Question

if the area of two similar triangles are equal. prove that they are tangent

Answer 1

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Answer 2

Heya !!

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QUESTION CORRECTION :- If the area of two similar ∆s are equal. Prove that they are congruent.

SOLUTION :-

Given :- Two ∆s ABC and PQR such that ∆ABC ~ ∆PQR and ar(∆ABC) = ar(∆PQR)

To prove :- ∆ABC ≈ ∆PQR

Proof :- ∆ABC ~ ∆PQR
So, angle A = angle P ,
angle B = angle Q ,
angle C = angle R ,

and

AB/PQ = BC/QR = AC/PR

Now, ar(∆ABC) = ar(∆PQR)
=> ar(∆ABC) / ar(∆PQR) = 1
=> (AB/PQ)² = (BC/QR)² = (AC/PR)² = 1
=> AB/PQ = BC/QR = AC/PR = 1

=> AB = PQ
=> BC = QR
=> AC = PR

Therefore, ∆ABC ≈ ∆PQR (by SSS congruency criterion)

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Hope my ans.'s satisfactory.☺