General Idea:
In any right triangle,
1. The altitude to the hypotenuse is the geometric mean between the segments into which is separates the hypotenuse.
2. Each leg is a geometric mean of the hypotenuse and the segment of the hypotenuse adjacent of the leg.
Applying the concept:
Step 1: We need to label the given diagram and set up a proportion by comparing the similar triangles as provided in attached figure.
Setting up the proportion, we get...
[tex]\frac{Shorter \; Leg}{Hypotenuse}=\frac{AB}{AC}=\frac{AD}{AB}\\\\\frac{x}{9+16}=\frac{9}{x}\\Cross \; Multiplying \; we \; get...\\\\x^2=9(9+16)\\x^2=9(25)\\x^2=225\\Take \; square \; root \; on \; both \; sides\\\\\sqrt{x^2}=\sqrt{225}\\x=15[/tex]
Conclusion:
[tex]x=15 \; cm[/tex]
