Converse of Pythagoras Theorem

The converse of Pythagoras theorem is the reverse of the Pythagoras theorem and it helps in determining if a triangle is acute, right, or obtuse if the sum of the squares of two sides of a triangle is compared to the square of its third side. The Pythagorean theorem is the most used in trigonometry. Let us learn more about the converse of the Pythagoras theorem, the proof, and solve a few examples.

The converse of Pythagoras theorem states 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, then the triangle is a right triangle. The converse is the complete reverse of the Pythagoras theorem. The main application of the converse of the Pythagorean theorem is that the measurements help in determining the type of triangle - right, acute, or obtuse. Once the triangle is identified, constructing that triangle becomes simple. There are three cases that occur:

1. If the sum of the squares of two sides of a triangle is considered equivalent to the square of the hypotenuse, the triangle is a right triangle.

2. If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle.

3. If the sum of the squares of two sides of a triangle is greater than the square of the hypotenuse, the triangle is an acute triangle.

Pythagoras Theorem

The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. In the given triangle ABC, we have BC² = AB² + AC²​​. Here, ​​​​AB is the base, AC is the altitude or the height, and BC is the hypotenuse. In other words, we can say, in a right triangle, (Opposite)² + (Adjacent)² = (Hypotenuse)²​​​​​​.

Statement: In a triangle, if the 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 (According to the SSS congruence)

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

∠Q = 90° (By construction)

So ∠B = 90°.

Hence, the converse of the Pythagoras theorem is proved.

The converse of Pythagoras theorem formula is c² = a² + b², where a, b, and c are the sides of the triangle.

Related Topics

Listed below are a few topics related to the converse of the Pythagoras theorem, take a look.

• Right Triangle Formulas
• Hypotenuse Leg Theorem
• What is Similarity
• Similarity in Triangles

What is the Converse of Pythagoras Theorem?

The coverse of the Pythagoras theorem states that, in a triangle, if the 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.

What is the Formula for Converse of Pythagoras Theorem?

The converse of Pythagoras theorem formula is c² = a² + b², where a, b, and c are the sides of the triangle.

How Do You Prove the Converse of Pythagoras Theorem?

The converse of the Pythagoras theorem states 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, then the triangle is a right triangle. If we consider two triangles ΔABC and ΔPQR where c² = a² + b² then we can say ∠C is a right triangle.

What is the Converse of the Pythagoras Theorem Useful For?

The converse of the Pythagoras theorem is useful in determining if a triangle is a right triangle or not. Whereas, a Pythagorean theorem helps in determining the length of the missing side of a right triangle.

What is the Application of the Converse of Pythagoras Theorem?

The application of the converse of the Pythagoras theorem is that the measurements help in determining what type of a triangle it is. There are three scenarios that we can determine, they are:

• If the sum of the squares of two sides of a triangle is considered equivalent to the square of the hypotenuse, the triangle is a right triangle.
• If the sum of the squares of two sides of a triangle is less than the square of the hypotenuse, the triangle is an obtuse triangle.
• If the sum of the squares of two sides of a triangle is greater than the square of the hypotenuse, the triangle is an acute triangle.

What is the Pythagorean Theorem?

The Pythagoras theorem states that if a triangle is right-angled (90 degrees), then the square of the hypotenuse is equal to the sum of the squares of the other two sides. We can say, (Opposite)² + (Adjacent)² = (Hypotenuse)²​​​​​​.