle M of diameter 14 cm, A is a point in its plane: i) if MA = 9 cm , then A lies .......

iii) if MA = 7 cm , then A lies ......

ii) if MA = 5 cm , then A lies ......

iv) If MA = 0, then A lies ......

b) If M is a circle of radius 5 cm and A is a point on the circle, then MA = ...... cm and the length of the longest chord of the circle = …… cm c) AB is a chord of circle M of diameter 26 cm, and AB = 24 cm, then the distance between M and AB = …… cm

2) Choose the correct answer: i) If N is a point inside circle M of radius 5 cm, then MN ∈ …… a) [ 0 , 5 ]

c) ] 5 , ∞ [

b) { 5 }

d) ] 0 , 5 [

ii) If N is a point on the circle M, then circle M ∩ MN = …… a) { N }

b) MN

c) { M , N }

d) ϕ

ii) If N is a point on the circle M, then surface of circle M ∩ MN = …… a) { N }

b) MN

c) { M , N } 54

d) ϕ

Positions of a staright line with respect to a circle If L is any straight line in the plane of a circle M of radius = r Then we have 3 cases of the positions of straight line L with respect to the circle M L B

Straight line L cuts Positions

circle M at 2 points

Straight line L touches circle M at one point

(secant)

MX , r

(tangent)

MX < r

MX = r

Straight line L is outside circle M (outside the circle)

MX > r

Circle M {X}

∩

Straight

{A,B}

(point of tangency)

Line L Surface of circle M ∩

{X}

Straight Line L

Fact: i) A tangent is perpendicular to the radius at the point of tangency. ii) If a straight line is perpendicular to a diameter of a circle at one point of its endpoints, then it is a tangent to the circle. 55

Exercises 1) Complete: a) A circle M of diameter 6 cm and L is a straight line in its plane, MA ⊥ L, A ∈ L: i) If MA = 7 cm , then L is ........

ii) If MA = 4 cm , then L is ......

iii) If MA = 6 cm , then L is .......

iv) If MA = 0 cm, then L is ……

b) If the radius of circle M = r cm, MX ⊥ straight line L, X ∈ L. i) If XM = r, then L is …… iii) If XM =

ii) If XM = 4 r, then L is ……

5 r, then L is …… 8

iv) If XM = 0, then L is ……

b) If the radius of circle M = r cm, MX ⊥ straight line L, X ∈ L. i) If XM = r, then L is …… iii) If XM =

ii) If XM = 4 r, then L is ……

5 r, then L is …… 8

iv) If XM = 0, then L is ……

e) In the opposite figure: AB and AC are two tangents to circle M at B and C m ( ∠ BAC ) = 25˚,

B A

m ∠ BMC = ……º

25˚

f) In the opposite figure: AB is a diameter of circle M, AC is a tangent to circle M at A, m ∠ AMD = 70º i) m ∠ CAD = ……º

iii) m ∠ ADB = ……º

ii) m ∠ ABD = ……º

iv) m ∠ ACB = ……º

g) In the opposite figure: A circle of centre M, where MA = AB, AC is a tangent to circle M at A, m ∠ CAB = …… h) In the opposite figure: AB touches circle M at A, If AM = 6 cm and MB = 10 cm, then AB = …… 56

i) In the opposite figure: If AB touches circle M at A, MB ∩ circle M = { C } and MC = BC m ∠ B = …… j) In the opposite figure: BX is a tangent to the circle M at B

AB // MC , m ∠ ABX = 50° m ∠ CBX = ……

2) In the opposite figure: BA is a tangent to a circle M at B, MC intersects the circle at C and AB at A, m ( ∠ BAC ) = 34˚, Find:

i) m ∠ BMA

ii) m ∠ BCA

3) In the opposite figure: A circle of centre M DC = MC, m ∠ MAC = 20˚, m ∠ AMC = 70˚, i) Prove that AD touches the circle at C ii) Find m ( ∠ MDC ) ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… 57

4) In the given figure: AB is a diameter of circle M, D ∈ BA If DC is a tangent to the circle at C, m ∠ B = 25º. Find m ∠ D ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

5) In the opposite figure: AB touches the circle M at B, X is a midpoint of DC , m ( ∠ CAM ) = 15˚, m ∠ AMB = 70˚ Find i) m ( ∠ A )

ii) m ( ∠ XMB )

ABCD is a quadrilateral, CD touches circle M at X

BC ∩ circle M = { E } where Y is a midpoint of BE

X D

MY // AB and MX // AD , m ∠ YMX = 70°, Find m ∠ A

M B

Positions of a circle with respect to a circle

M and N are 2 circles of radii r1 and r2 respectively, where r1 > r2

Fact: M and N are two intersecting circles at A and B AB is a common chord MN is an axis of symmetry of AB

Exercises 1) The lengths of the radii of circles M and N are 2.37 cm and 6.42 cm. Determine the position of each of them with respect to the other if: 1) MN = 8.8 cm 4) MN = 0

1 cm 20 5) MN = 8.79 cm

3) MN = 2.8 cm

2) MN = 4

6) MN = 4.1 cm

2) M and N are two circles of radii 4 cm and 6 cm respectively. Find the length of MN if the 2 circles are: i) Touch each other externally

ii) Touch each other internally

iii) Intersecting

iv) Concentric

v) Excentric

vi) Distant

3) Complete: a) If the surface of the circle M ∩ the surface of circle N = ϕ, then circle M and circle N are …… b) If the circle M ∩ the circle N = ϕ, then circle M and circle N are either …… or ...... c) If the surface of the circle M ∩ the surface of circle N = { A }, then circle M and circle N are …… d) If the circle M ∩ the circle N = { A } then circle M & circle N are either …… or ...... e) If the surface of the circle M ∩ the surface of circle N = the surfaces of circle M, then the two circles M and N are either …… or …… 60

f) In the opposite figure: Circles M and circle N touch at A MN = 12 cm and NB = 7 cm MA = …… cm g) In the opposite figure: Circles M and N touch at A, MC = 4.5 cm and ND = 9.1 cm MN = …… cm h) If M and N are 2 circles with radii 7 cm and 4 cm, then the needed condition to be one of them inside the other is MN ∈ ] ..... , ..... [ i) If M and N are 2 touching circles where MN = 6 cm, the radius length of circle M is 10 cm, then the radius length of the other circle = ...... cm

4) In the opposite figure: Two concentric circles at M, CD is a chord of the big circle cutting the small circle at X, Y H ∈ CD where HX = HY. AB touches the big circle at B

m ∠ A = 50°. Find m ∠ HMB ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… Student Security: 1- The school will provide every appropriate measure to ensure the safety of its students. 2- The school employs security guards that patrol the campus. There are also permanent posts at each gate. 3- Parents who wish to take their children off campus during school hours must complete the permission form and obtain the relevant Division Supervisor's signature. Student Handbook 61

5) In the opposite figure: 2 circles with centers M and N intersected at A and B.

