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UNIT 3UNIT 3UNIT 3UNIT 3: : : : COORDINATE GEOMETRYCOORDINATE GEOMETRYCOORDINATE GEOMETRYCOORDINATE GEOMETRY

Unit 3.1Unit 3.1Unit 3.1Unit 3.1: : : : Formulae for Gradient, MidFormulae for Gradient, MidFormulae for Gradient, MidFormulae for Gradient, Mid----point & Distancepoint & Distancepoint & Distancepoint & Distance

(A) Gradient of a Straight Line

Examples:

1. Find the gradient of the line passing through A (-3, 2) and B (2, 3).

2. Find the gradient of the lines shown below:

(a)

(b)

3. Find the gradient of the line joining the points M (-3, 1) and

N (4, 7).

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(B) Midpoint of a Straight Line

Examples:

1. The diagram shows a line segment AB. Find the midpoint of line

AB.

2. Find the coordinates of the midpoint of the line segment joining

the following pair of points:

(a) A (3, 4) and B (5, 6) (b) P (-1, 2) and Q (3, - 4)

3. M (-1, 2) is the midpoint of the line segment joining the points A

and B. If the coordinates of A are (2, 3), find the coordinates of B.

4. The point M (s, 2) is the midpoint of the interval from (-3, -4) and

(5, t). Find the values of s and t.

5. A triangle has vertices A (1, 4), B (6, 0) and C (12, 4). Calculate

(a) the gradient of AB,

(b) the coordinates of the midpoint of BC.

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(C) Distance / Length of a Straight Line

Examples:

1. Calculate the distance between points U (2, 4) and V (-2, 1).

2. B is the point (2, 6) and C is the point (-3, 0). Calculate the length

of the line segment BC correct to 2 decimal places.

Unit 3.2Unit 3.2Unit 3.2Unit 3.2: : : : Gradient Gradient Gradient Gradient of Special Lines of Special Lines of Special Lines of Special Lines & Parallel Lines& Parallel Lines& Parallel Lines& Parallel Lines

(A) Gradient of a Horizontal Line

Gradient is always zero for a

horizontal line.

The equation of a

horizontal line is y = c

where c = y-intercept

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(B) Gradient of a Vertical Line

(C) Gradient of Parallel Lines

Gradient is always undefined for a

vertical line.

The equation of a vertical

line is x = k where k = x-

intercept

When two lines are parallel, they have

equation with the same gradient.

Equation of AB is 1y mx c= + and equation of PQ is

2y mx c= + where m is the gradient of the line.

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Examples:

1. Find the equation of a line which passes through the point (2, - 4)

and has a gradient of 0.

2. Find the equation of a line which passes through the point (3, - 1)

and has an undefined gradient.

3. Line PQ, is parallel to line AB. Given that line AB passes through

the points A (1, - 1) and B (2, 3), find the gradient of line PQ.

4. Line MN has a gradient of 5. Find the equation of another line PQ

passing through the origin and parallel to MN.

Classwork # ____ :

Gradient, Midpoint & Distance of a Straight Line Graph 1. Find the midpoint of the straight line joining the following pairs of

points:

(a) (2, 5) and (3, 7) (b) (- 2, - 1) and (- 4, - 7)

2. In each of the following, M is the midpoint of AB. Find the unknown

(a) M(3, 5), A(0, 4), B(x, y) (b) M(1, 4), A(- 2, a), B(b, 1)

3. For each of the pair of points below, find the length of the line segment

joining them. Give your answers correct to 2 decimal places.

(a) (2, 5) and (6, 10) (b) (- 2, - 3) and (4, 6)

4. Find the gradient and y-intercept of each of the following lines:

5. Line MN is parallel to the x – axis.

(a) What is the gradient of line MN?

(b) Write down the equation of line MN which passes through the

point (2, 4).

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Homework # ____ :

Gradient, Midpoint & Distance of a Straight Line Graph

1. Find the midpoint of the straight line joining the following pairs of

points:

(a) (- 3, - 4) and (4, 5) (b) (- 2, 5) and (6, - 3)

2. In each of the following, M is the midpoint of AB. Find the

unknown

(a) M(1, 3), A(- 1, 1), B(x, y) (b) M(8, 5), A(a, b), B(10, 2)

3. For each of the pair of points below, find the length of the line

segment joining them. Give your answers correct to 2 decimal

places.

(a) (0, - 1) and (5, - 2) (b) (- 3, - 2) and (- 1, 5)

Unit 3.3Unit 3.3Unit 3.3Unit 3.3: : : : Equation of a Straight LineEquation of a Straight LineEquation of a Straight LineEquation of a Straight Line

(A) Equation of a Straight Line given its Gradient and Y-intercept

Examples:

1. Write down the equation of the straight line given that

(a) the gradient is 3 and y-intercept is – 2 ,

(b) the gradient is 1

2− and y-intercept is 3.

2. Find the gradient and the y-intercept of the following straight

line equations:

(a) 2 4y x= + (b) 2

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y x= −

(c) 2 6y x+ = (d) 2 5 15x y− − =

(e) 2 3 10 0y x− − = (f) 2 3 1

5 10 5x y− =

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(B) Equation of a Straight Line given its Gradient and a Point Examples:

1. Find the equation of line passing through the following point and

gradient

(a) 1

,2

5

; gradient = – 6. (b) (4, - 2); gradient = 3

4− .

