Question

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


(REQUIRED QUALITY ANSWER)​

Answer 1

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Given that:

ar△ABC=ar△DEF

Prove that:

△ABC≅△DEF

Proof:

Two triangles ABC and DEF are shown in figure.

ar△ABC=ar△DEF (given)

$\frac{ar \:∆ ABC }{ar \:∆ DEF } = 1$

[ Using Theorem of Area of similar Triangles ]

$\frac{ {AB }^{2} }{ { DE}^{2} } = \frac{ {BC }^{2} }{ { EF}^{2} } = \frac{ {CA }^{2} }{ { FD}^{2} } = 1$

[ By SSS criterion of congruence ]

AB = DE, BC = EF, CA = FD

Thus, △ABC≅△DEF