Describes how to find the angle between 2 lines

1. Angle between 2 Lines
Preliminary Extension Mathematics
Date: Tuesday 10th May 2011

2. Angle between 2 lines
y
line l1 has gradient m1
l2 l1 line l2 has gradient m2
∴ m1 = tan α and m2 = tan β
θ
α β
x
0

3. Angle between 2 lines
y
line l1 has gradient m1
l2 l1 line l2 has gradient m2
∴ m1 = tan α and m2 = tan β
θ
α β
x
and α +θ =β (Why?)
0

4. Angle between 2 lines
y
line l1 has gradient m1
l2 l1 line l2 has gradient m2
∴ m1 = tan α and m2 = tan β
θ and α +θ =β
α β
x (Exterior angle of V)
0

5. Angle between 2 lines
y
So
l2 l1 θ = β −α
θ
α β
x
0

6. Angle between 2 lines
y
So
l2 l1 θ = β −α
∴ tan θ = tan(β − α )
θ
α β
x
0

7. Angle between 2 lines
y
So
l2 l1 θ = β −α
∴ tan θ = tan(β − α )
θ tan β − tan α
=
α β 1 + tan β tan α
x
0
You will learn this formula later

8. Angle between 2 lines
y
So
l2 l1 θ = β −α
∴ tan θ = tan(β − α )
θ tan β − tan α
=
α β 1 + tan β tan α
x
0
m1 − m2
=
1 + m1m2
Why?

9. Angle between 2 lines
y
So
l2 l1 θ = β −α
∴ tan θ = tan(β − α )
θ tan β − tan α
=
α β 1 + tan β tan α
x
0
m1 − m2
=
1 + m1m2
When tan θ is positive, θ is acute.
When tan θ is negative, θ is obtuse.

10. Angle between 2 lines
y
Thus for two lines of gradient
l2 l1 m1 and m2
the acute angle between them is given by
θ m1 − m2
tan θ =
α β
x 1 + m1m2
0
Note that m1m2 ≠ −1 what does this mean?

11. Angle between 2 lines
y
Thus for two lines of gradient
l2 l1 m1 and m2
the acute angle between them is given by
θ m1 − m2
tan θ =
α β
x 1 + m1m2
0
Note that m1m2 ≠ −1
the formula does not work for perpendicular lines

12. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)

13. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3

14. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2

15. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2
2+3
∴ tan θ =
1− 6

16. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2
2+3
∴ tan θ =
1− 6
∴ tan θ = −1

17. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2
2+3
∴ tan θ =
1− 6
∴ tan θ = −1
∴ tan θ = 1

18. Example 1
Find the acute angle between y = 2x + 1 and y = −3x − 2
(to nearest degree)
∴ m1 = 2 and m2 = −3
m1 − m2
tan θ =
1 + m1m2
2+3
∴ tan θ =
1− 6
∴ tan θ = −1
∴ tan θ = 1 → θ = 45°

19. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0

20. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
∴2y = 3x + 7

21. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
∴2y = 3x + 7
3 7
∴y = x +
2 2

22. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
∴2y = 3x + 7
3 7
∴y = x +
2 2
3
∴ m1 =
2

23. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
∴2y = 3x + 7
similarly ∴2y = −4x + 3
3 7
∴y = x + 3
2 2 ∴ y = −2x +
2
3
∴ m1 = ∴ m2 = −2
2

24. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
applying the formula
m1 − m2
tan θ = 3
1 + m1m2 m1 = m2 = −2
2

25. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
applying the formula
m1 − m2
tan θ = 3
1 + m1m2 m1 = m2 = −2
2
3
+2
∴ tan θ = 2
1− 3

26. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
applying the formula
m1 − m2
tan θ = 3
1 + m1m2 m1 = m2 = −2
2
3
+2
∴ tan θ = 2
1− 3
−7
∴ tan θ =
4

27. Example 2
Find the acute angle between 3x − 2y + 7 = 0 and
(to nearest degree) 2y + 4x − 3 = 0
applying the formula
m1 − m2
tan θ = 3
1 + m1m2 m1 = m2 = −2
2
3
+2
∴ tan θ = 2
1− 3
−7 7
∴ tan θ = ∴ tan θ = → θ = 60°
4 4

28. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)

29. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
y = −3x + 5 y
y= x+3
θ
α β
x
0

30. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
m1 = 1 → α = 45°
y = −3x + 5 y
y= x+3
θ
α β
x
0

31. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
m1 = 1 → α = 45°
y = −3x + 5 y
y= x+3 m2 = −3 → β = 108°
θ
α β
x
0

32. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
m1 = 1 → α = 45°
y = −3x + 5 y
y= x+3 m2 = −3 → β = 108°
But α +θ =β
θ
α β
x
0

33. Example 3 - by thinking and
drawing....
Find the acute angle between y= x+3 and y = −3x + 5
(to nearest degree)
m1 = 1 → α = 45°
y = −3x + 5 y
y= x+3 m2 = −3 → β = 108°
But α +θ =β
θ ∴θ = 63°
α β
x
0