Proof of the Converse of the Pythagorean Theorem

The Pythagorean Theorem states that if ABC is a triangle right angled at C, then,

a^2 + b^2 = c^2.

The converse of the Pythagorean Theorem states that if

a^2 + b^2 = c^2 holds,

then triangle ABC is a right triangle right angled at C.

right triangle

This means that in order to prove the converse of the Pythagorean Theorem, we need to prove that \angle C in the figure above is a right angle.  Now, we discuss the proof.

Theorem

In a triangle with sides a, b and c (see figure above), if

a^2 + b^2 = c^2 holds,

then ABC is a right triangle with a right angle at C.

Proof

Let DEF be a triangle such that EF = a, DF = b and right angled at F (see figure below). If we let DE = x, since DEF is a right triangle, by the Pythagorean Theorem

a^2 + b^2 = x^2 (1).

But from the supposition,

a^2 + b^2 = c^2 (2).

right triangle 3.

From (1) and (2)

x^2 = c^2.

Since x and c are both positive (Can you see why?), we can therefore conclude that

x = c.

This means that length of the three corresponding pairs of sides of triangle ABC and triangle DEF are equal.

Therefore, by SSS Congruence, \triangle ABC \cong \triangle DEF.

Since F and C are corresponding angles, \angle F = \angle C = 90 degrees.

And hence we have proved that triangle ABC is right angled at C

* Note that in the proof, a is the opposite of angle A, b is the opposite of angle B, and c is the opposite side of angle C.

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