MATHEMATICS Learner’s Study and Revision Guide for Grade 12 STRAIGHT LINE
Revision Notes, Exercises and Solution Hints by
Roseinnes Phahle Examination Questions by the Department of Basic Education
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Contents Unit 13 All you need to know: Revision notes
3
Exercise 13
4
Answers
5
Examination questions with solution hints and answers
7
More questions from past examination papers
12
Answers
19
How to use this revision and study guide 1. Study the revision notes given at the beginning. The notes are interactive in that in some parts you are required to make a response based on your prior learning of the topic from your teacher in class or from a textbook. Furthermore, the notes cover all the Mathematics from Grade 10 to Grade 12. 2. “Warm‐up” exercises follow the notes. Some exercises carry solution HINTS in the answer section. Do not read the answer or hints until you have tried to work out a question and are having difficulty. 3. The notes and exercises are followed by questions from past examination papers. 4. The examination questions are followed by blank spaces or boxes inside a table. Do the working out of the question inside these spaces or boxes. 5. Alongside the blank boxes are HINTS in case you have difficulty solving a part of the question. Do not read the hints until you have tried to work out the question and are having difficulty. 6. What follows next are more questions taken from past examination papers. 7. Answers to the extra past examination questions appear at the end. Some answers carry HINTS and notes to enrich your knowledge. 8. Finally, don’t be a loner. Work through this guide in a team with your classmates.
2
Analytical geometry of the straight line
REVISION UNIT 13: ANALYTICAL GEOMETRY All you need to know: What follows below is all you need to know in order to answer the question on the analytical geometry of the straight. Consider two points A ( x1 , y1 ) and B ( x 2 , y 2 ) . 1. Distance between A and B is [( x 2 − x1 ) + ( y 2 − y1 ) ] 2
2
⎛ x1 + x2 y1 + y 2 ⎞ ; ⎟ 2 ⎠ ⎝ 2
2. Midpoint of the line joining A and B has coordinates ⎜ 3. The gradient of the line joining A and B is m =
y 2 − y1 x 2 − x1
4. If two lines are parallel, they have equal gradients. 5. If two lines are perpendicular, the product of their gradients is ‐1. 6. All straight line equations can be written in the form y = mx + c where m is the gradient and
(0; c ) is the point at which the line cuts the y ‐ axis or the y ‐intercept. 7. The gradient of a line can also be measured by the tangent of the angle θ between the positive direction of the x ‐axis and the line, measured anti‐clockwise. That is:
m = tanθ 8. The equation of the line joining the points A and B is given by:
y − y1 = m(x − x2 )
or
y − y1 =
y 2 − y1 .( x − x1 ) x 2 − x1
or
y − y1 y 2 − y1 = x − x1 x 2 − x1
9. Intersection of lines: In order to find the point in which two lines intersect we have to find a point with coordinates which satisfy both equations. We find this point by solving the equations of the lines simultaneously. 10. Intersection of a straight line and a curve: A straight line may intersect a curve at more than one point. Thus solving the equations of the line and the curve simultaneously could give more than one answer, these being the points in which the line and curve intersect. 11. Collinearity: Points A, B and C are collinear if they are joined by lines of equal slopes.
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EXERCISE 13 13.1.
A triangle has vertices at A(0;8), B(1;1) and C(5;3). Depict this triangle in a sketch. Show that the triangle is isosceles and find: 13.1.1 the equation of the straight line through A and C, 13.1.2 the coordinates of the foot of the perpendicular from B to AC, 13.1.3 the length of the perpendicular from B to AC.
13.2 ` Find the coordinates of the vertices of the triangle whose sides are given by the equations 2 y − x = 7 , 2 y = −5 x + 19 and 2 y + 3 x = 3 . Illustrate this question with a diagram. 13.3 Determine 13.3.1 the equation of the straight line passing through the point (8;‐2) and which is at right angles to the line y = 2 x + 7 , 13.3.2 the coordinates of the point where this line intersects the line y = 2 x + 7 , 13.3.3 the distance of the point (8;‐2) from the line y = 2 x + 7 . 13.4.
Determine 13.4.1 the equation of the perpendicular from (7;3) to the line 6 x − y = 2 , 13.4.2 the length of the perpendicular from (7;3) to the line 6 x − y = 2 .
13.5
The line y = 2 x + 4 cuts the curve y = x 2 − 3 x at the points A and B.
