Pythagorean Theorem

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

Theorem

a² + b² = c²

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,
x² = 4² + 3².

4² = 16
3² = 9

16 + 9 = 25

So x² = 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 = 5²

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,
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 > 0
So you don't have to write (±)
in front of the square root sign.

To solve the square root,
make a square.
64 = 8²

Then take 8
out from the square root sign.

So x = 8.

Simplify a radical

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