Question

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

Explanation:

Step1: Identify known values

Let hypotenuse $c=10$, leg $a=4$, unknown leg $b$.

Step2: Apply Pythagorean theorem

$$a^2 + b^2 = c^2$$
Substitute values: $4^2 + b^2 = 10^2$

Step3: Calculate squared terms

$16 + b^2 = 100$

Step4: Isolate $b^2$

$b^2 = 100 - 16 = 84$

Step5: Solve for $b$, simplify radical

$b = \sqrt{84} = \sqrt{4 \times 21} = 2\sqrt{21}$

Answer:

$2\sqrt{21}$