Question

Use area theorem of similar triangles to prove congruency of two similar triangles with equal areas.

Answer 1

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.