Question

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

Explanation:

Step1: Identify triangle type (right triangle)

This is a right triangle, so we use the Pythagorean theorem: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse, and \(a, b\) are the legs. Here, hypotenuse \(c = 10\), one leg \(b = 6\), find the other leg \(a\).

Step2: Apply Pythagorean theorem

Rearrange the formula to solve for \(a\): \(a = \sqrt{c^2 - b^2}\). Substitute \(c = 10\) and \(b = 6\): \(a = \sqrt{10^2 - 6^2}\).

Step3: Calculate the values

First, calculate the squares: \(10^2 = 100\), \(6^2 = 36\). Then subtract: \(100 - 36 = 64\). So \(a = \sqrt{64}\).

Step4: Simplify the square root

\(\sqrt{64} = 8\).

Answer:

8