(Q1) Prove that if the areas of two similar triangles are equal, then they are congruent.

Solution :

∆ABC ∼ ∆PQR

So

ar(∆ABC)

ar(∆PQR)

= [

PQ

]² = [

BC

]² = [

PR

]²
But

ar(∆ABC)

ar(∆PQR)

= (∵ areas are Equal)
[

AB

PQ

]² = [

BC

QR

]² = [

AC

PR

]² =

So, AB² =

BC² =

AC² =

From which we get

AB =

BC =

AC =

∴ ∆ ≅ ∆ (by SSS Congruency)