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The Straight Line

Maths4Scotland Higher

The Straight Line

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Maths4Scotland Higher

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

and is perpendicular to the line with equation 4 1 0x y

Find gradient of given line: 4 1 0 4 1 4x y y x m

Find gradient of perpendicular:1

4m

Find equation:1 3

1 4( 3) 1 4 124 ( 1)

yx y x y

x

4 13 0y x

Maths4Scotland Higher

Hint

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Find the equation of the straight line which is parallel to the line with

equation and which passes through the point (2, –1).

Find gradient of given line:

Gradient of parallel line is same:2

3m

Find equation: 2 ( 1)2 4 3 3

3 2

yx y

x

3 2 1y x

2 3 5x y

2 2

3 33 2 5 5y x y x m

Maths4Scotland Higher

Hint

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Find gradient of the line:

1tan

3

Use table of exact values1 1

tan 303

2 ( 1) 3 1

3 3 0 3 3 3m

Use tanm

Find the size of the angle a° that the line joining the

points A(0, -1) and B(33, 2) makes with the

positive direction of the x-axis.

Maths4Scotland Higher

Hint

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A and B are the points (–3, –1) and (5, 5).Find the equation ofa) the line AB.

b) the perpendicular bisector of AB

Find gradient of the AB: 4 3 5y x

Find mid-point of AB 1, 3

3

4m Find equation of AB

Gradient of AB (perp):4

3m

Use gradient and mid-point to obtain perpendicular bisector AB

3 4 13y x

Maths4Scotland Higher

Hint

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The line AB makes an angle of radians with

the

y-axis, as shown in the diagram.

Find the exact value of the gradient of AB.

Find angle between AB and x-axis:2 3 6

Use table of exact values

3

Use tanm tan6

m

1

3m

(x and y axes are perpendicular)

Maths4Scotland Higher

Hint

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A triangle ABC has vertices A(4, 3), B(6, 1)

and C(–2, –3) as shown in the diagram.

Find the equation of AM, the median from A.

Find mid-point of BC: (2, 1)

Find equation of median AM

Find gradient of median AM 2m

2 5y x

Maths4Scotland Higher

Hint

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P(–4, 5), Q(–2, –2) and R(4, 1) are the vertices

of triangle PQR as shown in the diagram.

Find the equation of PS, the altitude from P.

Find gradient of QR:1

2m

Find equation of altitude PS

Find gradient of PS (perpendicular to QR) 2m

2 3 0y x

Maths4Scotland Higher

Hint

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The lines and makeangles of a and b with the positivedirection of the x-axis, as shown in the diagram.a) Find the values of a and bb) Hence find the acute angle

between the two given lines.

2m

Find supplement of b 180 135 45

2 4y x 13x y

Find gradient of 2 4y x

Find gradient of 13x y 1m

Find a° tan 2 63a a

Find b° tan 1 135b b

Angle between two lines

Use angle sum triangle = 180°

72°

Maths4Scotland Higher

Hint

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Triangle ABC has vertices A(–1, 6), B(–3, –2) and C(5, 2)Find: a) the equation of the line p, the median from C of triangle ABC. b) the equation of the line q, the perpendicular bisector of BC. c) the co-ordinates of the point of intersection of the lines p and q.

Find mid-point of AB

Find equation of p 2y

Find gradient of p(-2, 2)

Find mid-point of BC (1, 0) Find gradient of BC1

2m

0m

Find gradient of q 2m Find equation of q 2 2y x

Solve p and q simultaneously for intersection (0, 2)

Maths4Scotland Higher

Hint

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Triangle ABC has vertices A(2, 2), B(12, 2) and C(8, 6). a) Write down the equation of l1, the perpendicular bisector of AB

b) Find the equation of l2, the perpendicular bisector of AC.

c) Find the point of intersection of lines l1 and l2 d) Hence find the equation of the circle passing through A, B and C.

7, 2Mid-point AB

Find mid-point AC (5, 4) Find gradient of AC2

3m

Equ. of perp. bisector AC

26r

Gradient AC perp.3

2m 2 3 23y x

Point of intersection (7, 1) This is the centre of circle

Find radius (intersection to A)

Equation of circle: 2 27 1 26x y

7x Perpendicular bisector AB

Maths4Scotland Higher

Hint

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A triangle ABC has vertices A(–4, 1), B(12,3) and C(7, –7). a) Find the equation of the median CM. b) Find the equation of the altitude AD. c) Find the co-ordinates of the point of intersection of CM and AD

4, 2Mid-point AB

Equation of median CM

Gradient of perpendicular ADGradient BC 2m1

2m

Equation of AD

3mGradient CM (median)

3 14y x

Solve simultaneously for point of intersection (6, -4)

2 2 0y x

Maths4Scotland Higher

Hint

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A triangle ABC has vertices A(–3, –3), B(–1, 1) and C(7,–3). a) Show that the triangle ABC is right angled at B.

b) The medians AD and BE intersect at M. i) Find the equations of AD and BE.

ii) Hence find the co-ordinates of M. 2mGradient AB

Product of gradients

Gradient of median ADMid-point BC 3, 11

3m Equation AD

1

2mGradient BC

12 1

2

Solve simultaneously for M, point of intersection

3 6 0y x

Hence AB is perpendicular to BC, so B = 90°

Gradient of median BEMid-point AC 2, 34

3m Equation AD 3 4 1 0y x

51,

3

Maths4Scotland Higher

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sin

cos

tan 1

6

4

3

1

2

1

23

2

3

2

1

21

21

3 3

Table of exact values