Everything Maths Grade 10

To derive the formula for gradient, we consider any right-
angled triangle formed fromA(x 1 ;y 1 )andB(x 2 ;y 2 )with
hypotenuseABas shown in the diagram alongside. The gra-
dient is determined by the ratio of the length of the vertical
side of the triangle to the length of the horizontal side of the
triangle. The length of the vertical side of the triangle is the
difference iny-values of pointsAandB. The length of the
horizontal side of the triangle is the difference inx-values of
pointsAandB.

 3  2  1 1 2 3

 3

 2

 1

1

2

3

A(x 1 ;y 1 )

B(x 2 ;y 2 )

C

x

y

Therefore, gradient is determined using the following formula:

Gradient(m) =

y 2 y 1

x 2 x 1

or

y 1 y 2

x 1 x 2

IMPORTANT!

Remember to be consistent:m̸=
y 1 y 2
x 2 x 1

VISIT:

To learn more about determining the gradient you can watch the following video.

See video:2GC6atwww.everythingmaths.co.za

Worked example 3: Gradient between two points

QUESTION

Find the gradient of the line between pointsE(2; 5)andF(3; 9).

SOLUTION

Step 1: Draw a sketch

 4  3  2  1 1 2 3 4

1

2

3

4

5

6

7

8

9

E(2; 5)

F(3; 9)

x

y

294 8.3. Gradient of a line