Linear Geometry

1

________________________________

Objective

Deadlines / Progress

Lin

ear

Geo

met

ry

Know the properties of linear graphs: the

intersections with axes; gradient and y-intercept;

Know the properties of parallel and

perpendicular lines

Find intersection points of lines, and lines and

curves, using simultaneous equation methods

Understand tangent line and normal line to a

curve at a point

Rearrange between general and other forms of a

linear equation e.g. 3𝑥 − 2𝑦 + 7 = 0 is

equivalent to 𝑦 =1

2𝑥 +

7

2

Cir

cle

Geo

met

ry

Find the centre and radius of a circle from its

equation

Convert from 𝑥2 + 𝑦2 + 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 to ‘centre, radius form’ of the equation of a circle

Find intersections of circles and lines; circles and

curves using simultaneous equation methods

Understand tangent line and normal line to a

circle at a point

Linear Geometry

2

Lines - Notes

One way of writing the equation of a line is 𝒚 = 𝒎𝒙 + 𝒄

Where m is the gradient and (0, c) is the y-intercept

The general equation of a line is when it is written in the

form 𝒂𝒙 + 𝒃𝒚 + 𝒄 = 𝟎 exam questions sometimes ask for the answer in

this form or the form 𝑎𝑥 + 𝑏𝑦 = 𝑐

Parallel lines – have the same gradient

Perpendicular lines – the gradients are negative reciprocals of each other

𝒎𝟏𝒎𝟐 = −𝟏

Given the gradient 𝑑𝑦

𝑑𝑥 (or m) and one point on a line we can use this

formula to get the equation of a line

𝒚 − 𝒚𝟏 = 𝒎(𝒙 − 𝒙𝟏)

Linear Geometry

3

WB1 For each of these equations, i) rearrange it into the form y = mx + c ii) give the gradient

iii) give the intercept on the y-axis

a) 2𝑥 + 𝑦 − 10 = 0

b) 5𝑥 − 2𝑦 + 6 = 0

WB2 There are three sets of parallel lines here.

Match them up and say what the gradient is for each set

𝑦 = 5𝑥 − 3

1

5𝑥 + 𝑦 −

1

5= 0

𝑦 =2

5𝑥 + 3

𝑦 = −5𝑥 − 1

𝑥 =1

5𝑦 +

3

5

𝑥 + 5𝑦 = 1

5𝑥 − 𝑦 − 3 = 0

𝑥 + 7.5 = 2.5𝑦

𝑦 = 0.2 −1

5𝑥

5𝑦 − 2𝑥 = 15

Linear Geometry

4

WB3 Which of these lines are parallel to the line 3x - 2y – 4 = 0

𝑦 = 3𝑥 − 2

2𝑦 − 3𝑥 + 4 = 0

2𝑦 + 4 =4

3𝑥

3𝑦 + 6 = 2𝑥

𝑦 =3

2𝑥 − 2

𝑦 = −3

2𝑥 + 2

−6𝑥 + 4𝑦 = 8

6𝑦 = 9𝑥 − 12

WB4 Draw a Perpendicular line to y = 3x Draw another one

y

x

Linear Geometry

5

WB5 What is the gradient of the lines perpendicular to these?

y = 2x + 1

y = 2 + 4x

y = 3x + 2

y + 2x = 2

2y = 3x – 2

5y + 2x = 3

WB6 Give the General equation of the perpendicular line to 2𝑥 + 𝑦 − 8 = 0 that goes through (4, 9)

Draw a sketch to go with your answer

Linear Geometry

6

WB7 Give the General equation of the perpendicular line to 𝑥 + 5𝑦 − 6 = 0 that goes

through (3

5, 7)

Draw a sketch to go with your answer

WB8 Two points A(1,2) and B(-3,6) are joined to make the line AB.

Find the equation of the perpendicular bisector of AB

Write your answer in the form 𝑎𝑥 + 𝑏𝑦 = 𝑐

Linear Geometry

7

WB9 Explain how you did each problem

a) Find the distance between these points: A (3, 7) and B (9, 19)

b) Find the gradient between these points: A (-1, 7) and B (19, 22)

WB10 Find the distance and gradient between each of these points

A (4, 9) and B (7, 14)

A (-2, -5) and B (-7, 7)

A (8, -1) and B (7, -11)

A (14, 3) and B (22, 13)

A (3, 18) and B (9, 2)

A (-7, -6) and B (14, -6)

Linear Geometry

8

WB11 Using 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1)

a) Find the line that joins these points (-2, 8) and (3,-7)

b) Find the line that joins these points (3, 7) and (9, 19)

c) Find the line that joins these points (4, 2) and (7, -4)

d) Find the line that goes through (3, 7) and has m = 3

e) Find the line that goes through (2, -3) and has m = -4

f) Find the line that joins (-1, -1) and (4, 14)

g) Find the line that joins (8, 1) and (18, 6)

h) Find the line that goes through (7, 3) and is parallel to 3y = x+7

i) Find the line that goes through (5, -5) and is parallel to x+ 2y =11

Linear Geometry

9

WB12a Find the equations of the three lines that join these points

a) (-6, 1) (2, 5) (-3, -5)

b) Choose your own 3 points

y

x

y

x

Linear Geometry

10

WB13a

Find the line that joins points (4, 9) and (8, 12)

in the form ax + by + c = 0

WB13b Find the line that joins points (-2, 8) and (3,-7)

in the form ax + by + c = 0

Linear Geometry

11

WB14

Find the general equation of the line through (3, 7) that is perpendicular to y = 2x + 8

