1. Starter: state the gradient and y intercept of each line
Coordinate Geometry
KUS objectives
BAT find distances between points
BAT explore equations of lines, know the general equation of a line
BAT use a new formula to find equations of lines
Starter: state the gradient and y intercept of each line

2. A (3, 7) and B (9, 19) 12 6 Find the distance between these points:
Notes 1
Find the distance between these points:
A (3, 7) and B (9, 19) 12 6

3. A (-1, 7) and B (19, 22) 15 20 Find the gradient between these points:
Notes 2
Find the gradient between these points:
A (-1, 7) and B (19, 22) 15 20

4. A (x1, y1) and B (x2, y2) 𝒅 y1 – y2 x1 – x2 𝑑= 𝑥− 𝑥 1 2 +( 𝑦 2 − 𝑦 1 2
Notes 3 Formula
Find the distance between these points:
A (x1, y1) and B (x2, y2) 𝒅
y1 – y2
x1 – x2
𝑑= 𝑥− 𝑥 ( 𝑦 2 − 𝑦 1 2
Generally:
Can you establish this result in a proof with clear steps?

5. Find the distance between these points
Practice 1
Find the distance between these points
Find the gradient between 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)

6. Objectives BAT find distances between points
BAT explore equations of lines, know the general equation of a line
BAT use a new formula to find equations of lines

7. Find the gradient between these points
Notes 4
What is gradient?
(x2, y2)
Change in y
Change in x
y2- y1
y2-y1
x2-x1
x2- x1
(x1, y1)
Find the gradient between these points
(4, 9) and (7, 12)

8. We can generalise y - y1 y- y1 x - x1 Rearrange to: x- x1
notes
We can generalise
(x, y)
y - y1
x - x1
y- y1
Rearrange to:
x- x1
(x1, y1)
How can we use this?

9. Find the line that joins these points
WB5
Find the line that joins these points
(-2, 8) and (3,-7)
Use this point (3, -7)
x1 is 3 and y1 is -7
y – (-7) = -3 (x – 3)
y + 7 = -3x + 9
y = -3 x + 2

10. Find the equation of the line that: Joins (3, 7) and (9, 19)
Practice 2
Find the equation of the line that:
Joins (3, 7) and (9, 19)
Joins (4, 2) and (7, -4)
Goes through (3, 7) and has m = 3
Goes through (2, -3) and has m = -4
Joins (-1, -1) and (4, 14)
Joins (8, 1) and (18, 6)
Goes through (7, 3) and parallel to 3y = x+7
Goes through (5, -5) and parallel to x+ 2y =11

11. y x (-6, 1) (2, 5) (-3, -5) Find the equations of the lines that
WB6
Find the equations
of the lines that
join these points
(-6, 1)
(2, 5)
(-3, -5)
activity: Geogebra
three points

12. y x Choose three points of your own Work out the general
Challenge
Choose three points
of your own
Work out the general
equations of each line

13. Objectives 1 find distances between points
2 explore equations of lines, know the general equation of a line
3 use a new formula to find equations of lines

14. The general equation of a line: ax + by + c = 0
Notes
The general equation of a line: ax + by + c = 0
y = 4x 
4x – y + 6 = 0
y = ½ x 
x – 2y + 4 = 0
y = - x 
x + y + 3 = 0
y = -5 x 
5x + y – 1 = 0

15. y – 9 = ¾ (x – 4) y – 9 = ¾ x – 3 y = ¾ x + 6 or 3x - 4y + 24 = 0
WB7
Find the line that joins points (4, 9) and (8, 12) in the form ax + by + c = 0
Use this point (4, 9)
x1 is 4 and y1 is 9
y – 9 = ¾ (x – 4)
y – 9 = ¾ x – 3
y = ¾ x + 6 or
3x - 4y + 24 = 0

16. y – (-7) = -3 (x – 3) y + 7 = -3x + 9 y = -3 x + 2 or 3x + y - 2 = 0
WB8
Find the line that joins points (-2, 8) and (3,-7) in the form ax + by + c = 0
Use this point (3, -7)
x1 is 3 and y1 is -7
y – (-7) = -3 (x – 3)
y + 7 = -3x + 9
y = -3 x + 2 or
3x + y - 2 = 0

17. y – 7 = -½ (x – 3) y - 7 = -½ x + 3/2 y = -½x + 17/2 or
WB9
Find the general equation of the line through
(3, 7) that is perpendicular to y = 2x + 8
Use gradient
Use the point (3, 7)
x1 is 3 and y1 is 7
y – 7 = -½ (x – 3)
y - 7 = -½ x + 3/2
y = -½x + 17/2 or
x + 2y - 17 = 0

18. y – 7 = -½ (x – 3) y - 7 = -½ x + 3/2 y = -½x + 17/2 or x + 2y = 17
WB10
Find the equation of the line that goes through
(3, 7) and is perpendicular to y = 2x + 8
Giving your answer in the form 𝑎𝑥+𝑏𝑦=𝑐
Use gradient
Use the point (3, 7)
x1 is 3 and y1 is 7
y – 7 = -½ (x – 3)
y - 7 = -½ x + 3/2
y = -½x + 17/2 or
x + 2y = 17

19. y – 3 = -3 (x – (-2)) l1 is y = -3 x - 3 y – 3 = 1/3(x – (-2))
WB11
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
y – 3 = -3 (x – (-2))
l1 is y = -3 x - 3
y – 3 = 1/3(x – (-2))
l2 is 3y = x + 11

20. Find the equation of the line in the form ax + by +c = 0
Practice
Find the equation of the line in the form ax + by +c = 0
Perpendicular to y = 3x + 2
And goes through (6, 12)
Perpendicular to y = -1/2x + 1
And goes through (-3, 8)
Find the equation of the line perpendicular to:
The midpoint of the line that joins (4, 9) and (10, 21)
The midpoint of the line that joins (-3,-6) and (9, 2)

21. 𝑦 = −2𝑥+ 4 3 𝑦−5= −2 𝑥−5 𝑦 =−2𝑥+15 2𝑥+𝑦 = 15 Gradient is -2
WB12
Giving your answer in the form 𝑎𝑥+𝑏𝑦=𝑐
𝑦 = −2𝑥+ 4 3
Gradient is -2
𝑦−5= −2 𝑥−5
𝑦 =−2𝑥+15
2𝑥+𝑦 = 15

22. WB13
𝑦−11=3 𝑥−3
y=3𝑥+2

23. Summary You should be able to: Rearrange equations of lines
Use to find the equation of a line from two points or gradient plus a point
Use to find the equation of a perpendicular line given enough information

24. One thing to improve is –
KUS objectives BAT solve linear geometry problems
self-assess
One thing learned is –
One thing to improve is –

25. END