Question

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

Answer 1

Let us assume two similar triangle 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)