# How to Solve Pythagorean Theorem Problems? (+FREE Worksheet!)

In mathematics, the Pythagorean Theorem is the relationship between three sides of a right triangle.

## Step by step guide to solve Pythagorean Theorem problems

• We can use the Pythagorean Theorem to find a missing side in a right triangle.
• In any right triangle: $$\color{blue}{a^2+b^2= c^2}$$

### The Pythagorean Theorem – Example 1:

Right triangle ABC has two legs of lengths $$9$$ cm (AB) and $$12$$ cm (AC). What is the length of the third side (BC)?

Solution:

Use Pythagorean Theorem: $$\color{blue}{a^2+b^2= c^2}$$
Then: $$a^2+b^2= c^2 →9^2+12^2= c^2 →81+144=c^2$$
$$c^2=225 →$$ $$c=\sqrt{225}=15$$ $$cm$$ → $$c=15 cm$$

### The Pythagorean Theorem – Example 2:

Find the hypotenuse of the following right triangle.

Solution:

Use Pythagorean Theorem: $$\color{blue}{a^2+b^2= c^2}$$
Then: $$a^2+b^2= c^2 →8^2+6^2= c^2 →64+36=c^2$$
$$c^2=100 →$$ $$c=\sqrt{100}=10$$ → $$c=10$$

### The Pythagorean Theorem – Example 3:

Find the hypotenuse of the following right triangle.

Solution:

Use Pythagorean Theorem: $$\color{blue}{a^2+b^2= c^2}$$
Then: $$a^2+b^2= c^2 →3^2+4^2= c^2 →9+16=c^2$$
$$c^2=25 →$$ $$c=\sqrt{25}=5$$ → $$c=5$$

### The Pythagorean Theorem – Example 4:

Right triangle ABC has two legs of lengths $$6$$ cm (AB) and $$8$$ cm (AC). What is the length of the third side (BC)?

Solution:

Use Pythagorean Theorem: $$\color{blue}{a^2+b^2= c^2}$$
Then: $$a^2+b^2= c^2 →6^2+8^2= c^2 →36+64=c^2$$
$$c^2=100 →$$ $$c=\sqrt{100}=10$$ $$cm$$ → $$c= 10 cm$$

## Exercises for Solveing the Pythagorean Theorem

### Find the missing side in each right triangle.

1. $$\color{blue}{13}$$
2. $$\color{blue}{5}$$
3. $$\color{blue}{15}$$
4. $$\color{blue}{8}$$