Question

find the length of the third side. if necessary, write in simplest radical form.
image of a right triangle with hypotenuse (2sqrt{34}) and one leg 6, and a right angle symbol

Explanation:

Step1: Identify the triangle type

This is a right - triangle, so we can use the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs. Here, the hypotenuse \(c = 2\sqrt{34}\) and one leg \(b = 6\), and we need to find the other leg \(a\).

Step2: Rearrange the Pythagorean theorem

We know that \(a^{2}=c^{2}-b^{2}\). Substitute \(c = 2\sqrt{34}\) and \(b = 6\) into the formula. First, calculate \(c^{2}=(2\sqrt{34})^{2}=4\times34 = 136\), and \(b^{2}=6^{2}=36\).

Step3: Calculate \(a^{2}\)

Then \(a^{2}=136 - 36=100\).

Step4: Find the value of \(a\)

Take the square root of both sides: \(a=\sqrt{100} = 10\).

Answer:

The length of the third side is \(10\).