Keisha is designing a wooden frame in the shape of a right triangle, as shown in Figure 8.3.14. The legs of the triangle are 3 ft and 4 ft. How long will the cut along the diagonal side be? Use the Pythagorean Theorem to find the length of the hypotenuse.

a right triangle with a base labeled a=3 ft; the height is labeled b=4 ft; the hypotenuse is labeled c
Figure8.3.14Wooden Frame

According to Pythagorean Theorem, we have:

\begin{align*} c^2\amp=a^2+b^2\\ c^2\amp=3^2+4^2\\ c^2\amp=9+16\\ c^2\amp=25\\ \end{align*}

Now we have a quadratic equation that we need to solve. We need to find the number that has a square of \(25\text{.}\) That is what the square root operation does.

\begin{align*} c\amp=\sqrt{25}\\ c\amp=5 \end{align*}

The diagonal cut Keisha will cut is 5 ft long.

Note that \(-5\) is also a solution of \(c^2=25\) because \((-5)^2=25\) but a length cannot be a negative number. We will need to include both solutions when they are relevant.