C ∈ XY where XY touches circle M at C, B

Prove that: i) AB // XY

ii) C is a midpoint of XY

……………………..…………………..……………………………………………… …………………………….…………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ……………………..…………………..……………………………………………… …………………………….…………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

6) In the opposite figure: Two circles M and N of radii 3 cm, 4 cm, are intersected at A and B. If NA is a tangent to circle M at A. i) Prove that MA is a tangent circle N at A

A M

ii) Find the length of the common chord B

To download the student handbook: www.nis-egypt.com 62

Identifying Circle

A circle is drawn if we identify 1) Its centre 2) Its radius length

Example 1: Draw a circle of radius length 5 cm 1) Drawing a circle passing through a given point

Example 2: If A is a point in the plane. Draw in your copybook a circle passes through point A. How many circles could be drawn passes through point A ? There are infinite numbers of circles that pass through a point A

2) Drawing a circle passing through two points

Example 3: If A and B are two points in the plane. Draw in your copybook a circle passes through points A and B. How many circles could be drawn passes through the two points There are infinite numbers of circles that pass through two points Note 1: The axis of symmetry of AB passes through the centers

3) Drawing a circle passing through three points Note 2: a) It is impossible to draw a circle passing through three collinear points b) There is ONE and ONLY ONE circle passes through three noncollinear points 63

Note 3: A and B are two points in plane. A circle is drawn passes through A and B where

1) Diameter of the circle > AB, then there are infinite number of circles could be drawn pass through A and B

Example 4: A and B are two points in plane where AB = 4 cm. Draw in your copybook a circle of radius length more than AB and passes through the two points A and B Note 4:

AB is a common chord of the circles The axis of symmetry of AB passes through the centers

2) Diameter of the circle = AB, then there is only one circle could be drawn pass through A and B

Example 5: A and B are two points in plane where AB = 4 cm. Draw in your copybook a circle of radius length 4 cm and passes through the two points A and B Note 5:

the centre of the circle is the midpoint of AB The axis of symmetry of AB passes through the centers

3) Diameter of the circle < AB, then there is no circles could be drawn pass through A and B

Definition:

The circle which passes through the vertices of a ▲ is called circumcircle of this ▲

We can say that the ▲ is inscribed in the circle if its vertices are on the circle

Example 6: In the opposite figure: Circle M is a circumcircle of ▲ ABC The centre of circle M is determine by The intersection of the axes of symmetries of the sides of ▲ ABC Fact: The perpendicular bisectors of the sides of a triangle intersect at a point which is the centre of the circumcircle of the triangle

Note 6:

Exercises

1) Complete: a) An infinite number of circles can be drawn to pass through …… b) The smallest circle which can be drawn to pass through two points of 8 cm apart, the length of its radius is …… cm c) The perpendicular bisectors of the sides of a triangle meet at one point which is …. d) The centers of the circles which pass through the two points A and B lie on …… e) We can draw …… passing through three non-collinear points f) It is possible to draw …… passing through a given point g) The number of circles that can be drawn passing through the 2 given points is …… h) The circle passing through the vertices of a triangle is called …… i) There exists ……and …… circle which passes through 3 non-collinear points. j) The number of circles that can be drawn passing through 3 collinear points is …… k) The centre of a circumcircle of a triangle is …… l)

The centre of a circumcircle of an equilateral triangle lies on the

intersection point of …… or …… or …… m) The centre of a circumcircle of an acute angled triangle lies on …… n) The centre of a circumcircle of an obtuse angled triangle lies on …… o) The centre of a circumcircle of a right angled triangle lies on …… p) The smallest circle that passes through the end-points of a line segment of length 14 cm is of radius equals …… cm q) A and B are 2 points on the plane where AB = 9.5 cm, then the number of circles whose radius length of each is 4.8 cm and passing through A and B is …… 66

2) Choose the correct answer: a) It is possible to draw a circle passing through the vertices of …… i) isosceles trapezium and square

ii) square and parallelogram

iii) rhombus and equilateral triangle

iv) trapezium and rectangle

b) If C is a midpoint of AB , then the number of circles that pass through A, B and C is …… i) infinite

ii) zero

iii) 2 circles

iv) 1 circle

3) A and B are two points in the same plane where AB = 5 cm. Draw a circle of radius 3 cm and passing through A and B. How many solutions are there?

4) A and B are two points in the same plane where AB = 6 cm. Draw a circle with the smallest possible radius passing through A and B . How many solutions are there?

5) Draw a ∆ ABC in which AB = 3 cm , BC = 4 cm and AC = 5 cm, then draw the circumcircle of the ∆ ABC

8) ABC is a triangle, where AB = 5 cm, BC = 6 cm and AC = 7 cm. Draw circumcircle M of triangle ABC

The relation between the chords of the circle and its center

Theorem:

In the opposite figure:

If CD and AB are two equal chords of circle M Where MX ⊥ CD and MY ⊥ AB

C A

Then MX = MY

Proof of the theorem: CD and AB are two equal chords of circle M MX ⊥ CD and MY ⊥ AB Join MC and MA

∵ MX ⊥ CD and MX passes through the centre M

∴ CX = ½ CD ∵ MY ⊥ AB and MY passes through the centre M

∴ AY = ½ AB ∵ AB = CD ∴ CX = AY In two triangles AMY and CMX

∵ CM = AM

( two radii )

∵ m ∠ MXC = m ∠ MYA = 90° ∵ AY = CX ▲ AMY ≡ ▲ CMX

∴ MX = MY

( that is the required to proof )

To download the student handbook: www.nis-egypt.com

Corollary: In the opposite figure:

Circles M and N are congruent ( equal radii )

