Question

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

Answer 1

Given that:

$ar△ABC=ar△DEF$

Prove that:

$△ABC\cong △DEF$

Proof:

Two triangles ABC and DEF are shown in figure.

$ar△ABC=ar△DEF$ (given)

$\frac{ar△ABC}{ar△DEF}=1$

$\frac{A{B}^2}{D{E}^2}=\frac{B{C}^2}{E{F}^2}=\frac{C{A}^2}{F{D}^2}=1$ [ USING THEOREM OF AREA OF SIMILAR TRIANGLES]

$AB=DE,BC=EF,CA=FD$

Thus, $△ABC\cong △DEF$ [BY SSS criterion of congruence]