Question

problem 5
what is the exact length of the unlabeled side of the triangle, in units?
use the sketch tool if it helps with your thinking
(there is a right triangle with one leg 5 and the other leg 8, and the hypotenuse is unlabeled)

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle (has a right - angle symbol). For a right - triangle, we can use the Pythagorean theorem, which states that for a right - triangle with legs of lengths \(a\) and \(b\) and hypotenuse of length \(c\), \(a^{2}+b^{2}=c^{2}\). Here, the legs are \(a = 5\) and \(b = 8\), and we need to find the length of the hypotenuse \(c\).

Step2: Apply the Pythagorean theorem

Substitute \(a = 5\) and \(b = 8\) into the Pythagorean theorem formula \(c^{2}=a^{2}+b^{2}\).
So, \(c^{2}=5^{2}+8^{2}\)
Calculate \(5^{2}=25\) and \(8^{2}=64\). Then \(c^{2}=25 + 64=89\).
To find \(c\), we take the square root of both sides: \(c=\sqrt{89}\) (since length cannot be negative, we take the positive square root).

Answer:

\(\sqrt{89}\)