Question

(99 POINTS) Using similarity solve for x. Show all work

Answer 1

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...

$\frac{Shorter \; Leg}{Hypotenuse}=\frac{AB}{AC}=\frac{AD}{AB}\\\\\frac{x}{9+16}=\frac{9}{x}\\Cross \; Multiplying \; we \; get...\\\\x^2=9(9+16)\\x^2=9(25)\\x^2=225\\Take \; square \; root \; on \; both \; sides\\\\\sqrt{x^2}=\sqrt{225}\\x=15$

Conclusion:

$x=15 \; cm$

Answer 2

Shorter Length/Hypotenuse = ab/ac = ad/ab

x/9+16 = 9/x

x^2 = 9 x (9 + 16)

= 9 x 25

= 225

square root of 225 = 15

x = 15