Question

given right triangle xyz, with altitude wz, find each length.
a. if xw = 3 and wy = 12, determine wz.
wz

b. if wz = 4 and xw = 2, determine wy.

c. if xy = 22 and xw = 9, determine wz.

d. if xz = 10 and xw = 4, determine xy.

e. if wz = 18 and xw = 9, determine xz

Explanation:

Part a

Step1: Recall geometric mean theorem

In a right triangle, the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse. So \( WZ^2=XW\times WY \).

Step2: Substitute values

Given \( XW = 3 \) and \( WY=12 \), then \( WZ^2=3\times12 = 36 \).

Step3: Solve for \( WZ \)

Take square root: \( WZ=\sqrt{36}=6 \).

Step1: Use geometric mean theorem

\( WZ^2=XW\times WY \), so \( WY=\frac{WZ^2}{XW} \).

Step2: Substitute values

Given \( WZ = 4 \) and \( XW = 2 \), then \( WY=\frac{4^2}{2}=\frac{16}{2}=8 \).

Step1: Find \( WY \)

First, \( WY=XY - XW \). Given \( XY = 22 \) and \( XW = 9 \), so \( WY=22 - 9=13 \).

Step2: Apply geometric mean theorem

\( WZ^2=XW\times WY \), so \( WZ^2=9\times13 = 117 \).

Step3: Solve for \( WZ \)

\( WZ=\sqrt{117}=3\sqrt{13}\approx10.82 \).

Answer:

\( WZ = 6 \)

Part b