Pythagoras Theorem

# Pythagoras Theorem

## Pythagoras Theorem

In a right-angled triangle, the side opposite to the right angle is called the hypotenuse. It is the longest side of the right-angled triangle. In the following figure, the side AB, which is opposite to C (right angle), is the hypotenuse.

We generally use the small letter of an angle in a triangle to denote the side opposite the angle. In the above figure, a, b, and c denote the sides opposite A, B, and C respectively.
Pythagoras theorem relates the lengths of the three sides of a right-angled triangle.

According to Pythagoras theorem,
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.”

Pythagoras theorem states that,

In ABC, if C = 90°, then AB2 = BC2 + AC2
or c2 = a2 + b2

Example 1: In ABC, C = 90°, AC = 8 cm, and BC = 6 cm. Find the length of AB.

Solution: C = 90° (given)
c2 = a2 + b2 (Pythagoras Theorem)
= (6)2 + (8)2 = 36 + 64 = 100
c = 100 = 10
The length of AB is 10 cm.

Example 2: In PQR, P = 90°, PQ = 24 cm, and QR = 25 cm. Find the length of PR.

Solution: P = 90° (given)
QR2 = PQ2 + PR2 (Pythagoras Theorem)
252 = 242 + PR2
PR2 = 252 – 242
= 625 – 576 = 49
PR = 49 = 7 cm
The length of PR is 7 cm.

## The Converse of Pythagoras Theorem

In a triangle, if the square of the longest side is equal to the sum of the squares of the other two sides, then the angle opposite the longest side is a right angle.

For any triangle with sides a, b, and c, if c2 = a2 + b2 , then the angle between a and b measures 90° and the triangle is a right-angled triangle. This is called the converse of Pythagoras theorem.

Example 3: Determine whether the following triangle is a right-angled triangle.

Solution: QR2 = 302 = 900
PR2 + PQ2 = 202 + 212 = 841
QR2 PR2 + PQ2
Hence, PQR is not a right-angled triangle.

Example 4: The sides of a triangle are 10 cm, 24 cm and 26 cm. Determine whether the triangle is a right-angled triangle or not.

Solution: Here, 102 + 242 = 100 + 576 = 676
Again, 262 = 676
102 + 242 262
In the given triangle, (Base)2 + (Perpendicular)2 (Hypotenuse)2
Hence, the triangle is a right-angled triangle.

Please do not enter any spam link in the comment box.