∵ AB = CD

X M

C N

∵ NX ⊥ CD and MY ⊥ AB

∴ MX = NY The converse of the theorem: In the opposite figure:

CD and AB are two chords of circle M

∵ MX ⊥ CD and MY ⊥ AB

∵ MX = MY

∴ AB = CD Exercises

1) Complete:

B X

In the opposite figure:

AB and CD are 2 chords of circle M. If MX ⊥ AB , Y is the midpoint of CD

MX = MY, and CD = 5 cm, then AX = …… cm

Questions and Concerns: 1- For classroom concerns, contact the Division Supervisor directly. 2- For fees or other financial concerns, contact the Finance Office. 3- For transportation questions, contact the General Administration department. 4- If you are still not satisfied, contact the school's secretary who will arrange a meeting with SIB Director, or if necessary with school's CEO. Student Handbook

2) In the opposite figure: AB , AC are two equal chords in circle M, N, F are the midpoints of AB , AC and m ( ∠ E ) = 30˚, Prove that:

i) DN = EF

ii) AXY is an equilateral ▲

3) In the opposite figure:

ABC is a triangle inscribed in circle M, m ∠ A = 40º, m ∠ B = 70º, MX ⊥ AB and MY ⊥ AC

ii) Find the area of circle M

X M

( π = 3.14 )

If MX = 5 cm, YC = 12 cm i) Prove that MX = MY

4) In the opposite figure: Two concentric circles at M. AB and AC are 2 chords in the big circle where AB and AC touch the small circle at X and Y. Prove that AB = DC ……………………..…………………..……………………………………………… …………………………….…………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… 70

Central Angles and Measuring Arcs A

Definitions: In the opposite figure: M

A circle of centre M A and B are points on circle M

1) Arc The set of points of the circle from A to B is arc AB and it is denoted by AB

2) Central Angle It is the angle whose vertex is the center of the circle and its two sides are radii.

Rules: 1) A and B â&#x2C6;&#x2C6; circle M, then

The length of AB The measure of AB = The circumference 360Â°

2) The measure of the arc equals the measure of the central angle opposite to it.

Corollary ( 1 ): = m CD If A, B, C and D are points on a circle where m AB = length CD and conversely Then length AB

Corollary ( 2 ):

= AB If CD and AB are two chords of a circle where CD Then CD = AB and conversely

To download the student handbook: www.nis-egypt.com 71

Corollary ( 3 ): If AD and CB are two chords of a circle where AD // CB

= CD Then AB

Corollary ( 4 ):

If AB is a chord of a circle, XY touches the circle at D where AB // XY

= BD Then AD

Exercises 1) Complete: 3 the measure of 4 the circle = …… °, and if the radius of the circle is 6 cm,

a) The measure of the arc which represents

then the length of this arc = …… π b) The measure of the arc which represents

3 of a circle = ……° 5

c) The measure of a semicircle = ……° d) The measure of

2 of a semicircle = ……° 3

2) If the measure of an arc in a circle is 60° and the radius of the circle is 12 cm, find the length of this arc. ( π =

22 ) 7

………………………………………………………………………………………… …………………………………………………………………………………………

3) If the measure of an arc in a circle is 120 cm and the radius is 21 cm, find the length of the arc.

(π=

22 ) 7

………………………………………………………………………………………… ………………………………………………………………………………………… 72

4) In the opposite figure: AB is a diameter in circle M. X , Y are two points on the same side of ) = m ( XY ) = m ( YB ) AB where m ( AX

Prove that: 1) MXY is an equilateral triangle ) = 1 m ( BAX ) 2) m ( XY 4 …………………………………………………………………………………………

5) In the opposite figure: ABCD is a quadrilateral inscribed in a circle where AD // CB DB ∩ AC = { E }. XY is a tangent to the circle at B where XY // AC Prove that DCB is an isosceles triangle ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

6) AB and CD are two diameters of circle M where m ∠ DMB = 35° ) E circle M where AB // CE . Find m ( BE

Relation Between The Inscribed Angle And Central Angle Subtended By The Same Arc

Inscribed Angle It is the angle whose vertex lies on the circle and its sides are chords.

Example 1: In the opposite figure: A circle of centre M. If A, B, C, D and E are points on circle M Write all central angles and Inscribed angles and their subtended arcs

Theorem ( 1 ): In a circle of centre M

If ∠ ACB is an inscribed angle subtended by AB If ∠ AMB is a central angle subtended by AB

Then m ∠ ACB =

1 m ∠ AMB 2

m ∠ AMB = 2 m ∠ ACB

Proof of Theorem ( 1 ):

∵ ∠ AMB is an exterior angle of triangle AMC ∴ m ∠ AMB = m ∠ A + m ∠ C ……… ( 1 ) C

∵ MA = MC ( radii ) M

∴ m ∠ A = m ∠ C ……… ( 2 ) B

By substituting from ( 2 ) in ( 1 )

∴ m ∠ AMB = 2 m ∠ C

Corollary ( 1 ): For any circle: m ∠ ACB =

1 m AB 2

Corollary ( 2 ): AB is a diameter of a circle

∠ AXB, ∠ AYB and ∠ AXB are inscribed angles subtended by AB m ∠ AXB = m ∠ AYB = m ∠ AXB = 90°

Exercises

1) In the opposte figures: A circle of centre M Complete: A

35° M

28°

56°

M B

C B

1) m ∠ CMB = …… °

2) m ∠ CBA = …… °

3) m ∠ CAB = ……°

B M

110° B

34°

140° C C

5) m ∠ ABM= …… °

6) m ∠ BAC = …… °

7) m ∠ ADC = ……°

A D

60°

M B

A B

8) m ∠ ADM = …… °

9) m ∠ AMB = …… °

10) m ∠ CBM = …… °

m ∠ CBM = …… ° C

50° M

D A

130°

11) m ∠ ADM = ……°

135°

12) m ∠ AMC = …… °

13) m ∠ ABC = …… ° A

2) In the given figure: AC is a diameter of circle M

60º

m ∠ ABD = 60º, m ∠ BCA = 50º

M 50º

Find m ∠ ACD, m ∠ BAD and m ∠ CBD D

………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… Our staff responsibilities: ( some items ) 1- To promote positive behavior in school at all times; 2- To maintain clear expectations. 3- To regularly praise positive behavior. 4- To take time to build relationship with students. 5- To enable student to develop self-esteem. 6- To communicate any issue or concerns with relevant staff. Student Handbook 76

