# Pythagorean Theorem

How to Use The Pythagorean Theorem

### The Formula

The picture below shows the formula for the Pythagorean theorem. In the pictures below, side C is always the hypotenuse. Remember that this formula only applies to right triangles.

### Examples of the Pythagorean Theorem

When you use the Pythagorean theorem, just remember that the hypotenuse is always 'C' in the formula above . Look at the following examples to see pictures of the formula.

### Step By Step Examplesof Using the Pythagorean Theorem

##### Example 1 (solving for the hypotenuse)

Use the Pythagorean theorem to determine the length of X

Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length '6 and '8' . 'X' is the hypotenuse because it is opposite the right angle.See Picture

Step 2

Substitute values into the formula (remember 'c' is the hypotenuse)

$A^2+ B^2= \red C^2 \\ 6^2+ 8^2= \red X^2$

Step 3

Solve for the unknown

$A^2+ B^2= \red X^2 \\ 100= \red X^2 \\ \sqrt {100} = \red X \\ 10= \red X$
##### Example 2 (solving for a Leg)

Use the Pythagorean theorem to determine the length of X

Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length '24' and 'X' are the legs. The hypotenuse is 26. See Picture

Step 2

Substitute values into the formula (remember 'c' is the hypotenuse)

$\red A^2+ B^2= C^2 \\ \red x^2 + 24^2= {26}^2$

Step 3

Solve for the unknown

$\red x^2 + 24^2= 26^2 \\ \red x^2 + 576^2= 676^2 \\ \red x^2 = 676^2 - 576^2 \\ \red x^2 = 100 \\ \red x = \sqrt { 100} \\ \red x = 10$

Remember our steps for how to use this theorem. This problems is like example 1 because we are solving for the hypotenuse.

Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length '14' and 48 are the legs. The hypotenuse is X. See Picture

Step 2

Substitute values into the formula (remember 'c' is the hypotenuse)

A2 + B2 = C2
142 + 482 = x2

Step 3

Solve for the unknown

Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length '9' and 'X'. The hypotenuse is 10.

Step 2

Substitute values into the formula (remember 'c' is the hypotenuse)

A2 + B2 = C2
92 + X2 = 102

Step 3

Solve for the unknown

Step 1

Identify the legs and the hypotenuse of the right triangle.

The legs have length '10' and 'X'. The hypotenuse is 20.

Step 2

Substitute values into the formula (remember 'c' is the hypotenuse)

A2 + B2 = C2
102 + X2 = 202

Step 3

Solve for the unknown

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