To find the length of the altitude to the hypotenuse of a right triangle, let's use the geometric mean theorem, also known as the altitude theorem. This theorem states that the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the two segments into which it divides the hypotenuse.
Let's denote the lengths of the two segments as [tex]\( seg1 \)[/tex] and [tex]\( seg2 \)[/tex]. According to the problem, these are 6 and 9 respectively.
The geometric mean of two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] is given by:
[tex]\[ \text{Geometric Mean} = \sqrt{a \cdot b} \][/tex]
Here, [tex]\( a = 6 \)[/tex] and [tex]\( b = 9 \)[/tex].
So, we calculate:
[tex]\[ \text{Altitude} = \sqrt{6 \cdot 9} = \sqrt{54} \][/tex]
We can simplify [tex]\( \sqrt{54} \)[/tex] as follows:
[tex]\[ \sqrt{54} = \sqrt{9 \cdot 6} = \sqrt{9} \cdot \sqrt{6} = 3\sqrt{6} \][/tex]
Therefore, the length of the altitude is:
[tex]\[ \text{Altitude} = 3\sqrt{6} \][/tex]
Given the options:
A. [tex]\( 9 \sqrt{2} \)[/tex]
B. [tex]\( 6 \sqrt{6} \)[/tex]
C. [tex]\( 3 \sqrt{6} \)[/tex]
D. [tex]\( 6 \sqrt{3} \)[/tex]
The correct answer is:
C. [tex]\( 3\sqrt{6} \)[/tex]
