**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.

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.”*

In ∆ABC, if ∠C = 90°, then AB

^{2}= BC^{2}+ AC^{2}
or c

^{2}= a^{2}+ b^{2}^{}

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

**Solution:**∠C = 90° (given)

c

^{2}= a^{2}+ b^{2}(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)

QR

^{2}= PQ^{2}+ PR^{2}(Pythagoras Theorem)
25

^{2}= 24^{2}+ PR^{2}
PR

^{2}= 25^{2}– 24^{2}
= 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 c

^{2}= a^{2}+ b^{2}, 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:**QR

^{2}= 30

^{2}= 900

PR

^{2}+ PQ^{2}= 20^{2}+ 21^{2}= 841
∴ QR

^{2}≠ PR^{2}+ PQ^{2}
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, 10

^{2}+ 24

^{2}= 100 + 576 = 676

Again, 26

^{2}= 676∴ 10

^{2}+ 24^{2}= 26^{2}In the given triangle, (Base)

^{2}+ (Perpendicular)^{2}= (Hypotenuse)^{2}Hence, the triangle is a right-angled triangle.