Question

If A (x1, y1), B (x2, y2) and C (x3, y3) are the vertices of a triangle ABC, then its area is given by \(\frac {1}{2}\) {x1 (y2 – y3 ) + x2 (y3 – y1 ) + x3 (y1 – y2 ) }

(a) True

(b) False

The question was posed to me in an interview for internship.

Query is from Geometry topic in portion Coordinate Geometry of Mathematics – Class 10

Answer 1

Correct answer is (a) True

Explanation: A (x1, y1), B (x2, y2) and C (x3, y3) are the vertices of a triangle ABC.

Draw AL, CM, BN perpendicular to x – axis.

Then, ML = (x1 – x2), LN = (x3 – x1) and MN = (x3 – x2).

∴ area of ∆ABC = ar(trap.BMLA) + ar(trap.ALNC) + ar(trap.BMNC)

 = {\(\frac {1}{2}\) (AL + BM) × ML} + {\(\frac {1}{2}\) (AL + CN) × LN} – {\(\frac {1}{2}\) (CN + BM) × MN}

 = {\(\frac {1}{2}\) (y1 + y2 ) × (x1 – x2)} + {\(\frac {1}{2}\) (y1 + y3 ) × (x3 – x1)} – {\(\frac {1}{2}\) (y2 + y3 ) × (x3 – x2)}

 = \(\frac {1}{2}\) {x1 (y1 + y2 – y1 – y3 ) + x2 (y2 + y3 – y1 – y2 ) – x3 (y1 + y3 – y2 – y3 ) }

 = \(\frac {1}{2}\) {x1 (y2 – y3 ) + x2 (y3 – y1 ) + x3 (y1 – y2 ) }