Question

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

Answer 1

Given, $\Delta ABC$ and $\Delta PQR$ are similar and equal in area.
To Prove that $\Delta ABC\cong \Delta PQR$
Proof
Since, $\Delta ABC\sim \Delta PQR$
$\therefore \frac{Areaof\Delta ABC}{Areaof\Delta PQR}=\frac{B{C}^2}{Q{R}^2}$
$\Rightarrow \frac{B{C}^2}{Q{R}^2}=1$ [Since,$Areaof\Delta ABC=Areaof\Delta PQR$]
$\Rightarrow B{C}^2=Q{R}^2$
$\Rightarrow BC=QR$
Similarly, we can prove that
AB = PQ and AC = PR
Thus, $\Delta ABC\cong \Delta PQR$ [By using SSS criterion of congruence]