# Pythagorean Theorem

How to find the side of a right triangle by using the Pythagorean theorem: formula, 2 examples, and their solutions.

## Formula

### Formula

The Pythagorean theorem (Pythagoras' theorem)

shows the relationship

between the sides of a right triangle.

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

a, b: Legs

c: Hypotenuse

## Example 1

### Example

### Solution

The legs are 4 and 3.

The hypotenuse is x.

So, by the Pythagorean theorem,

x^{2} = 4^{2} + 3^{2}.

4^{2} = 16

3^{2} = 9

16 + 9 = 25

Square root both sides.

Then x = √25.

x is the lenght of the hypotenuse.

So x is plus.

So you don't have to write ± sign.

Quadratic Equation: Square Root

25 = 5^{2}

√5^{2} = 5

Simplify a Radical

So x = 5.

## Example 2

### Example

### Solution

The legs are x and 5.

The hypotenuse is √89.

So, by the Pythagorean theorem,

x^{2} + 5^{2} + (√89)^{2}.

+5^{2} = +25

(√89)^{2} = 89

Move +25 to the right side.

Then x^{2} = 64.

Square root both sides.

Then x = √64.

x is the lenght of the leg.

So x is plus.

So you don't have to write ± sign.

64 = 8^{2}

√8^{2} = 8

So x = 8.