2. Given that 3y x c= + passes through the point (1, 2), find c.

(C) Equation of a Straight Line given two points. Examples:

1. Find the equation of the straight line passing through the following

points:

(a) (4, 3) and (5, 5) (b) ( ),1 2− and (0, 5) (c) 1

2, 4

and 3

2,2

−

(D) Equation of a Straight Line given one point and the equation

of a parallel line. Examples:

1. Find the equation of the line parallel to 5 2 0y x+ + = and passing

through 1

,12

.

2. Find the equation of the line parallel to 3 4 3x y= − and passing

through (6,11) .

3. Find the equation of the line parallel to 8 0y + = and passing

through ( 3, 4)− .

4. Find the equation of the line parallel to 4 3 0x − = and passing

through (6,3) .

(E) Equation of a Straight Line of a given diagram. Examples:

Write down the equations of the lines AB, AC, AD, AE, BC and BE as

shown.

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Classwork # ____ : Equation of a Straight Line 1. Find the gradient and the y-intercept for each of the following

equations:

(a) 4y x= − + (b) 2

3y x= (c) 3 9 6 0y x− + =

2. Write down the equation of the line in general form, given

(a) gradient = – 3 and y-intercept = 2

(b) gradient = 1

4 and y-intercept =

2

3−

3. Find the equation of the straight line that passes through the

point (4, -2) with gradient 3

4− .

4. If 2y x c= + passes through a point (4, 1), find the value of c.

5. The equation of a straight line is 2 4x y+ = .

(a) Find the gradient of the line.

(b) Given that the point (5, k) lies on the line, find the value of k.

6. The straight line y mx c= + is parallel to the straight line

3 2y x= + and passes through the point (1, 2). Find the values of

m and c.

7. Find the equation of the line which passes through the points

(6, 5) and (4, 3).

8. Find the equation parallel to 2 3 0x y− + = and passing through

(5, 2).

Homework # ____ : Equation of a Straight Line 1. Find the gradient and the y-intercept for each of the following:

(a) 4 6x y− = (b) 4 2 3 0x y+ − =

2. Write down the equation of the line given its gradient is -1 and its

y-intercept is 3.

3. A straight line of gradient 3 passes through the point (0, 7). Write

down the equation of the line.

4. Find the equation of the line which passes through the points

(- 2, - 8) and (2, 4).

5. The line 2 3 0y x c+ − = passes through the point (- 1, 3). Find c.

6. Find the equation parallel to 2 3y x= − + and passing through

(1, 2).

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Unit 3.4Unit 3.4Unit 3.4Unit 3.4:::: Miscellaneous Problems on Coordinate GeometryMiscellaneous Problems on Coordinate GeometryMiscellaneous Problems on Coordinate GeometryMiscellaneous Problems on Coordinate Geometry

Examples:

1. The vertices of ∆PRT are P(0, 4), R(2, 5) and

T(4, - 4). TP is a line segment that is

perpendicular to RP.

Find the area of ∆PRT.

2. Three points A, B and C form an isosceles triangle where

AB = BC. Find

(a) the length of AB,

(b) the coordinates of C,

(c) the equation of the line

joining A and C.

3. ABCD is a trapezium which is symmetrical about the y-axis.

Given that 1

2AD BC= and the height of the trapezium is 4 units,

find

(a) the coordinates of D,

(b) the equation of the diagonal

joining B and D,

(c) the area of trapezium ABCD.

4. P is the point (4, 0), Q is the point (10, 4), R is the point (2, 6) and

O is the origin. Find

(a) PQ 2,

(b) the gradient of the line QR,

(c) the equation of the line through Q parallel to OR.

5. On the graph, O is the origin and l is the line which passes

through the points P(–2, 1) and R(4, 4). T is the point (1, 4).

(a) Find

(i) the gradient of l,

(ii) the equation of l,

(iii) the equation of the line through T parallel to l.

(b) (i) Write down the coordinates of M, the midpoint of PR.

(ii) Calculate the area of ∆TMR.

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Classwork # ____ : Problems on Coordinate Geometry

1. The vertices of a triangle are I(0, – 5), J(–9, 7) and K(16, 7).

Calculate

(a) the length of IJ,

(b) the length of JK,

(c) the length of KI,

(b) the perimeter of the triangle IJK.

2. In the diagram, the line BA meets the y-axis at C. Given that A is (2,

3) and B is (6, 5), calculate

(a) the coordinates of C,

(b) the gradient of AB,

(c) the length of AB,

(d) the area of trapezium ABNM.

3. A triangle has vertices A(1, 4), B(6, 0) and C(12, 4). Calculate

(a) the gradient of AB,

(b) the coordinate of the midpoint of BC.

4. The points A and B have coordinates (– 6, 2) and (6, 6)

respectively.

(a) Find the coordinates of the midpoint of AB.

(b) Calculate the length of AB.

(c) Find the gradient of the line AB.

(d) Find the equation of the line AB.

Homework # ____ : Problems on Coordinate Geometry

1. ABCD is a parallelogram.

(a) Find the coordinates of the midpoint of AC.

(b) Using the result in (a), find the coordinates of point D.

2. The line 2 6y x+ = cuts the x-axis at A and y-axis at B. The point

P is the midpoint of AB and O is the origin.

Find

(a) the coordinates of the point A,

(b) the coordinates of the point B,

(c) the length of AB,

(d) the equation of the line through P which is parallel to the

x-axis.