Question

Without using distance formula, prove that the points A(1, 4), B(3, –2) and C(–3, 16) are collinear.

Answer 1

Given: Points A(1, 4), B(3, –2) and C(–3, 16).

To prove: A, B, C are collinear.

Proof:

1. Three points A, B, C are collinear if and only if slope of AB = slope of BC. [Condition for collinearity]

2. Slope of AB:

$$m_{AB} = \frac{-2 - 4}{3 - 1}$$

$$= \frac{-6}{2}$$

$$= -3$$

3. Slope of BC:

$$m_{BC} = \frac{16 - (-2)}{-3 - 3}$$

$$= \frac{18}{-6}$$

$$= -3$$

4. Since slope of AB = slope of BC = –3, AB is parallel to BC. Since both lines pass through B, they coincide. Hence A, B, C are collinear.

Hence proved.