Pythagorean Theorem

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

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

Find the value of x.

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

Find the value of x. The lengths of the right triangle's legs: 5, x. The length of the right triangle's hypotenuse: square root 89.

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