The gradient of a line is defined to be the gradient of any interval within the line.

This definition depends on the fact that two intervals on a line have the same gradient.

Suppose AB and PQ are two intervals on the same straight line.

Draw right-angled triangles ABX and PQY with sides AX and PY parallel to the x-axis and sides BX and QY parallel to the y-axis.

Triangle ABX is similar to triangle PQY since the corresponding angles are equal. Therefore:

$$\dfrac{QY}{PY}=\dfrac{BX}{AX}$$.

That is, the intervals have the same gradient.

#### Example 4

Find the gradient of the line passing through the points:

1. A(1, 4) and B(5, 12)
2. C(6, –5) and D(–4, 2)

#### Solution

1. The gradient of the interval $$AB =\dfrac{y_2 − y_1}{x_2 − x_1} =\dfrac{12 − 4}{5 −1} = \dfrac{8}{4} = 2$$
Therefore the gradient of the line AB = 2.
2. The gradient of the interval $$CD =\dfrac{y_2 − y_1}{x_2 − x_1} =\dfrac{2 −(−5)}{−4 − 6} = \dfrac{7}{−10}$$
Therefore the gradient of the line $$CD = −\dfrac{7}{10}$$.