Question

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

Answer 1

Given:-

$ar\left(△ABC\right)=ar\left(△DEF\right)△ABC\sim △DEF$

To prove $△ABC\cong △DEF$

Proof:-

$\because △ABC\sim △PQR\left(\text{Given}\right)$

$\therefore \frac{{AB}^2}{{DE}^2}=\frac{{BC}^2}{{EF}^2}=\frac{{CA}^2}{{DF}^2}=\frac{ar\left(△ABC\right)}{ar\left(△DEF\right)}$

$\because ar\left(△ABC\right)=ar\left(△DEF\right)$

$\therefore \frac{{AB}^2}{{DE}^2}=1$

$\Rightarrow AB=DE$

Similarly,

$BC=EF$

$AC=DF$

Now,

In $△ABC$ and $△DEF$,

$AB=DE$

$BC=EF$

$AC=DF$

By SSS congruency,

$△ABC\cong △DEF$

Hence proved.