Question

the altitude to the hypotenuse of a right triangle divides the hypotenuse into segments of lengths 6 and 9. what is the length of the altitude? a. $9\sqrt{2}$ b. $6\sqrt{6}$ c. $3\sqrt{6}$ d. $6\sqrt{3}$

Explanation:

Step1: Recall geometric mean theorem

For a right triangle, the altitude $h$ to the hypotenuse satisfies $h = \sqrt{p \times q}$, where $p$ and $q$ are the lengths of the hypotenuse segments.

Step2: Substitute given values

Here, $p=6$ and $q=9$. Substitute into the formula:
$h = \sqrt{6 \times 9}$

Step3: Simplify the expression

Calculate the product inside the square root:
$h = \sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$

Answer:

C. $3\sqrt{6}$