Inscribed Angles Subtended by the Same Arc

Theorem (2): If ∠ A and ∠ D are inscribed angles subtended by the same arc

∴m∠A=m∠D Proof of Theorem (2):

∠ A and ∠ D are inscribed angles subtended by the same arc BC 1 ) …………… ( 1 ) m ∠ A = m BC ( where ∠ A is subtended by BC 2 1 ) …………… ( 2 ) ( where ∠ A is subtended by BC m ∠ D = m BC 2 From ( 1 ) and ( 2 ): m ∠ A = m ∠ D Corollary:

= m EF If m CB

B A

Then m ∠ A = m ∠ D C

Exercises 1) In the opposte figures: A circle of centre M Complete: A

A D

40°

30°

A 30°

42° C

35°

B B

1) m ∠ ABD = ……°

2) m ∠ DBX = ……°

m ∠ BDC = ……°

m ∠ BDA = ……° 77

3) m ∠ ADB = ……°

45°

D B

M C

4) m ∠ CBD = ……° D

6) m ∠ CBD = ……°

D B

M E C

B D

5) m ∠ BAC = ……°

20°

40°

140°

70° A

130° D

7) m ∠ A = ……°

8) m ∠ CDB = ……°

7) m ∠ C

m ∠ C = ……°

2) In the given figure:

BC is a diameter in circle M,

) = 80°, m ( BD ) = 20° m ( AC C

Choose the correct answer: a) m ∠ ABC = ……

[ 20° , 40° , 60° , 80° ]

) = …… b) m ( AB

[ 40° , 60° , 80° , 100° ]

c) m ∠ E = ……

[ 10° , 20° , 30° , 40° ]

3) In the given figure:

D N

AB is a diameter in circle M AC and BD are 2 chords intersect at N,

m ∠ ANB = 110°, m ∠ CAB = 30° Complete: a) m ∠ CDB = ……°

b) m ∠ ADB = ……°

d) m ∠ DCA = ……°

b) m ∠ CMB = ……°

) = ……° e) m ( CB

f) m ∠ DCA = ……°

4) In the given figure: CB ∩ ED = { A }, m ∠ CDE = 50°

m ∠ BED = 25°, CD ∩ BE = { N }

) Find: a) m ( CE b) m ∠ A c) m ∠ CBE

………………………………………………………………………………………… …..………………………………………….........…………………………………… ………………………………………………………………………………………… …..………………………………..…………………………………………………… ……………………….………………………………………………………………… ….……………………………………………………………………………………… …...……………..………………………………………………………………………

5) In the given figure: AD ∩ BC = { E }, CA ∩ BD = { F }

A D E

m ∠ DBC = 25°, m ∠ E = 36°

N C

Find: a) m ∠ ADB

b) m ∠ DAC c) m ∠ ANB ………………………………………………………………………………………… …………………………………………….........…………………………………..… ………………………………..………………………………………………………... ……………………….………………………………………………………………… …………………………………………………………………………………………. ………………………………………………………………………………………… ………………………………………………….........……………………………… …………………………………………………………………………………………. ………………………………………………….........……………………………… 79

The Converse of Theorem ( 2 ): ABCD is a quadrilateral a) If m ∠ ADB = m ∠ ACB, then there is a circle passes through A, C, B and D b) If m ∠ …… = m ∠ ……, then there is a circle passes through A, C, B and D c) If m ∠ …… = m ∠ ……, then there is a circle passes through A, C, B and D d) If m ∠ …… = m ∠ ……, then there is a circle passes through A, C, B and D Note: ABCD is called cyclic quadrilateral

Exercises 1) In each of the following figures, say whether ABCD is a cyclic quadrilateral. Mention the reason: A

60°

70° C

30° B

A B

C A

120° E

50°

25° 85°

35°

70°

60° 60° A

30°

C 80

D E

2) Complete:

In the opposite figure: WXYZ is a cyclic quadrilateral, where YZ = YX m ∠ Y = 70º and m ∠ ZXW = 41º m ∠ XWY = ……º

m ∠ XZW = ……º m ∠ XNY = ……º

W D

3) In the opposite figure:

ABCD is a quadrilateral BC = CM, m ∠ CMD = 130° and m ∠ DAC = 50°

Prove that ABCD is a cyclic quadrilateral …………………………………………………………… …………………………………………….........…………

………………………………..………………………………………………………... ……………………….………………………………………………………………… …………………………………………………………………………………………. ………………………………………………………………………………………… ………………………………………………….........………………………………

4) ABC is a triangle in which AB = AC and m ∠ A = 70°. BX bisects ∠ B and intersect AC at X and BY bisects ∠ C and intersect AB at Y Prove that AXDY is a cyclic quadrilateral ………………………………………………………………………………………… …………………………………………….........…………………………………..… ………………………………..………………………………………………………... ……………………….………………………………………………………………… ………………………………………………………………………………………….

5) In the given figure: ABCD is a trapezium in which AD // BC , AC ∩ BD = { F }. If m ∠ DBC = 40° and m ∠ AFB = 80°. Prove that ABCD is a cyclic quadrilateral ………………………………………………………………………………………… …………………………………………….........…………………………………..… ………………………………..………………………………………………………... ……………………….………………………………………………………………… …………………………………………………………………………………………. ………………………………..………………………………………………………...

