# Hypotenuse of a Right Triangle

Sub Topics
Right Triangle: A right triangle is a triangle in which one of the angle is 90°.

In a right triangle, one side is the base, the longest side is called the hypotenuse and the third side is the height of the triangle.

Pythagorean Theorem:
The sum of the squares of the two sides (base and height) is equal to the square of the hypotenuse.

In the above right triangle, the base is denoted by ‘b’, height by ‘a’ and the hypotenuse by ‘c’.

Applying the Pythagorean theorem, we get
a2 + b2 = c2
[(height)2 + (base)2 = (hypotenuse)2 ]  (hypotenuse of a right triangle formula)

Pythagorean Theorem Statement: In a right triangle, the square of the hypotenuse is equal to the squares of the sum of the other two sides

Proof:
We are given a right triangle with right angle at B
We need to prove that: AC2 = AB2 + BC2
Let us draw a perpendicular, BD $\perp$  AC

Now,  $\bigtriangleup$ ADB is similar to $\bigtriangleup$ ABC

(According to the theorem, If a perpendicular is drawn from the vertex at the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other).

So,   $\frac{AD}{AB}$ = $\frac{AB}{AC}$ (sides are proportional)

Also, $\bigtriangleup$ BDC is similar to $\bigtriangleup$ ABC

$\frac{CD}{BC}$ = $\frac{BC}{AC}$

CD. AC = BC2                 -------------------(2)

AD. AC + CD.AC = AB2 + BC2
AC (AD + CD) = AB2 + BC2
AC.AC = AB2 + BC2
AC2 = AB2 + BC2
Hence the Pythagorean Theorem is proved.

## Finding the Hypotenuse of a Right Triangle

To find the hypotenuse of a right triangle, we need to find the square root of the sum of the squares of the base and height (the two shorter legs). The hypotenuse is the longest side of a right triangle.

c2 = a2 + b2
c = $\sqrt{a^{2}+b^{2}}$

## How to Calculate the Hypotenuse of a Right Triangle

Below you could see examples to calculate the hypotenuse of a right triangle

Example 1:
If one side of a right triangle is 12 and its hypotenuse is 15, what is the length of the other side?

Solution:
Given hypotenuse c = 15
and side a =12

Applying the Pythagorean theorem,
=>a2 + b2 = c2

=>122 + b2 = 152

=>144 + b2 = 225

subtract 144 on both sides
=>b2 = 81
square root both sides
=>b=9

Length of other side is 9 in

Example 2:
Given two sides of the right triangle are 3 and 4 , what is the length of the
hypotenuse?

Solution:
Given side a=3
and side b=4

Applying the Pythagorean theorem,
=>c2 =a2 + b2
=>c2=32+42
=>c2 =9+16
=>c2=25
=>c=$\sqrt{25}$
=>c=5
The hypotenuse is 5