# Pythagorean Theorem

How to use the Pythagorean theorem to solve the sides of a given right triangle: theorem, examples, and their solutions.

## Theorem

*a*^{2} + *b*^{2} = *c*^{2}*a*, *b*: Legs of a right triangle*c*: Hypotenuse of a right triangle

(Hypotenuse: The longest side)

The Pythagorean theorem (Pythagoras' theorem)

shows the relationship

between the sides of a right triangle.

## Example 1

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

So *x*^{2} = 25.

Square root both sides.

Then *x* = √25*x* > 0

So you don't have to write (±)

in front of the square root sign.

To solve the square root,

make a square.

25 = 5^{2}

Then take 5

out from the square root sign.

So *x* = 5.

Simplify a radical

## Example 2

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* > 0

So you don't have to write (±)

in front of the square root sign.

To solve the square root,

make a square.

64 = 8^{2}

Then take 8

out from the square root sign.

So *x* = 8.

Simplify a radical