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Home » Two similar triangles are equal in area. Prove that the triangles are congruent.
Two similar triangles are equal in area. Prove that the triangles are congruent.
Class 10 | Exercise 15E | ICSE | Maths | Selina | Similarity (With Applications to Maps and Models)
Two similar triangles are equal in area. Prove that the triangles are congruent.

Let’s consider two similar triangles as

    \[\blacktriangle ABC\text{ }\sim\text{ }\blacktriangle PQR\]

So,

    \[Ar\left( \blacktriangle ABC \right)/\text{ }Ar\left( \blacktriangle PQR \right)\]

    \[=\text{ }{{\left( AB/PQ \right)}^{2}}~=\text{ }{{\left( BC/QR \right)}^{2}}~\]

So,

    \[=\text{ }{{\left( AC/PR \right)}^{2}}\]

Since,

    \[Area\text{ }of\text{ }\blacktriangle ABC\text{ }=\text{ }Area\text{ }of\text{ }\blacktriangle PQR\]

[Given]

Hence,

    \[AB\text{ }=\text{ }PQ\]

    \[BC\text{ }=\text{ }QR\]

And,

    \[AC\text{ }=\text{ }PR\]

So, as the respective sides of two similar triangles are all of same length.

We can conclude that,

    \[\blacktriangle ABC\cong \blacktriangle PQR\]

[By SSS rule]

– Hence Proved

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