# Pythagorean Theorem

How to solve the Pythagorean theorem problems: theorem, proof, examples, and their solutions.

## Theorem

For a right triangle,

if legs: *a*, *b* and hypotenuse: *c*,

then *a*^{2} + *b*^{2} = *c*^{2}.

This is the Pythagorean theorem.

(= Pythagoras' theorem)

## Proof

Start from the right triangle

whose lengths of the legs are *a* and *b*,

and whose length of the hypotenuse is *c*.

Draw two squares using four of the right triangles.

The center quadrilateral's angle is a right angle,

because m∠(plain) + m∠(dot) + 90 = 180.

(See the right triangle's interior angles.)

Find the areas of both squares:

(purple square) = (*a* + *b*)^{2}

(right square) = 4 × (1/2)*ab* + *c*^{2}

= (4 × blue triangle) + (red square)

The areas of both squares are equal:

(*a* + *b*)^{2} = 4 × (1/2)*ab* + *c*^{2}

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

Square of a sum

## Example 1

Legs: 3, 4

Hypotenuse: *x*

## Example 2

Legs: 5, *x*

Hypotenuse: √89