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If the Areas of Two Similar Triangles Are Equal, Prove that They Are Congruent - Mathematics

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

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Solution

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)

Concept: Areas of Similar Triangles
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