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. [tex]9 \sqrt{2}[/tex]
B. [tex]6 \sqrt{6}[/tex]
C. [tex]3 \sqrt{6}[/tex]
D. [tex]6 \sqrt{3}[/tex]



Answer :

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]