6) In the opposite figure:

AYZ is a triangle where m ∠ YLZ = 100°, m ∠ YZM = 50° and m ∠ MYZ = 30° Prove that:

a) LYZM is a cyclic quadrilateral 1 b) m ∠ ALM = m ∠ YLZ 2

…………………………………………….........…………………………………….. ………………………………..………………………………………………………... ……………………….………………………………………………………………… …………………………………………………………………………………………. ………………………………………………………………………………………… ………………………………………………….........……………………………… Parents Responsibilities: ( some items ) 1- To actively discuss the code of conduct with them. 2- To support the school when dealing with poor behavior. 3- To fully support positive behavior in their children through praise and encouragement. 4- To inform the school of any issues or concerns which may affect behavior in school. 5- To treat all staff with professional courtesy. Student Handbook

Properties of Cyclic Quadrilaterals

Theorem (3) ABCD is a cyclic quadrilateral, then 1) m ∠ A + m ∠ C = 180° 2) m ∠ B + m ∠ D = 180°

The proof of Theorem (3): ABCD is a cyclic quadrilateral m∠A=

1 m …… 2

−−−−−−−− ( 1 )

m∠C=

1 m …… 2

−−−−−−−− ( 2 )

By adding ( 1 ) and ( 2 ): ∴ m ∠ A + m ∠ C = …… + …… ∴m∠A+m∠C=

1 ( …… + …… ) 2

∴m∠A+m∠C=

1 × …… = …… 2

Similarly m ∠B + m ∠ D = 180°

Corollary ( 1 ) ABCD is a cyclic quadrilateral If E ∈ CD , then m ∠ EDA = m ∠ B

Possible consequences would be: 1- Detentions: Less serious offenses would require a student to give up a lunch break to do a writing assignment. More serious offenses would require the student to stay after school for a writing assignment. Such detentions take place on Wednesdays after school and it would be the parent's responsibility to arrange for transportation home. Student Handbook 83

Exercises 1) In each of the following figures ABCD is a cyclic quad., complete: B A

B A

70°

C C

95°

70°

110°

D E

m ∠ B = ……°

m ∠ BCA = ……°

m ∠ B = ……°

m ∠ C = ……° A A

D 5x

B 4x

M 60°

80° B

m ∠ A = ……°

x = ……°

m ∠ A = ……°

m ∠ D = ……°

m ∠ C = ……°

2) In the opposite figure:

ABCD is a cyclic quadrilateral a circle M m ∠ A = 3 x°, m ∠ M = 4 x°and m ∠ C = y ° Find the numerical value of x and y

4x M y C

………………………………………………………………………………………… …………………………………………….........…………………………………..… ………………………………..………………………………………………………... ……………………….………………………………………………………………… …………………………………………………………………………………………. ………………………………..………………………………………………………...

The converse of the theorem ( 3 ) ABCD is a quadrilateral If m ∠ A + m ∠ C = 180°, then ABCD is cyclic

Converse of corollary ( 1 ) ABCD is a quadrilateral where E ∈ CD

If m ∠ EDA = m ∠ B, then ABCD is cyclic

Exercises

A 55°

1) In the opposite figure: ABCD is a quadrilateral where AB = AD AE // DB , m ∠ C = 110° and m ∠ BAE = 55° Prove that ABCD is cyclic quadrilateral

D 110° C

2) ABC is a triangle. A circle with diameter BC intersects AB at D and intersects AC at E. BE ∩ CD = { F }. Prove that: a) ADFE is a cyclic quadrilateral

b) m ∠ DAF = m ∠ BCD

3) ABCD is a quadrilateral where AB // CD , DB ∩ AC = { M }, MD = MC and m ∠ CAB = 30°. i) Find m ∠ CDB ii) Prove that ABCD is a cyclic quadrilateral ………………………………………………………………………………………… …………………………………………….........…………………………………..… ………………………………..………………………………………………………... ……………………….………………………………………………………………… …………………………………………………………………………………………. ………………………………..………………………………………………………... ……………………….………………………………………………………………… ………………………………………………………………………………………….

4) In the opposite figure:

A circle of centre M.

A X

X is the midpoint of BC and Y is the midpoint of CD Prove that i) MXCY is a cyclic quadrilateral

M C Y

ii) m ∠ XMY = m ∠ BAD

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5) In the opposite figure: A

Circle M ∩ circle N = { A , B }, AB ∩ NM = { Z } X and Y are points on circle M, XY ∩ BA = { C } D ∈ XY where DX = YD

X D

Y M

Prove that CZMD is cyclic quadrilateral ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. D

6) In the given figure:

A E

ABCD is a parallelogram, E ∈ AB A circle passes through the points B, C, D and E C

Prove that ADE is an isosceles triangle ……………………….……………………………………………………………… ………………………………………………………………………………………… ….……………………….…………………………………………………………… ………………………………………………………………………………………… …….……………………….………………………………………………………… ………………………………………………………………………………………… ……….……………………….……………………………………………………… ………………………………………………………………………………………… ……….……………………….……………………………………………………… …………………………………………………………………………………………

The Relation between the Tangents of a Circle

Remember: In the opposite figure: Straight line L touches circle M at point X Straight line L is called tangent to the circle Point X is called point of tangency

Remember: In the opposite figure: Straight line L is a tangent to circle M Then MX ⊥ L

Remember: In the opposite figure: Straight lines L1 and L2 are tangents to circle M Then L1 // L2

Definition: In the opposite figure: L1 and L2 touch the circle at B and C L1 ∩ L 2 = { A } AC and AB are called tangent-segments

Theorem ( 4 ):: In the opposite figure: L1 and L2 touch the circle at B and C L1 ∩ L 2 = { A } AB = AC 88

Corollary ( 1 ): In the opposite figure:

L1 and L2 touch the circle at B and C

L1 ∩ L 2 = { A }

AM bisects ∠ A and ∠ M

Note: ABMC is a cyclic quadrilateral

Corollary ( 2 ): X

In the opposite figure: WX is a tangent segment of circle N at X

WY is a tangent segment of circle N at Y WN is an axis of symmetry of XY WN is an axis of symmetry of WXNY

Y A X

Explanation: B

In the opposite figure : A circle is called the inscribed circle of the triangle ABC

Y Z

Because it touches all of its sides internally at X, Y and Z. C

Rule: The center of the inscribed circle for any triangle the intersection point of the bisectors of its interior angles

Exercises 1) AB , AC are tangent-segment to circle M A

Complete: C

50°

135°

60° C

m ∠ ACB = ……°

m ∠ A = ……°

m ∠ BAC = ……° A

A B 40°

40° M

50°

M D

m ∠ MAB = ……°

m ∠ A = ……°

m ∠ BAC = ……°

C 20° C

50°

55° C

M 130°

A A

B E

70°

B B

m ∠ BAD = ……°

m ∠ BAC = ……°

m ∠ A = ……°

2) In the opposite figure: AB , AC are 2 tangent-segments to a circle M from point A

