An application of squaring numbers and square roots is the Pythagorean Theorem. The Pythagorean Theorem states that, for a right angled triangle, the square of the hypotenuse is equal to the sum of the square of the other two sides. The following triangle is a right triangle since there is a right angle (90°) represented by the red square. The hypotenuse is the longest side of the triangle and is across from the right angle in purple.

For the given triangle, the Pythagorean Theorem can be written as:

\( \textcolor{blue}{c}^2 = \textcolor{red}{a}^2 + \textcolor{green}{b}^2 \)

If we know two sides of the right triangle we can use the Pythagorean Theorem equation to solve for the other one.

For this triangle, \( \textcolor{red}{a} = \textcolor{red}{4} \) and \( \textcolor{green}{b} = \textcolor{green}{3} \).

Using the Pythagorean Theorem, the length of side c, the hypotenuse, is:

\( \textcolor{blue}{c}^2 = \textcolor{red}{a}^2 + \textcolor{green}{b}^2 \)

\( \textcolor{blue}{c}^2 = \textcolor{red}{4}^2 + \textcolor{green}{3}^2 \)

\( \textcolor{blue}{c}^2 = \textcolor{red}{16} + \textcolor{green}{9} \)

\( \textcolor{blue}{c}^2 = 25 \)

\( \textcolor{blue}{c} = \sqrt{25} \)

\( \textcolor{blue}{c} = 5 \)

Notice that it does not matter how you label a and b:

\( \textcolor{blue}{c}^2 = \textcolor{red}{a}^2 + \textcolor{green}{b}^2 \)

\( \textcolor{blue}{c}^2 = \textcolor{red}{3}^2 + \textcolor{green}{4}^2 \)

\( \textcolor{blue}{c}^2 = \textcolor{red}{9} + \textcolor{green}{16} \)

\( \textcolor{blue}{c}^2 = 25 \)

\( \textcolor{blue}{c} = \sqrt{25} \)

\( \textcolor{blue}{c} = 5 \)

Just be sure to label the hypotenuse correctly!

Pythagorean Theorem can be used to solve for side \( \textcolor{red}{a} \) or \( \textcolor{green}{b} \) too:

\( \textcolor{blue}{c}^2 = \textcolor{red}{a}^2 + \textcolor{green}{b}^2 \)

\( \textcolor{blue}{10}^2 = \textcolor{red}{6}^2 + \textcolor{green}{b}^2 \)

\( \textcolor{blue}{100} = \textcolor{red}{36} + \textcolor{green}{b}^2 \)

\( \textcolor{blue}{100} - \textcolor{red}{36} = \cancel{\textcolor{red}{36}} - \cancel{\textcolor{red}{36}} + \textcolor{green}{b}^2 \)

\( \textcolor{blue}{100} - \textcolor{red}{36} = \textcolor{green}{b}^2 \)

\( 64 = \textcolor{green}{b}^2 \)

\( \textcolor{green}{b} = \sqrt{64} \)

\( \textcolor{green}{b} = 8 \)

Enter 2 out of 3 sides of the triangle and use Pythagorean Theorem to solve for the third: