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.

a2 + b2 = c2

a, b: Legs
c: Hypotenuse

Pythagorean Theorem: Proof

Example 1

Example

Solution

The legs are 4 and 3.
The hypotenuse is x.

So, by the Pythagorean theorem,
x2 = 42 + 32.

42 = 16
32 = 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 = 52

So x = 5.

Example 2

Example

Solution

The legs are x and 5.
The hypotenuse is √89.

So, by the Pythagorean theorem,
x2 + 52 + (√89)2.

+52 = +25
(√89)2 = 89

Move +25 to the right side.

Then x2 = 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 = 82

82 = 8

So x = 8.