Converse of Pythagoras theorem – Proof, Applications

Converse of Pythagoras theorem – Proof, Applications

Converse of Pythagoras theorem – Proof

• Consider a right-angle triangle ABC, with its three sides namely the opposite, adjacent and the hypotenuse. In a right-angled triangle we generally refer to the three sides in order to their relation with the angle θ⁰. The little box in the right corner of the triangle given below denotes the right angle which is equal to 90⁰.
• The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse(H). The side that is opposite to the angle θ is known as the opposite(O). And the side which lies next to the angle θ is known as the Adjacent(A)

The Pythagoras theorem states that,

Converse of Pythagorean Theorem proof:

The converse of the Pythagorean Theorem proof is:

Converse of Pythagoras theorem statement: The Converse of Pythagoras theorem statement says that 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 of a triangle, then the triangle is known to be a right triangle.

That is, in ΔABC if c² = a² + b² then

∠C is a right triangle, the ΔPQR being the right angle.

We can prove this by contradiction.

Let us assume that ,

c² = a² + b² 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 = a² + b²

But we know that , a² + b² = c² and c=AB.

So, (PQ)² = a² + b² = (AB)²

That is, (PQ)² = (AB)²

Since the lengths of the sides are PQ and AB, we can take positive square roots.

PQ=AB

That is, all the three sides of the triangle PQR are congruent to the three sides of the triangle ABC. 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.This is the converse of Pythagoras theorem proof.

Formula of converse of the Pythagorean Theorem:

As per the converse of the Pythagorean theorem, the formula for a right-angled triangle is given by:

Where the variables a, b and c are the sides of a triangle.

Applications of the Converse of Pythagoras Theorem:

Basically, the converse of the Pythagoras theorem is used to find whether the measurements of a given triangle belong to the right triangle or not. If we come to know that the given sides belong to a right-angled triangle, it helps in the construction of such a triangle. Using the concept of the converse of Pythagoras theorem, one can determine if the given three sides form a Pythagorean triplet.