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Question
Use area theorem of similar triangles to prove congruency of two similar triangles with equal areas.
Solution
Let two similar triangles be ΔABC and ΔPQR.
So, ΔABC ∼ ΔPQR
By area theorem of similar triangles,
`(ar(ΔABC))/(ar(ΔPQR)) = ((BC)/(QR))^2` ......(i)
According to question,
ar(ΔABC) = ar(ΔPQR)
Substituting the values in equation (i),
`(ar(ΔABC))/(ar(ΔABC)) = ((BC)/(QR))^2`
⇒ 1 = `((BC)/(QR))^2`
⇒ (QR)2 = (BC)2
⇒ BC = QR
Similarly, AB = PQ and AC = PR.
Since sides of one triangle are equal to corresponding sides of another triangle
So, by SSS rule of congruence,
ΔABC ≅ ΔPQR
Hence proved.
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