1. Linear
Geometry

2. A (3, 7) and B (9, 19) A (-4, 1) and B (6, -9)
Linear coord geometry
KUS objectives
BAT solve linear geometry problems
BAT x
Starter: Find the midpoint of these points:
A (3, 7) and B (9, 19)
19−7 9−3 =2
A (-4, 1) and B (6, -9)
−9−1 6−(−4) =−1
−8−3 −7−(−5) =5.5
A (-5, 3) and B (-7, -8)

3. WB16
(4, −5)
Discussion question

4. Perpendicular line has gradient m2 =
WB17
Line l1 joins points A (3, 6) and B (6, 4)
What is the equation of the perpendicular line through midpoint of AB ?
Show this line goes through (3, 11/4)
L1 has gradient m1 =
Perpendicular line has gradient m2 =
Midpoint is =
Perpendicular line:
b) check:
QED

5. b) Gradient L1 = -2, gradient perpendicular = ½
WB18
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?
a) p = 6 and q = 3
b) Gradient L1 = -2, gradient perpendicular = ½
c) Area = ½ base x height = ½ x 6 x 3 = 9

6. Line L1 goes through points A(-3, 2) and B(3, -1) Find distance AB
WB19
Line L1 goes through points A(-3, 2) and B(3, -1)
Find distance AB
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 A B

7. D (-1, 1) a) AB = b) m = -½ , c) Solve simultaneous equations , Q A D
WB 19 solution
II Answer
Line L1 goes through points A(-3, 2) and B(3, -1)
Find distance AB
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
a) AB =
b) m = -½ ,
c) Solve simultaneous equations , Q
D (-1, 1) A D
Q(0, 3) QD =
Area AQB = B

8. Show that CD is perpendicular to AB
WB20
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
Show that CD is perpendicular to AB
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
[7]
D = (-2, -4)
Gradient of CD = -4, Gradient of AB= ¼
Hence product of gradient is m1 x m2 = -1, QED
Y + 5 = ¼ (x + 6)  2x – 8y – 28 = 0 or equivalent
B1
M2
A1
M2
A1

9. The straight line L1 ha equation 4y +x = 0
WB21
The straight line L1 ha 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
[6]

10. The points A and B have coordinates (5, -1) and (10, 4)
WB22
The points A and B have coordinates (5, -1) and (10, 4)
AB is a chord of a circle with centre C
Find the gradient of AB
The midpoint of AB is point M
Find an equation for the line through C and M
Given that the x-coordinate of point C is 6,
Find the y coordinate of C
Show that the radius of the circle is 17
[13]
gradient AB = 4−−1 10−5 =1
Midpoint AB = 7.5, and gradient is -1
Line through CM 𝑦− 3 2 =−1 𝑥− 15 2
 𝑥+𝑦=9
Point C 6, 3
radius AC = − −−1 2 = 17 7

11. Fins the values of p and q
WB23
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
Fins the values of p and q
The line L, which passes through D and is perpendicular to AC, intersects AB at E
Find an equation for line L in the form 𝑎𝑥+𝑏𝑦+𝑐=0
Find the exact x-coordinate of E
[9]

12. The straight line L1 has equation 𝑦=2𝑥+4
WB24
The straight line L1 has equation 𝑦=2𝑥+4
The straight line L2 has equation 6x-3y-9=0
Show that L1 is parallel to L2
Find an equation of the line L3 that is perpendicular to L1 and passes through the point (3, 10)
Find the point of intersection between lines L2 and L3
Find the shortest distance between lines L1 and L2
L1 has gradient 2
L2 y=2x−3 has gradient 2
L3 has gradient − and equation 𝑦−10=− 1 2 𝑥−3
 𝑥+2𝑦=23
Intersection L2 and L3 solving simultaneously gives , 43 5
Intersection L1 and L3 3, 10
Distance between 3− − =

13. Meet at (2, 1) Are perpendicular One of the gradients is 3 Challenge
Find PAIRS of lines for each of the eight regions in the Venn Diagram
Write the equations in the form 𝑦=𝑚𝑥+𝑐
Meet at
(2, 1)
Are perpendicular
One of the gradients is 3

14. Summary You should be able to: Rearrange equations of lines
Use to find the equation of a line and a perpendicular line
Solve problems involving midpoints, distances, areas, intersections and equations of lines

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

16. END