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: \(\color{buel}{a^2+b^2= c^2}\)

### 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: \(a^2+b^2= c^2 →9^2+12^2= c^2 →81+144=c^2\)

\(c^2=225 →c=15\) cm

### Example 2:

Find the hypotenuse of the following right triangle.

**Solution**:

Use Pythagorean Theorem: \(\color{blue}{a^2+b^2= c^2}\)

Then: \(a^2+b^2= c^2 →8^2+6^2= c^2 →64+36=c^2\)

\(c^2=100 →c=10 \)

### Example 3:

Find the hypotenuse of the following right triangle.

**Solution**:

Use Pythagorean Theorem: \(\color{blue}{a^2+b^2= c^2}\)

Then: \(a^2+b^2= c^2 →3^2+4^2= c^2 →9+16=c^2\)

\(c^2=25 →c=5 \)

### 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: \(a^2+b^2= c^2 →6^2+8^2= c^2 →36+64=c^2\)

\( c^2=100 →c=10\)

## Exercises

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

### Download Pythagorean Relationship Worksheet

- \(\color{blue}{13}\)
- \(\color{blue}{5}\)
- \(\color{blue}{15}\)
- \(\color{blue}{8}\)