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

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

Answer

Let ΔABC ~ ΔPQR. 

$\frac{ar\triangle ABC}{ar\triangle PQR} = \frac{AB^2} {PQ^2} = \frac{BC^2} {QR^2} = \frac{AC^2} {PR^2}$

Given ar ΔABC = ar ΔPQR

$\frac{AB^2} {PQ^2} = 1 = \frac{BC^2} {QR^2} = \frac{AC^2} {PR^2}$ 

AB = PQ, BC = QR, AC = PR 

Therefore, ΔABC $\cong$ ΔPQR. (sss congruence rule)