Question

use the geometric mean (altitude) theorem. what is the value of m?
write a proportion containing altitude m and the segments with known lengths.
\\( \frac { 2 0 } { m } = \frac { m } {? } \\)

Explanation:

Step1: Apply the geometric mean (altitude) theorem

The geometric mean (altitude) theorem states that in a right - triangle (formed by the altitude to the hypotenuse of a larger right - triangle), the altitude \(m\) to the hypotenuse is the geometric mean of the segments of the hypotenuse. The proportion is \(\frac{8}{m}=\frac{m}{20}\).

Step2: Cross - multiply the proportion

Cross - multiplying gives \(m\times m = 8\times20\), so \(m^{2}=160\).

Step3: Solve for \(m\)

Take the square root of both sides: \(m=\sqrt{160}\). Simplify \(\sqrt{160}=\sqrt{16\times10}=4\sqrt{10}\).

Answer:

\(4\sqrt{10}\)