ADDITIONAL MATHEMATICS KSSM

1. COORDINATE
GEOMETRY
7.2 Parallel Lines &
Perpendicular Lines
Prepared By : Muslimah Binti Mahbob

2. Objective for today’s lesson:
Make and verify conjectures about
gradient of parallel lines and
perpendicular lines, and hence, make
generalisation.
Solve problems involving equations of
parallel and perpendicular lines.
01
02

3. EXPLORATION
TIME

4. LET EXPLORE

5. PARALLEL
LINES

6. What is it about parallel lines?

7. Example 1
Since the gradients for both lines are equal,
thus, both lines are parallel to each other.
Since the gradients for both lines
are not equal, both lines are not
parallel lines.
Determine whether the following pairs of straight lines are
parallel lines.
(a) (b)
9
5
1
6





x
y
x
y
3
4
6
4
3
2




y
x
x
y
2
1 m
m 
5
6
5
,
9
5
6
,
1
6
)
(
2
1
2
1











m
m
m
x
y
m
x
y
a
2
3
2
3
2
3
4
2
4
3
2
)
(
1 






m
x
y
x
y
x
y
b
2
3
4
3
2
3
3
6
4
3
4
6
2 






m
x
y
x
y
y
x
2
1 m
m 

8. Example 2
Find the equation of the straight line by
using the coordinate (1, 4) with the
gradient of -2.
Find the gradient
of the straight line.
Find the equation of straight line which passes through (1, 4)
and is parallel to line 5
4
2 
 x
y
2
2
2
5
4
5
2
5
4
2
1 







m
x
y
x
y
x
y
 
 
6
2
4
2
2
1
2
4
1
1














x
y
x
y
x
y
x
x
m
y
y
 
1
1 x
x
m
y
y 



9. PERPENDICULAR
LINES

10. What is it about perpendicular lines?

11. Gradient of Parallel &
Perpendicular Lines by graphing

12. Proof of Gradient of
Perpendicular Lines

13. Example 3
Since the product of the gradients for both
lines is -1, thus, both lines are perpendicular
to each other.
Determine whether the following pairs of straight lines are
perpendicular lines.
0
4
3
0
2
3






x
y
x
y
1
2
1 

 m
m
3
1
3
2
3
1
2
3
0
2
3
1 







m
x
y
x
y
x
y
3
4
3
0
4
3
2 







m
x
y
x
y
  1
3
3
1




1
2
1 

 m
m

14. Example 4
Find the equation of the normal line by
using the coordinate (1, 4) with the
gradient of 0.5.
Find the gradient
of the straight line.
Find the equation of straight line which passes through (1, 4)
and is perpendicular to line 6
2 

 x
y
2
6
2
1 




m
x
y
 
 
2
7
2
1
4
2
1
2
1
1
2
1
4
1
2
1











x
y
x
y
x
y
x
x
m
y
y
 
1
1 x
x
m
y
y 


Find the gradient
normal of the
straight line.
2
1
2
1
1
1
2
1
2
2
1









m
m
m
m
m

15. Example 5
3
1
5
3
1
15
3
15
3
)
(







mCD
x
y
x
y
x
y
a  
 
 
3
4
3
1
2
3
2
3
1
2
3
1
2
3
1
,
2
,
1
1
1












x
y
x
y
x
y
x
x
m
y
y
mAB
A
 
 
 
15
3
6
9
3
3
3
6
6
,
3
,
3
3
1
1
1
1
2
1




















x
y
x
y
x
y
x
x
m
y
y
D
mAB
mDE

16. Example 5
4
40
10
5
45
9
45
9
5
15
3
3
5
3
1
15
3
:
,
3
5
3
1
:
)
(


















x
x
x
x
x
x
x
x
x
y
DE
x
y
AB
b
 
 
3
,
4
3
15
4
3
?
,
4
15
3
:










E
y
y
y
x
when
x
y
AB
 
 
4
,
7
4
7
6
2
8
1
3
2
2
,
4
2
1
2
2
,
2
1
3
,
4
2
,
2
2
2
2
2
2
2
2
2
2
1
2
1
















 







 


B
y
x
y
x
y
x
y
x
y
y
x
x
M

17. SOLVING
PROBLEMS

18. Example 6

19. Example 6

20. Example 7
2
3
24
2
3
48
3
2
48
2
3
1 









m
x
y
x
y
y
x
3
2
2
3
1
1
2
2
1
2
2







m
m
m
m
mAB
m
 
 
 
3
20
3
2
12
3
16
3
2
8
3
2
12
12
,
8
,
3
2
1
2
1












x
y
x
y
x
y
x
x
m
y
y
B
mAB

21. Example 8
 
5
,
5
2
10
,
2
10
2
2
8
,
2
7
3
2
,
2
)
( 2
1
2
1













 







 


M
M
M
y
y
x
x
M
a    
   
   
606
.
3
13
4
9
5
7
5
8
7
,
8
,
5
,
5
,
?
)
(
2
2
2
1
2
2
1
2













MP
MP
MP
y
y
x
x
MP
P
M
MP
b

22. EVALUATION
TIME

23. TRY
DISCUSS THE ANSWERS WITH YOUR TEACHER

24. HOMEWORK

25. Self Practice 7.4, page 187,
Q1-Q4

26. Intensive Practice 7.2, page 190,
Q1–Q6 & Q10–Q11

27. Intensive Practice 7.2, page 190,
Q1–Q7 & Q10-Q11

28. A problem can be real,
but it is your choice
whether you want to stay
in it or you get out of it.
GOOD LUCK & ALL THE BEST