The document discusses the slope (gradient) of a line and how to calculate it. It provides four methods to calculate slope: (1) The rise over the run between two points (vertical change over horizontal change) (2) The change in y-values over the change in x-values between two points using a formula (3) The slope of a line is equal to the tangent of the angle of inclination (4) The relationship between slopes of parallel and perpendicular lines. Two lines are parallel if their slopes are equal, and perpendicular if the product of their slopes is -1. An example problem demonstrates finding the value of a that results in two lines being parallel or perpendicular.

1. The Slope (Gradient)

2. The Slope (Gradient)
vertical rise
(1) m 
horizontal run

3. The Slope (Gradient)
vertical rise y
(1) m 
horizontal run
x

4. The Slope (Gradient)
vertical rise y
l
(1) m 
horizontal run  5,5 
 2,1
x

5. The Slope (Gradient)
vertical rise y
l
(1) m 
horizontal run  5,5 
 2,1
y2  y1
(2) m  x
x2  x1

6. The Slope (Gradient)
vertical rise y
l
(1) m 
horizontal run  5,5 
 2,1
y2  y1
(2) m  x
x2  x1
5 1
ml 
52

7. The Slope (Gradient)
vertical rise y
l
(1) m 
horizontal run  5,5 
 2,1
y2  y1
(2) m  x
x2  x1
5 1
ml 
52
4

3

8. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1
(2) m  x
x2  x1
5 1
ml 
52
4

3

9. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 120
(2) m  x
x2  x1
5 1
ml 
52
4

3

10. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 120
(2) m  x
x2  x1
5 1
ml 
52
4

3
(3) m  tan 

11. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 120
(2) m  x
x2  x1
5 1
ml 
52
4

3
(3) m  tan 
 is the angle of inclination
with the positive x axis

12. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 120
(2) m  x
x2  x1
5 1
ml 
52
4
 mL  tan120
3
(3) m  tan 
 is the angle of inclination
with the positive x axis

13. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 120
(2) m  x
x2  x1
5 1
ml 
52
4
 mL  tan120
3
 3
(3) m  tan 
 is the angle of inclination
with the positive x axis

14. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 
120
(2) m  x
x2  x1
5 1
ml 
52
4
 mL  tan120
3
 3
(3) m  tan 
 is the angle of inclination
with the positive x axis

15. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 
120
(2) m  x
x2  x1
5 1
ml 
52
4
 mL  tan120
3
 3
(3) m  tan 
 is the angle of inclination ml  tan 
with the positive x axis

16. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 
120
(2) m  x
x2  x1
5 1
ml 
52
4
 mL  tan120
3
 3
(3) m  tan 
 is the angle of inclination ml  tan 
4
with the positive x axis tan  
3

17. The Slope (Gradient) L y
vertical rise l
(1) m 
horizontal run  5,5 
 2,1
y2  y1 
120
(2) m  x
x2  x1
5 1
ml 
52
4
 mL  tan120
3
 3
(3) m  tan 
 is the angle of inclination ml  tan 
4
with the positive x axis tan  
3
  53

18. Two lines are parallel iff they have the same slope
i.e. m1  m2

19. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other

20. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1

21. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;

22. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD

23. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD
3 1
mAB 
2 1
2

24. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD
3 1
mAB 
2 1
2
42
mCD 
a 3
2

a 3

25. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD If AB  CD then
3 1
mAB  mAB  mCD
2 1
2
42
mCD 
a 3
2

a 3

26. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD If AB  CD then
3 1
mAB  mAB  mCD
2 1
2 2
2
42 a 3
mCD 
a 3
2

a 3

27. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD If AB  CD then
3 1
mAB  mAB  mCD
2 1
2 2
2
42 a 3
mCD  a 3 1
a 3
2 a4

a 3

28. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD If AB  CD then (b) AB  CD
3 1
mAB  mAB  mCD
2 1
2 2
2
42 a 3
mCD  a 3 1
a 3
2 a4

a 3

29. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD If AB  CD then (b) AB  CD
3 1
mAB  mAB  mCD If AB  CD then
2 1
2 2 mAB  mCD  1
2
42 a 3
mCD  a 3 1
a 3
2 a4

a 3

30. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD If AB  CD then (b) AB  CD
3 1
mAB  mAB  mCD If AB  CD then
2 1
2 2 mAB  mCD  1
2
42 a 3 2
2
 1
mCD  a 3 1 a 3
a 3
2 a4

a 3

31. Two lines are parallel iff they have the same slope
i.e. m1  m2
Two lines are perpendicular iff their slopes are the
negative inverse of each other
i.e. m1  m2  1
e.g. A, B, C and D are the points (1,1), (2,3), (3,2) and (a,4)
Find a such that;
(a ) AB  CD If AB  CD then (b) AB  CD
3 1
mAB  mAB  mCD If AB  CD then
2 1
2 2 mAB  mCD  1
2
42 a 3 2
2
 1
mCD  a 3 1 a 3
a 3
4  a  3
2 a4
 a  1
a 3

32. (ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram
ABCD. Find the coordinates of D, the fourth vertex of the
parallelogram.

33. (ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram
ABCD. Find the coordinates of D, the fourth vertex of the
parallelogram. y
B 1,5 
A  4,0 
x
C  3, 5 

34. (ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram
ABCD. Find the coordinates of D, the fourth vertex of the
parallelogram. y
B 1,5 
3
5
A  4,0 
x
C  3, 5 
D  x, y 

35. (ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram
ABCD. Find the coordinates of D, the fourth vertex of the
parallelogram. y
B 1,5 
3
5
A  4,0 
x
C  3, 5 
3
5
D  x, y 

36. (ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram
ABCD. Find the coordinates of D, the fourth vertex of the
parallelogram. y
B 1,5 
3
5
A  4,0 
x
C  3, 5 
D   3  3, 5  5 
3
5
D  x, y 

37. (ii) A(4,0), B(1,5) and C(–3,–5) are three vertices of a parallelogram
ABCD. Find the coordinates of D, the fourth vertex of the
parallelogram. y
B 1,5 
3
5
A  4,0 
x
C  3, 5 
D   3  3, 5  5 
3
  0, 10 
5
D  x, y 

38. To prove three points (A, B, C) are collinear;

39. To prove three points (A, B, C) are collinear;
(i ) find mAB

40. To prove three points (A, B, C) are collinear;
(i ) find mAB
(ii ) find mAC

41. To prove three points (A, B, C) are collinear;
(i ) find mAB
(ii ) find mAC
(iii ) if mAB  mAC then A, B, C are collinear

42. To prove three points (A, B, C) are collinear;
(i ) find mAB
(ii ) find mAC
(iii ) if mAB  mAC then A, B, C are collinear
Exercise 5B; 2ace, 3bd, 4 ii, iv, 5ae, 8, 9a, 11ac,
13, 15, 18, 19, 20, 24*