# Prove Converse of Pythagorean Theorem

Discussion in 'Algebra' started by nycmathguy, Jul 6, 2022.

1. ### nycmathguy

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College Algebra
Chapter 1/Section 3

Prove the converse of the Pythagorean Theorem. Explain each step along the way.

nycmathguy, Jul 6, 2022

2. ### MathLover1

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The converse of the Pythagorean Theorem is:

If the square of the length of the longest side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

That is, in ΔABC, if c^2=a^2+b^2 then ∠C is a right triangle, ΔPQR being the right angle.

proof;
We can prove this by contradiction.

Let us assume that c^2=a^2+b^2 in ΔABC and the triangle is not a right triangle.

Now consider another triangle ΔPQR. We construct ΔPQR so that PR=a, QR=b and ∠R is a right angle. By the Pythagorean Theorem, (PQ)^2=a^2+b^2.

But we know that a^2+b^2=c^2 and c=AB.

So, (PQ)^2=a^2+b^2=(AB)^2.

That is, (PQ)^2=(AB)^2.

Since PQ and AB are lengths of sides, we can take positive square roots and say:

PQ=AB

That is, all the three sides of ΔPQR are congruent to the three sides of ΔABC.

So, the two triangles are congruent by the Side-Side-Side Congruence Property.

Since ΔABC is congruent to ΔPQR and ΔPQR is a right triangle, ΔABC must also be a right triangle.

This is a contradiction. Therefore, our assumption must be wrong.

MathLover1, Jul 6, 2022
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