Complete: A

a) AB = ……

b) m ∠ ABC = …… c) MA is ………………… of BC and quadrilateral ………… 90

3) In the opposite figure: AB , AC are tangents to the circle M at B and C m ∠ A = 50° Complete:

a) The quadrilateral ABMC is …… b) m ∠ BMC = ……°

c) m ∠ CDB = ……° B

d) m ∠ MCA = ……° e) m ∠ ACB = ……°

4) In the given figure:

XY , XZ are 2 tangents and m ∠ ZYM = 20°

20°

Complete:

i) XY = …… ii) m ∠ X = ……°

5) In the opposite figure: M, N are 2 external tangential circles at D AD is a common tangents

A 40° 30°

AB , AC are 2 tangent-segments m ∠ BAD = 30°, m ∠ CAD = 40° C

Complete: a) AD = …… = ……

B M

b) MN ⊥ …… c) m ∠ ACD = ……°, m ∠ CDM = ……° d) The figure ABND is …………………… e) The measure of each angle of ▲ AMN are …… , …… , ……

6) In the opposite figure AB , AC are 2 tangent-segments and m ∠ AMB = 50°

m ∠ BAC = ……° B

7) In the opposite figure: AB and AC are 2 tangents from A to a circle M. BD // AC and m ∠ BAC = 40° Find m ∠ CMD

D B

……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….…………………………………………………………………

8) In the opposite figure: The inscribed circle of ▲ ABC touches AB , BC and CA at X, Y and Z AX = 3 cm, XB = 5 cm, BC = 9 cm Find the perimeter of ▲ ABC ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….…………………………………………………………………

8) In the opposite figure: A

M, N are 2 circles intersecting at D and E AB is a tangent to the circle M AC is a tangent to the circle N

C D B

m ∠ BED = 60° and m ∠ CED = 40° i) Find the measures of the angles of ▲ ABC

M E

ii) Prove that ABEC is a cyclic quadrilateral ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….…………………………………………………………………

9) In the opposite figure AD , AE touch the circle at B, C

D B

BC // DE

Prove that BCED is a cyclic quadrilateral

A C

E ……………………….…………………………………………………………………

…………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… ………………………………………………………………………………………… ………………………………………………………………………………………… …………………………………………………………………………………………

Angles of Tangency

Definition: The angle of tangency is the angle whose vertex is the point of tangency and it is the union of two rays one is a tangent to the circle and the other contains a chord of the circle.

Example ( 1 ): In the opposite figure: XY is a tangent to circle M at A i) Write all angles of tangency. ii) Name the subtended arcs of the angles of tangency

Theorem ( 5 ) In the opposite figure: BD is a tangent to the circle at B ∠ ABD is an angle of tangency and ∠ C is an inscribed angle

m ∠ ABD = m ∠ C

The proof of Theorem (3): In the opposite figure: BD is a tangent to the circle at B ∠ ABD is an angle of tangency subtend …… ∴ m ∠ ABD =

1 m …… 2

∠ C is an inscribed angle subtend …… ∴m∠C=

1 m …… 2

Then m ∠ ABD = m ∠ C

Corollary: In the opposite figure: BD is a tangent to the circle M at B 1 m ∠ AMB 2

Then m ∠ ABD =

Exercises

1) In each of the following figures M is the centre of the circle and AB is a tangent, then complete: A

B D 70°

50°

M M

C D

1) m ∠ DBC = ……°

30°

110°

M C

40 °

2) m ∠ MDC = ……°

E M

3) m ∠ CAE = ……°

C E

35°

A E

4) m ∠ CDA = ……°

5) m ∠ CBA = ……°

6) m ∠ DAE = ……°

B C 50°

M 20° A

7) m ∠ D = ……°

8) m ∠ ACB = ……°

2) In the opposite figure:

BC is a diameter in circle M and DB is a tangent to it at B

m ∠ ACB = 35° such that A ∈ CD Complete:

a) m ∠ BAC = ……°

c) m ∠ ABD = ……°

b) m ∠ CBD = ……°

d) m ( AB ) = ……°

3) In the given figure: A circle M, DX is a tangent, m ∠ A = 50°

Complete:

a) m ∠ DMB = ……°

b) m ∠ DCB = ……° B

c) m ∠ XDB = ……°

4) In the opposite figure: AE is a diameter in circle M, AC , BC are two tangents

m ∠ E = 50° A

Choose the correct answer:

a) m ∠ ABE = ……

[ 90° , 50° , 100° ]

b) m ( AB ) = ……

[ 50° , 130° , 100° ]

c) m ∠ ACB = ……

[ 130° , 100° , 80° ]

5) In the opposite figure:

ABCD is an inscribed quadrilateral in a circle, BD is a diameter B XY is a tangent to the circle at A X

m ∠ ADB = 20° and m ∠ CBD = 60°

Complete: a) m ∠ CDB = ……°

b) m ∠ BAC = ……° 96

c) m ∠ DAY = ……°

6) In the opposite figure: ABCD is a cyclic quadrilateral where AB // DC EA and EB are two tangents from E to touch the circle at A and B, m ∠ ADC = 100°, m ∠ ACD = 50° Find

a) m ∠ ABC

b) m ∠ CBF

c) m ∠ AEB

7) In the opposite figure: An inscribed circle of triangle ABC Z

It touches all of its sides internally at X, Y and Z m ∠ XZY = 64° and m ∠ XYZ = 40°

64°

40°

Find the measure of the angels of ▲ ABC

B Y

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8) In the given figure: C

EA and ED are two tangent-segments of a circle m ∠ ABC = 110°, m ∠ ACD = 70° E

i) Find m ∠ E

ii) Prove that DA bisects ∠ EDC

……………………….……………………………………………………………… ………………………………………………………………………………………… ….……………………….…………………………………………………………… ………………………………………………………………………………………… …….……………………….………………………………………………………… ………………………………………………………………………………………… ……….……………………….……………………………………………………… …………………………………………………………………………………………

9) In the opposite figure: AB and AD are 2 tangents to circle M from point A to touch the circle at B and C

C F M

m ∠ A = 70° and m ∠ CFD = 125°

i) Prove that points A, B, M, and C belong to the same circle.

ii) Find m ∠ BMC, m ∠ BDC and m ∠ ABD ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………

10) In the opposite figure:

D 120°

ABCD is a cyclic quadrilateral C BC is a diameter, EA is a tangent and m ∠ ADC = 120°

