Question

If the area of two similar triangles are equal, then they are congruent". Prove.

Answer 1

Data: $△ABC\sim △DEF$
$ar\left(△ABC\right)=ar\left(△DEF\right)$
To Prove: $△ABC\cong △DEF$
Proof: $\frac{ar\left(△ABC\right)}{ar\left(△DEF\right)}=\frac{A{B}^2}{D{E}^2}=\frac{B{C}^2}{E{F}^2}=\frac{C{A}^2}{F{D}^2}$ [$\because$ Theorem of areas of similar triangles]
$1=\frac{A{B}^2}{D{E}^2}=\frac{B{C}^2}{E{F}^2}=\frac{C{A}^2}{F{D}^2}$ [$\because$ By data areas are equal]
$\Rightarrow A{B}^2=D{E}^2\therefore AB=DE$
$B{C}^2=E{F}^2\therefore BC=EF$
$C{A}^2=F{D}^2\therefore CA=FD$
$\therefore △ABC\cong △DEF$ [$\because$ SSS]