Question

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

Explanation:

Step1: Identify the type of triangle

It's a right - triangle, so use the Pythagorean theorem $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a$, $b$ are the legs. Here, assume the two given sides are the legs.

Step2: Apply the Pythagorean theorem

Let $a = 4$ and $b$ be the unknown side, $c = 7$. Then $b^{2}=c^{2}-a^{2}$. Substitute $a = 4$ and $c = 7$ into the formula: $b^{2}=7^{2}-4^{2}=49 - 16=33$.

Step3: Solve for $b$

Take the square root of both sides. Since we are looking for a length, $b=\sqrt{33}$.

Answer:

$\sqrt{33}$