Question

Areas of two similar triangles are equal then prove that triangles are congruent

Answer 1

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]