To determine the length of the altitude of a right triangle that divides the hypotenuse into segments of lengths 6 and 9, we use a method related to the geometric mean.
Given:
- Segment 1 (length) = 6
- Segment 2 (length) = 9
To find the length of the altitude \( h \), we use the geometric mean formula for the lengths of the segments:
[tex]\[ h = \sqrt{\text{segment1} \times \text{segment2}} \][/tex]
[tex]\[ h = \sqrt{6 \times 9} \][/tex]
[tex]\[ h = \sqrt{54} \][/tex]
Simplifying the square root:
[tex]\[ \sqrt{54} = \sqrt{9 \times 6} = \sqrt{9} \times \sqrt{6} = 3 \sqrt{6} \][/tex]
Thus, the length of the altitude, in its simplest form, is \( 3 \sqrt{6} \).
Now, we need to compare this calculated altitude with the given options:
1. \( 9 \sqrt{2} \)
2. \( 6 \sqrt{6} \)
3. \( 3 \sqrt{6} \)
4. \( 6 \sqrt{3} \)
From our calculations, we see that the altitude \( 3 \sqrt{6} \) matches one of the provided options exactly.
Therefore, the correct answer is:
[tex]\[ 3 \sqrt{6} \][/tex]
