Question

geometric mean (altitude to hypotenuse)
question
given right triangle abc with altitude bd drawn to hypotenuse ac. if bd = 6 and dc = 9, what is the length of ad?
triangle diagram with points a, b, c, d and segments labeled
answer
ad = blank

Explanation:

Step1: Recall the geometric mean theorem (altitude-on-hypotenuse theorem)

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. So, \( BD^2 = AD \times DC \).

Step2: Substitute the known values

We know \( BD = 6 \) and \( DC = 9 \). Substitute these into the formula: \( 6^2 = AD \times 9 \).

Step3: Solve for \( AD \)

First, calculate \( 6^2 = 36 \). Then the equation becomes \( 36 = 9 \times AD \). To find \( AD \), divide both sides by 9: \( AD=\frac{36}{9} = 4 \).

Answer:

\( 4 \)