13.5.1 What are the coordinates of A and B?
13.5.2 What are the coordinates of the midpoint of AB?
13.5.3 What is the equation of the line perpendicular to AB and passing through the midpoint of AB?
13.5.4 Illustrate your answer with a sketch.
13.6
The curve y = x 2 + 2 x is met by the line y = 3 x + 6 at two points A and B. 13.6.1 Calculate the coordinates of A and B. 13.6.2 Find the equation of the line joining A and B. 13.6.3 Find the length of AB. 4
Analytical geometry of the straight line
ANSWERS EXERCISE 13 8
y A
(0,8)
7
13.1.1 y=-x+8 6
13.1.2 HINT: First find the equation of the perpendicular. Then 5 solve its equation simultaneously with the equation of AC to find the coordinates of its foot on AC. 4 Answer: (4;4) 13.1.3 3sq root 3 3
(4,4)
C (5,3)
2
B(1,1)
1
x -13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
-1 -2 -3 -4 -5 -6 -7 -8
13.2 The coordinates are (‐1;3), (2;4,5) and (8;‐9,5). Illustration: y
f(x)=0.5(x+7)
9 8
f(x)=0.5(-5x+19) f(x)=0.5(-3x+3)
2y=-5x+19 2y-x=7
7 6
2y+3=3
5 4 3 2 1
x -4
-3
-2
-1
1 -1 -2 -3
Coordinates of the vertices are: (-1;3), (2;-4,5) and (8;-9,5)
-4 -5 -6 -7 -8 -9 -10 -11
2
3
4
5
6
7
8
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13.3.1 2 y = − x + 4 13.4.1 6 y = − x + 25 13.3.2 (‐2;3) 13.4.2 37 13.3.3 5 5 13.5.1 Coordinates are A(‐0,7;2,6) and B(5,7;15,4) 13.5.2 Coordinates of midpoint of AB are (2,5; 9) 13.5.3 Equation of line perpendicular to AB and passing through the midpoint of AB is 2 y = − x + 23 13.5.4 Sketch: B
y 15 14
Line perpenicular to AB and passing through midpoint of Ab
13 12 11 10 9 8 7 6 5 4
A
3 2 1
x -13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
-1 -2 -3
13.6.1 The coordinates are A(‐2;0) and B(3,15) 13.6.2 y = 3( x + 2) 13.6.3 Length of AB = 5 10
6
Analytical geometry of the straight line
PAPER 2 QUESTION 1
DoE/ADDITIONAL EXEMPLAR 2008
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PAPER 2 QUESTION 1
Number Hints and answers 1.1 Use the distance formula. 1.2
1.3
1.4
1.6
3 2
There are two formulae on the formula sheet. Choose either one. Recall that the product of the slopes of lines that are perpendicular is ‐1. Answer: y = −
1.5
Work out the solutions in the boxes below
Answer: AC= 208 Use the midpoint formula. Answer: M(‐1; 3) Use the gradient formula. Answer: m AC =
2 3
Area of a triangle is
1 base x height. 2
The diagram shows the height and which side is base. Find their lengths and substitute into the formula for area. Answer: Area of ΔABC =52 sq units Find a way of using the fact that the tangent of the angle which a line makes with the positive direction of the x ‐axis is equal to the slope of the line. Or, you could look at ΔABN . Answer: θ ≈ 33,7 o
8
DoE/ADDITIONAL EXEMPLAR 2008
Analytical geometry of the straight line
PAPER 2 QUESTION 1
DoE/NOVEMBER 2008
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PAPER 2 QUESTION 1
Number Hints and answers 1.1 Use the midpoint formula.
⎛ 1 1⎞ ;− ⎟ ⎝ 2 2⎠
1.3
Find the midpoint of BD and see if it is the same as the midpoint of AC you found in 1.1. Or, show that AC and BD bisect each other. Answer: Write down your conclusion. Use the product of gradients of perpendicular lines is equal to ‐1. That is, find the gradients of AD and AC and see if their product is ‐1. There are other ways of proving that
ˆ C = 90o . AD Can you find these other ways? Answer: Write down your conclusion. 10
DoE/NOVEMBER 2008
Work out the solutions in the boxes below
Answer: M⎜ − 1.2
Analytical geometry of the straight line
Number Hints and answers 1.4 There is more than one way to answer this question. One way: Using your knowledge of the properties of the square: 1. Show using the distance formula that the diagonals are equal in length; 2. Use what you proved in 1.2 that the diagonals bisect each other; 3. Use 1.3 that they are right angles. Can you find other ways of showing that ABCD is a square? Answer: Write down your conclusion. 1.5 Use tan θ = m to find the size of θ . This means that you must m the gradient of DC. Answer: θ = 123,7 o 1.6 Is the length of OC equal to radius 2? Find out. Answer: Write down your conclusion.
Work out the solutions in the boxes below
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MORE QUESTIONS FROM PAST EXAMINATION PAPERS Exemplar 2008
12
Analytical geometry of the straight line
Preparatory Examination 2008
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Feb – March 2009
14
Analytical geometry of the straight line
November 2009 (Unused)
Preparation for the Mathematics examination brought to you by Kagiso Trust
November 2009 (1)
16
Analytical geometry of the straight line
Feb – March 2010
Preparation for the Mathematics examination brought to you by Kagiso Trust
Feb – March 2010
18
Analytical geometry of the straight line
ANSWERS Exemplar 2008 1.1 AC = 2 10 1.2 M(‐1; 4) 1.3 Proof required. Provide a proof and check with the teacher if it is correct. 1.4 Area Δ ABC = 20 1.5 y = − x − 1 o 1.6 θ = 135
ˆ C = 53,1o 1.7 AB Preparatory Examination 2008 1.1.1 1.1.2 1.1.3
BC = 2 26 M(1; ‐1) mBC = 5
1.1.4
θ = 78,69o
1.2
1 4 y =− x− 5 5
1.3 1.4
A(11; ‐3) C’(0; ‐12)
1.5
Area ΔA' B' C' 4 = Area ΔABC 1
Feb/March 2009 1.1 m BC =
1 3
1 17 1.2 y = x + 3 3 1.3 t = 8 1.4 AB = 2 10 1.5 Proof required. Provide a proof and check with the teacher if it is correct. 1.6 Area of ABCD = 30 sq units
θ = 18,43o
November 2009 (Unused paper) 1.1 mAC = 1 1.2 y = x − 4 1.3 Proof required. Provide a proof and check with the teacher if it is correct. 1.4 Proof required. Provide a proof and check with the teacher if it is correct. 1.5 Area Δ ABC = 36 sq units November 2009(1) 4.1 mAB = 1 or
3 1- t
4.2 t = ‐2 4.3 Midpoint of BC = (0; ‐20 4.5 y = x − 6 Feb/March 2010 4.1 Proof required. Provide a proof and check with the teacher if it is correct. 4.2 A(2; 1) 4.3 y = 2 x − 3 4.4 BQ = 5 4.5 Proof required. Provide a proof and check with the teacher if it is correct. 4.6 R(4; 5) 5.1 mCD = 4 5.2 y = 4 x − 16