Question

find the length of the third side. if necessary, write in simplest radical form.
(image of a right triangle with one leg 4, the other leg 2√29, and the right angle between them)

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 \(a\), \(b\) are the legs. Let the given leg be \(a = 4\), the hypotenuse \(c=2\sqrt{29}\), and we need to find the other leg \(b\).

Step2: Rearrange the Pythagorean theorem

We know that \(b^{2}=c^{2}-a^{2}\). Substitute \(a = 4\) and \(c = 2\sqrt{29}\) into the formula. First, calculate \(c^{2}=(2\sqrt{29})^{2}=2^{2}\times(\sqrt{29})^{2}=4\times29 = 116\), and \(a^{2}=4^{2}=16\).

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

Then \(b^{2}=116 - 16=100\).

Step4: Find \(b\)

Take the square root of both sides, \(b=\sqrt{100} = 10\) (we take the positive root because length is non - negative).

Answer:

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