Prove that the ratio of the areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

Question

Philoid · Accepted Answer

Let ΔPQR and ΔABC be two similar triangles,

[Corresponding sides of similar triangles are in the same ratio] [1]

And as corresponding angles of similar triangles are equal

∠A = ∠P

∠B = ∠Q

∠C = ∠R

Construction: Draw PM ⏊ QR and AN ⏊ BC

In ΔPQR and ΔABC

∠PMR = ∠ANC [Both 90°]

∠R = ∠C [Shown above]

ΔPQR ~ ΔABC [By Angle-Angle Similarity]

[Corresponding sides of similar triangles are in the same ratio] [2]

Now, we know that

Area of a triangle

Therefore,

[From 2]

[From 1]

Hence, Proved.