Question

Converse of Pythagoras therom proof​

Answer 1

1. A Proof for the Converse of the Pythagorean Theorem. Converse of the Pythagorean Theorem: If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

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Answer 2

Answer:

Converse of Pythagoras Theorem.

Statement:

In a triangle, if square of one side is equal to the sum of the squares of the other two sides, then the angle opposite the first side is a right angle.

Proof:

Here, we are given a triangle ABC in which AC²= AB² + BC²

We need to prove that ∠B = 90°

To start with, we construct a ΔPQR right-angled at Q such that PQ = AB and QR = BC.

Now, from Δ PQR, we have:

PR² = PQ² + QR² (Pythagoras Theorem, as ∠Q=90°)

or PR² = AB² + BC² (By construction)........ (1)

But AC² =AB² + BC² (Given).......... (2)

So, AC=PR (From (1) and (2)).............. (3)

Now, in ΔABC and ΔPQR,

AB=PQ (By construction)

BC=QR (By construction)

AC=PR (Proved in (3))

So, ΔABC≃ΔPQR (By SSS congruence)

∠B=∠Q (Corresponding angles of congruent triangles)

∠Q=90° (By construction)

So ∠B=90°

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