Question

find the length of the third side. if necessary, write in simplest radical form.

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem, which states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(c^{2}=a^{2}+b^{2}\) (if we are finding the hypotenuse) or \(a^{2}=c^{2}-b^{2}\) (if we are finding a leg). Here, the two given sides are the legs (since the right - angle is between them), and we need to find the hypotenuse. Let \(a = 7\), \(b=4\sqrt{2}\), and we want to find \(c\).

Step2: Apply the Pythagorean theorem

According to the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\). Substitute \(a = 7\) and \(b = 4\sqrt{2}\) into the formula:
\[

$$\begin{align*} c^{2}&=7^{2}+(4\sqrt{2})^{2}\\ &=49 + 16\times2\\ &=49+32\\ &=81 \end{align*}$$

\]

Step3: Solve for \(c\)

Take the square root of both sides of the equation \(c^{2}=81\). Since \(c\) represents the length of a side of a triangle, \(c>0\), so \(c=\sqrt{81} = 9\).

Answer:

The length of the third side is \(9\).