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 $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a,b$ are the legs. Let the hypotenuse $c = 7$, one leg $a=2\sqrt{10}$, and the other leg be $b$.

Step2: Apply the Pythagorean theorem

From $a^{2}+b^{2}=c^{2}$, we can re - arrange it to solve for $b$: $b=\sqrt{c^{2}-a^{2}}$.
Substitute $c = 7$ and $a = 2\sqrt{10}$ into the formula:
First, calculate $c^{2}=7^{2}=49$ and $a^{2}=(2\sqrt{10})^{2}=2^{2}\times(\sqrt{10})^{2}=4\times10 = 40$.
Then, $b=\sqrt{49 - 40}=\sqrt{9}$.

Step3: Simplify the square root

Since $\sqrt{9}=3$, the length of the third side is 3.

Answer:

3