Question

if the area of two similar triangles are equal prove that they are congruent

Answer 1

Use the theorem that the ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides , then prove that they are congruent.________________________________________________________
Solution: 
[Fig is in the attachment]
Given: ΔABC ~ ΔPQR. & 
ar ΔABC =ar ΔPQR
To Prove: ΔABC ≅ ΔPQR
Proof: Since, ΔABC ~ ΔPQRar ΔABC =ar ΔPQR. (given)
ΔABC / ar ΔPQR = 1
⇒ AB²/PQ² = BC²/QR² = CA²/PR² = 1
[ USING THEOREM OF AREA OF SIMILAR TRIANGLES]
⇒ AB= PQ , BC= QR & CA= PR
Thus, ΔABC ≅ ΔPQR[BY SSS criterion of congruence]

Answer 2

ar(∆ABC)=ar(∆pqr). - 1
√ar(∆abc)\ar(pqr)=ab\pq
from both,
√1=ab\pq
1=ab\pq
ab=pq
therefore, ∆abc=~∆pqr