Question

If the areas of two similar triangles are equal, prove that they are congruent.

Answer 1

Let us assume two similar triangles as ΔABC ~ ΔPQR

`(ar(triangleABC))/(ar(trianglePQR)) = ((AB)/(PQ))^2 = ((BC)/(QR))^2 = ((AC)/(PR))^2`      ...(1)

Given that, ar(ΔABC) = ar(ΔABC)

`=> (ar(triangleABC))/(ar(trianglePQR)) = 1`

Putting this value in equation (1), we obtain:

`1 = ((AB)/(PQ))^2 = ((BC)/(QR))^2 = ((AC)/(PR))^2`

⇒ AB = PQ, BC = QR and AC = PR

∴ ΔABC ≅ ΔPQR   (By SSS congruence criterion)