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

• Geometry (C10)

If the area of two similar triangles are equal, prove that they are congruent.

Answer

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.

Given: ΔABC ~ ΔPQR

ar ΔABC = ar ΔPQR

To Prove: ΔABC ≅ ΔPQR

Proof:

ar ΔABC / ar ΔPQR = 1

AB²/PQ² = BC²/QR² = CA²/PR² = 1

AB = PQ, BC = QR and CA = PR

Therefore, ΔABC ≅ ΔPQR (By SSS criterion of congruence)