Question

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

Answer 1

Given data and similiar triangle property:

$∆ABC~∆PQR$

We know that the ratio of the area of two similar triangles is equal to the squares of the ratio of corresponding sides.

$\therefore \frac{ar\left(∆ABC\right)}{ar\left(∆PQR\right)}=\frac{A{B}^2}{P{Q}^2}=\frac{B{C}^2}{Q{R}^2}=\frac{A{C}^2}{P{R}^2}$

Given $ar\left(∆ABC\right)=ar\left(∆PQR\right)$

$\therefore \frac{A{B}^2}{P{Q}^2}=\frac{B{C}^2}{Q{R}^2}=\frac{A{C}^2}{P{R}^2}=1$

$\Rightarrow AB=PQ,BC=QR,AC=PR$

Therefore $∆ABC\cong ∆PQR$ [By $SSS$ congruency criteria]

Hence, it is proved that if the areas of two similar triangles are equal, prove that they are congruent