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