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

Q4   If the areas of two similar triangles are equal, prove that they are congruent.

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Let $\triangle ABC\, \sim \, \triangle DEF,$ therefore,

$ar(\triangle ABC\,) = \,ar( \triangle DEF)$                (Given )

$\frac{ar(\triangle ABC)}{ar(\triangle DEF)}=\frac{AB^2}{DE^2}=\frac{BC^2}{EF^2}=\frac{AC^2}{DF^2}................................1$

$\therefore \frac{ar(\triangle ABC)}{ar(\triangle DEF)}=1$

$\Rightarrow \frac{AB^2}{DE^2}=\frac{BC^2}{EF^2}=\frac{AC^2}{DF^2}=1$

$AB=DE$

$BC=EF$

$AC=DF$

$\triangle ABC\, \cong \, \triangle DEF$                    (SSS )

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