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

For Competitive Exams

Question 4 If the areas of two similar triangles are equal, prove that they are congruent.

Solution:

Say ΔABC and ΔPQR are two similar triangles and equal in area

Now let us prove ΔABC ≅ ΔPQR.

Since, ΔABC ~ ΔPQR

$\dfrac{Area \: of \: (ΔABC)}{Area \: of \: (ΔPQR)} = \dfrac{BC^2}{QR^2}$

$\dfrac{BC^2}{QR^2} =1 [Since, Area(ΔABC) = (ΔPQR)$

$\dfrac{BC^2}{QR^2}$

BC = QR

Similarly, we can prove that

AB = PQ and AC = PR

Thus, ΔABC ≅ ΔPQR [SSS criterion of congruence]

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