Complete: a) m ∠ BAC = ……°

d) m ∠ ACB = ……°

b) m ∠ ABC = ……°

e) m ∠ AEM = ……°

c) m ∠ BAE = ……°

11) In the given figure: AB , AC are two chords in a circle where m ∠ BAC = 60° B

DB , DC are two tangents Complete:

a) m ∠ DBC = ……° b) m ∠ DCB = ……°

c) m ∠ BDC = ……° d) The kind of ▲ DBC according to its sides is ……

12) In the given figure: BX is a tangent to the circle M at B

M C

AB // MC, m ∠ ABM = 50°

Find m ∠ CBX

13) In the opposite figure:

AB is a diameter in circle M, CF is a tangent at C D ∈ AB where DE ⊥ BC = { E } and DE ∩ CF = { F } Prove that:

F E

a) ADEC is a cyclic quadrilateral b) FE = FC

Possible consequences would be: 2- Suspensions: There are two types of suspensions. In-School Suspension: the student will attend school but not classes. He/she would spend the day in the administrative office working on assignments given by his / her teachers. Out-of-School Suspension: the student would not be allowed to attend school and assignments would have to be completed at home before returning to classes. Student Handbook

100

Converse of Theorem ( 5 ): If m ∠ C = m ∠ ABD, Then BD is a tangent to the circle that passes through the points A, B and C

Exercises

1) In which of the following figures: BD is a tangent to circle passes through the vertices of triangle ABC

D B

120°

B B

75° 60°

60°

30°

65° C

5 x°

3 x°

2) In the opposite figure:

E 55°

ABCD is a quadrilateral, AB = AD, AE // DB m ∠ C = 110° and m ∠ BAE = 55°

Prove that: AE is a tangent to the circle passing through A, B, C and D

110° C

……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… ………………………………………………………………………………………… 101

3) ABCD is a rhombus in which m ∠ A = 60°. Prove that AB is a tangent to the circle passing through the vertices of ▲ BCD ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….…………………………………………………………………

4) A is a point outside M where AB , AC are 2 tangent-segments to the circle at B, C. CD is a chord of the circle where CD = CB Prove that CD is a tangent-segment to the circle passing through the vertices of

▲ ABC ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… …………………………………………………………………………………………. ……………………….………………………………………………………………… ……………………….………………………………………………………………… …………………………………………………………………………………………. Expulsions: Students who are determined to be a danger or an offense to themselves, other students or staff members will be dismissed from the program. Student Handbook

102

REVISION SHEET

1) Choose the correct answer: 1) The length of the arc which represents half of the circumference of the (2πr, πr, πr,πr)

circle = ……………

2) In the opposite figure: ) = …………… m ( AD

(60° , 50° , 30° , 100°)

80°

3) In the opposite figure:

If AB is a diameter in circle M,

50 °

m (∠ ABD) = 25° then First: m (∠ DAB) = …………… Second: m (∠ DCB) = ………

(25° , 50° , 65° , 90°)

25 • ° M

(50° , 100° , 115° , 125°)

4) We can draw circle pass through vertices of …………… (trapezium, rhombus, parallelogram, rectangle) B

5) In the opposite figure: m (∠ ABC) = 35°

M •

then m (∠ AMC) = ……………

(100° , 60° , 35° , 70°)

6) If the measure of an arc in a circle = 60° then its length = …… the circumference of the circle.

(

1 1 1 1 , , , ) 2 3 4 6 B

7) In the opposite figure:

D A

) = 110° , m ( DE ) = 40° If m ( BC E

then m (∠ A) = …………… C

8) The inscribed angle which is subtended by an arc of measure greater than the measure of a semicircle is …………… angle. (straight , acute , right , obtuse)

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103

9) In the opposite figure: AB and CD are two parallel chords,

) = 30° m ( AC

then m (∠ BED) = ……………

(10° , 15° , 30°, 60°) A

10)

In the opposite figure: M

A circle of centre M If m (∠ MBC) = 40°

then m(∠ BAC) = …………… 11)

(40° , 50° , 60° , 100°) A

In the opposite figure:

AB is a diameter in circle M M

) = m ( BD ) m (∠ BMC) = 40° , m ( BC then m (∠ BAD) = …………… 12)

(20° , 40° , 50° , 80°)

1 An arc in a circle, its length = π r then it is opposite to a central angle 3 (30° , 60° , 120° , 240°)

of measure…………… 13)

The measure of the angle of tangency is ……………… the

measure of the central angle subtended by the same arc. ( equal to , double , half , quarter ) 14)

In the oppsoite figure:

A D

If AD is a tangent to the circle at A,

110°

m (∠ DAC) = 110 ° then m (∠ ACB) = ……………°

( 35° , 55° , 60° , 70° ) 15)

The number of common tangents for two distant circles is

…………… 16)

( 4 , 3 , 2 , infinite )

The measure of the exterior angle at a vertex of a cyclic

quadrilateral is ………… to the measure of the interior angle at the (=,>,≠,<)

opposite vertex.

104

17)

The quadrilateral in which the measure of two opposite angles are

100° and 80° is ……… ( square , rectangle , rhombus, cyclic quad ) 18)

The figure which the circle doesn’t passing through its vertices is ( square , rectangle , rhombus , triangle )

19)

In the cyclic quad, each two opposite angles are ……………… ( complementary , supplementary , equals , corresponding )

20)

The two tangent segments drawn to a circle from a point outside it

……… ( equal in length , different in length , parallel , perpendicular ) 21)

If the measure of a tangency angle = 40°, then measure the

inscribed angle subtended by the same arc equals ………… ( 40° , 20° , 90° , 80° ) 22)

The ratio between the measure of the central angle and the

measure of the inscribed angle subtended by the same arc =………… (3:1, 2:1, 1:2, 1:3) 2) Complete: 1) The longest chord is ……….. 2) Any straight line passing through the center is ………. 3) The straight line passing through the center and bisect any chord ………. 4) The perpendicular bisector of any chord ……….. 5) The straight line which perpendicular to a diameter of a circle at one of its end points is ………. 6) The line of centers of two intersecting circles is ………. 7) If the radius length of the circle is 8cm , the straight line L is distant from its center by 4cm. then L is ……. 8) If the surface of circle M ∩ surface of circle N = { A } then the two circles are ……… 9) M and N are two intersecting circles the two radii length are 3cm and 4 cm then MN ∈ …………. 105

10)

The area of a circle M = 16 π cm², A is a point where MA = 8 cm then A

is ……….. circle M 11)

The circle M of radius 6cm if the straight line L out side the eircle then

the distance of the center of the circle from the straight line L ∈ ……… 12)

a circle of diameter length ( 2 x + 5 ) cm, then straight line L is a distant

from is center by ( x + 2 ) cm, then the straight line is …….. 13)

We can draw ……….. number of circles passing through a given point.

