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(ΔABC))/(A(ΔPQR)) = 1`   ...(i) [Given]

Also, `(A(ΔABC))/(A(ΔPQR)) = (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]