WB15 The line L1 has gradient -3 goes through (-2, 3)

Line L2 is perpendicular to L1 and goes through (-2, 3)

Find the equations of lines L1 and L2

Linear Geometry

12

WB16 Find the equation of the line in the form ax + by +c = 0

a) parallel to 6𝑥 + 3𝑦 = 4 which passes through point (5, 5)

b) parallel to 6𝑦 = 3𝑥 + 13 which passes through point (3, 11)

c) Perpendicular to y = 3x + 2 and goes through (6, 12)

d) Perpendicular to y = -1/2x + 1 and goes through (-3, 8)

e) Perpendicular to the midpoint of the line that joins (4, 9) and (10, 21)

f) Perpendicular to the midpoint of the line that joins (-3,-6) and (9, 2)

Linear Geometry

13

Problem solving

WB17 Suggest possible equations for the lines in this diagram

Linear Geometry

14

WB18

Line l1 joins points A (3, 6) and B (6, 4)

a) What is the equation of the perpendicular line through midpoint of AB ? b) Show this line goes through (3, 11/4)

WB19

L1 has equation 2x + y - 6 = 0 and goes through points A(0, p) and B(q, 0)

a) Find the values of p and q

b) What is the equation of the perpendicular line from point C (4, 5) to line L1 ?

c) What is the area of triangle OAB?

Linear Geometry

15

WB20 Line L1 goes through points A(-3, 2) and B(3, -1) a) Find distance AB b) Find the equation of L1 in the form ax + by + c = 0

Perpendicular Line L2 has equation 2x – y + 3 = 0 and crosses L1 at point D.

c) Find coordinates of point D

Line L2 crosses the y-axis at point Q

d) Find the area of triangle AQB

WB21 The points A(-6, -5), B(2, -3) and C(4, -28) are the vertices of triangle ABC. Point D is the midpoint of the line joining A to B

a) Show that CD is perpendicular to AB b) Find the equation of the line passing through A and B in the form ax + by + c = 0, where a, b and c

are integers

Linear Geometry

16

WB22 The straight line L1 has equation 4y +x = 0 The straight line L2 has equation y = 5x - 4

a) The lines L1 and L2 intersect a the point A. Calculate, as exact fractions the coordinates of A

b) Find an equation of the line though A which is perpendicular to L1.

Give your answer in the form ax + by = c

WB23 The points A and B have coordinates (5, -1) and (10, 4) AB is a chord of a circle with centre C

a) Find the gradient of AB The midpoint of AB is point M

b) Find an equation for the line through C and M Given that the x-coordinate of point C is 6,

b) Find the y coordinate of C

c) Show that the radius of the circle is 17

Linear Geometry

17

WB24 The points A(3, 7) B(22, 7) and C(p, q) form the vertices of a triangle. Point D(9, 2) is the midpoint of AC a) Find the values of p and q

The line L, which passes through D and is perpendicular to AC, intersects AB at E

b) Find an equation for line L in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0 c) Find the exact x-coordinate of E

WB25 The straight line L1 has equation 𝑦 = 2𝑥 + 4 The straight line L2 has equation 6x-3y-9=0

a) Show that L1 is parallel to L2 b) Find an equation of the line L3 that is perpendicular to L1 and passes through the point (3, 10) c) Find the point of intersection between lines L2 and L3 d) Find the shortest distance between lines L1 and L2

Linear Geometry

18

WB26 Find PAIRS of lines for each of the eight regions in the Venn Diagram

Write the equations in the form 𝑎𝑥 + 𝑏𝑦 + 𝑐 = 0

Linear Geometry

19

WB27 The graph shows the extension 𝐸 of a spring when different masses, 𝑚, are attached to the end

a) Write a direct proportion equation connecting the two variables b) Interpret your result

𝐸 (𝑐𝑚)

𝑚 (𝑔𝑟𝑎𝑚𝑠) 100 200 300 400 500

5

10

15

20 (400, 20)

(0, 0)

Linear Geometry

20

WB28

The scatter graph shows the Oil production 𝑃 and carbon dioxide emissions, 𝐶, for various years since 1960 A line of best fit has been added to the scatter graph.

a) Formulate a linear model linking 𝑃 and 𝐶. Giving the relationship in the form 𝐶 = 𝑎𝑃 + 𝑏 b) Interpret the gradient of the line c) comment on the validity of the model for small values of 𝑃

𝐶𝑂

2

Bill

ion

to

nne

s

𝑂𝑖𝑙 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 (𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑡𝑜𝑛𝑛𝑒𝑠)

500 1000 1500 2500 2000

5

10

15

20