Question

Three points A (x1, y1), B (x2, y2) and C (x3, y3) are collinear only when \(\frac {1}{2}\) {x1 (y2 – y3 ) + x2 (y3 – y1 ) + x3 (y1 – y2 ) } = 0.

(a) False

(b) True

The question was asked in a national level competition.

The above asked question is from Geometry in portion Coordinate Geometry of Mathematics – Class 10

Answer 1

The correct answer is (b) True

Easiest explanation: Consider three points (-3, 3), (-1, 2) and (1, 1)

We know that, area of triangle = \(\frac {1}{2}\){x1 (y2 – y3 ) + x2 (y3 – y1 ) + x3 (y1 – y2)}

The area of triangle = \(\frac {1}{2}\) {-3(2 – 1) + (-1)(1 – 3) + 1(3 – 2)} = \(\frac {1}{2}\) {-3 – 1 + 3 + 3 – 2} = \(\frac {0}{2}\) = 0

Hence if the points are collinear the area of triangle is zero.