Question

use the geometric mean (leg) theorem. what is the value of a?
7√2
2√70
20√5
70√5

Explanation:

Step1: Apply the geometric mean (leg) theorem

The geometric mean (leg) theorem states that in a right - triangle, the length of a leg of the right - triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to that leg.
For a right - triangle with altitude \(h\) drawn to the hypotenuse, if the hypotenuse is divided into segments of lengths \(m = 6\) and \(n=14\), and the leg length is \(a\), the formula is \(a=\sqrt{(m + n)\times n}\)

Step2: Substitute the values of \(m\) and \(n\)

Here, \(m = 6\) and \(n = 14\). Then \(m + n=6 + 14=20\)
Substitute into the formula \(a=\sqrt{(m + n)\times n}\), we get \(a=\sqrt{20\times14}\)

Step3: Simplify the square - root

\(a=\sqrt{20\times14}=\sqrt{4\times5\times14}=2\sqrt{70}\)

Answer:

\(2\sqrt{70}\)