# Pythagorean Theorem

How to Use The Pythagorean Theorem

### The Formula

The picture below shows the formula for the Pythagorean theorem. For the purposes of the formula, side $$\overline{c}$$ 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 Examples of 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.

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.

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.

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

### Practice Problems

##### Problem 1

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. The hypotenuse is X.

Step 2

Substitute values into the formula (remember 'C' is the hypotenuse).

$A^2 + B^2 = C^2 \\ 14^2 + 48^2 = x^2$

Step 3

Solve for the unknown.

$14^2 + 48^2 = x^2 \\ 196 + 2304 = x^2 \\ \sqrt{2500} = x \\ \boxed{ 50 = x}$

##### Problem 2
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).

$A^2 + B^2 = C^2 \\ 9^2 + x^2 = 10^2$

Step 3

Solve for the unknown.

$9^2 + x^2 = 10^2 \\ 81 + x^2 = 100 \\ x^2 = 100 - 81 \\ x^2 = 19 \\ x = \sqrt{19} \approx 4.4$

##### Problem 3
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).

$A^2 + B^2 = C^2 \\ 10^2 + \red x^2 = 20^2$

Step 3

Solve for the unknown.

$10^2 + \red x^2 = 20^2 \\ 100 + \red x^2 = 400 \\ \red x^2 = 400 -100 \\ \red x^2 = 300 \\ \red x = \sqrt{300} \approx 17.32$