14)

The number of circles that pass through two given points is ……….

15)

The number of circles passing through three non. Collinear points is

……… 16)

The circle passing through a vertices of a triangle is called………

17)

The center of the circle passing a vertices of a triangle is the point of

intersection of …………. 18)

ABC is aright triangle at B. then the center of the circle passing through

∆ ABC is ………

19)

We can't draw a circle passing through the vertices of ………. or

…..……. or …………… 20)

The number of circles passing through two points of radius r cm is …….

21)

The radius of the smallest circle passing through AB equals ………

22)

If two chords are equal is length then they are …………..

23)

If two chords equidistant from the center then ……….

24)

The Line of centers of two touching externally circles is ……..

25)

If M and N are two intersecting circles then the axis of symmetry of the

two circles is ……… 26)

If AB = 8 cm . then the area of the smallest circle passing through AB

equals…….. 27)

If m is a circle with circumference 8 π cm, A is a point on the circle then

MA = …….. 28)

A chord with 8 cm length the length of its radius is 5 cm , then it is

distant from its center by …………… cm . 106

29)

The straight line perpendicular to a diameter in a circle at one of its end

points is ………. 30)

If M and N are two intersecting circles at x and y then the axis of

symmetry of the two circles is ……. 31)

If M and N are two touching externally circle of radii 6 cm and 7cm then

MN = ……… 32)

If M and N are two intersecting circles of radii 8 cm and 6 cm then MN ∈ ………..

33)

If M and N are two distant circles of radii 7 cm and 4 cm then MN ∈ ………..

34)

The two tangents drawn from endpoints of any diameter are …….

35)

If M and N are two circles of radii 8 cm and 3cm if MN = 5 cm then the

two circles are ……… 36)

If surface of circle M ∩ surface of circle N = ϕ then the two circles are

…………. 37)

If surface of circle M ∩ surface of circle N = surface of circle N then the A

two circles are ……………… 38)

In the opposite figure: \\

If m (∠ ABD) = 70° AB = AD then m(∠ C) = ………° 39)

70 °

In the same circle (or in congruent circles), if the measures of arcs are

equal, then their chords are …………… 40)

A //

In the opposite figure:

AB is a diameter of the circle C

) = 50° ) = m ( CB m ( AD

\\ B

) = …………… ° then m ( DC 41)

In the opposite figure:

AB is a diameter in circle M m (∠ A) = 65° then m (∠ B) = ……………° 107

65°

• M

42)

E /

) ) = m ( CD If m ( AB

• 40°

m (∠ E) = 40°

C B

) = ……………° then m ( CD 43)

In the opposite figure:

If m (∠ E) = 40° B

m (∠ C) = 25° E

then m (∠ ADC) = ……………°

40 °

? D

25 °

44)

The measure of inscribed angle drawn in a semi circle equals …………

45)

The center of the inscribed circle of any triangle is the point of

intersection of ……………… 46)

In the opposite figure:

ABCD is a cyclic quadrilateral AB = BD = DA

m (∠ C) = …………° 47)

In the cyclic quadrilateral each two angles drawn on one base and on

one side of it ………… 48)

Cyclic quadrilateral is a quadrilateral figure whose four vertices belong

to one ………… 49)

ABCD is a cyclic quadrilateral where m ∠ D = 95° and m ∠ C = 2 x – 5

then x = …………° 50) In the opposite figure: AD is a tangent to the circle which pass

A D

through the vertices of ∆ ABC B

then m (∠ DAB) = m (∠ ………)

3) In the opposite figure: A circle of two parallel chords AB // CD

Prove that: m (∠ AEC) = m (∠ BED) D

108

4) In the opposite figure: A circle of centre M. MC ∩ AB = { C }, MC intersects the circle at D. m (∠ MAB) = 20°

E M B

) Find: 1) m ( AD

2) m (∠ DEB)

5) Prove that: the inscribed angles subtended by the same arc in the same circle are equal in measure.

6) In the opposite figure:

AB is chord in the circle M

• E

AB // CM, BC ∩ AM = { E } Prove that: BE > AE

B A

7) In the opposite figure:

D \

ABC is equilateral triangle in which AD = DE , E ∈ DC D ∈ AB

E C

prove that: ADE is equilateral triangle A

8) In the opposite figure: 30 ° M

M is the centre of the circle, m (∠ A) = 30°, BC = 7 cm Find the radius length of the circle M

7 cm

9) In the opposite figure:

109

AE = CE Prove that: m (∠ ACB) = m (∠ CAD)

AB and CD are two intersected chords at E

E • M B

10) In the opposite figure: m (∠ B) = 80°, AD // BE CF bisects ∠ DCE, m (∠ DCF) = 50° prove that ABCD is cyclic quadrilateral

50° × × C E

11) In the opposite figure: AB and AC are two tangent-segments to the circle at B and C m (∠ A) = 50°, m ( CDE) =115° Prove that: 1) BC bisects ∠ ABE

B E 50 °

•

2) CB = CE

115 D °

C D

12) In the opposite figure:

// A

∆ ACD is equilateral triangle and m (∠ ABC) = 120° Prove that ABCD is cyclic quadrilateral

\\ 120°

C C

13) In the opposite figure:

X /

AB is a diameter in a circle M

• M

X is the midpoint of AC BY is a tangent to the circle at B where XM cuts BY at Y

Prove that AXBY is cyclic quadrilateral Y

14) In the opposite figure:

D A

AB and CD are two chords in a circle M

M •

AB ∩ CD = { E } where EB = ED Prove that ∆ ECA is isosceles

15) In the opposite figure:

ABCD is quadrilateral inscribed in a circle, AB // CD Prove that AC = BD 110