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Question
Areas of two similar triangles are equal then prove that triangles are congruent
Solution
Given: ΔABC ~ ΔPQR and A(ΔABC) = A(ΔPQR)
To prove: ΔABC ≅ ΔPQR
Proof:
`("A"(Δ"ABC"))/("A"(Δ"PQR"))` = 1 ......(i) [Given]
Also, `("A"(Δ"ABC"))/("A"(Δ"PQR")) = "AB"^2/"PQ"^2 = "BC"^2/"QR"^2 = "AC"^2/"PR"^2` ......[Theorem of areas of similar triangles]
∴ 1 = `"AB"^2/"PQ"^2 = "BC"^2/"QR"^2 = "AC"^2/"PR"^2` .....[From (i)]
∴ 1 = `"AB"^2/"PQ"^2`
∴ AB2 = PQ2
∴ AB = PQ ......[Taking square root of both sides]
i.e., seg AB ≅ seg PQ
Similarly, seg BC ≅ seg QR and seg AC ≅ seg PR
∴ ΔABC ≅ ΔPQR ......[SSS test of congruency]
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