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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?
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.
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]
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]
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