Question

find the length of the third side. if necessary, write in simplest radical form.
image of a right triangle with one leg labeled 5, hypotenuse labeled 10, and the right angle marked

Explanation:

Step1: Identify triangle type

It's a right triangle, so use Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is hypotenuse, \(a,b\) are legs. Here, hypotenuse \(c = 10\), one leg \(a = 5\), find the other leg \(b\).

Step2: Apply Pythagorean theorem

Rearrange formula: \(b^2 = c^2 - a^2\). Substitute \(c = 10\), \(a = 5\):
\(b^2 = 10^2 - 5^2 = 100 - 25 = 75\).

Step3: Simplify radical

\(b = \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}\).

Answer:

\(5\sqrt{3}\)