The converse of Pythagoras theorem is related to the right-angled triangles. It states that if the square of a side is equal to the sum of the squares of the remaining two sides, the triangle is the right-angled triangle. Here we are giving the proof, examples, formula, applications and problems related to the converse of Pythagoras’ theorem.

## What is the Converse of Pythagoras’ Theorem?

The converse of Pythagoras theorem statement is 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 to the first side is a right angle and the triangle is called a right-angled triangle.

Pythagorean theorem states that the sum of the squares of the two legs is equal to the square of the hypotenuse of a right-angled triangle. And it can also be stated as if this relation satisfies, then it is a right-angled triangle. If the sides of the triangle are a, b, c and the condition is a² + b²= c², then the triangle is a right-angled triangle.

The converse of Pythagoras theorem formula is a² = b² + c².

Where,

a, b, c are the sides of the right-angle triangle.

Also, Check:

### The Converse of Pythagorean Theorem Proof

Statement: If the length of the triangle is a, b, c and c² = a²+ b², then the triangle is a right-angled triangle.

Proof:
Draw a triangle XYZ, such as XZ = AC = b and BC = YZ = a.

In △XYZ, by Pythagoras Theorem:
XY² = XZ² + YZ² = b² + a² …………(1)
In △ABC, by Pythagoras Theorem:
AB² = AC² + BC² = b² + a² …………(2)
From equations (1) and (2), we have;
XY² = AB²
XY = AB
△ABC ≅ △XYZ (By SSS postulate)
∠G is the right angle
Thus, △XYZ is a right triangle.
Hence, we can say that the converse of Pythagorean theorem also holds.
Hence Proved.

### Questions on Converse of Pythagoras’ Theorem

Question 1:
The sides are the triangle are 7, 11, and 13. Check whether the given triangle is a right triangle or not?

Solution:
Given that,
a = 7, b = 11, c = 13
By using the converse of Pythagoras’ theorem
c² = a² + b²
13²= 7²+ 11²
169 = 49 + 121
169 = 170
So, it is not satisfied with the above equation.
Therefore, the given triangle is not a right triangle.

Question 2:
If the sides of a triangle are in the ratio 13:12:5, prove that the triangle is a right triangle. Also, state which angle is the right triangle.

Solution:
Let the trinagle be ABC
The sides are AB = 12k, BC = 5k, AC = 13k
AB²+ BC² = (12k)² + (5k)²
= 144k² + 25k²
= 169k²
= (13k)²
= AC²
Therefore, by the converse of Pythagoras theorem, ABC is a right-angled triangle in which ∠B = 90°.

Question 3:
The side of a triangle is of length 8 cm, 15 cm and 17 cm. Is this triangle a right triangle? If so, which side is the hypotenuse?

Solution:
Given that,
a = 15 cm, b = 8 cm, c = 17 cm
According to the converse of Pythagoras theorem,
c² = a² + b²
17² = 15² + 8²
289 = 225 + 64
289 = 289
As the condition is satisfied. The triangle is a right-angled triangle.
The hypotenuse is the longest side. So, it is 17 cm.

### FAQ’s on Converse of Pythagorean Theorem

1. What is the difference between the Pythagorean theorem and its converse?

The Pythagoras theorem is used to find the length of the missing side of a right triangle. Whereas the converse of the Pythagoras theorem is used to find if the triangle is a right-angled triangle or not.

2. How to find a hypotenuse?

The hypotenuse is the longest side of the right triangle or it is the side that is opposite to the 90 degrees of a triangle. From the Pythagoras theorem, hypotenuse = √[(base)² + (perpendicular)²].

3. What are the applications of the converse of Pythagora’s theorem?

The converse of Pythagora’s theorem is used to find whether the measurements of a triangle belong to the right triangle or not. You can also determine, three sides form a Pythagorean triplet.