Question

given that δuvw is similar to δxyz, find the length of \overline{zx}.
a 14
b 16
c 18
d 20
e 22

Explanation:

Step1: Identify corresponding sides

In similar triangles \( \triangle UVW \) and \( \triangle XYZ \), the corresponding sides are proportional. For \( \triangle UVW \), the sides are \( UV = 3 \), \( VW = 4 \), \( UW = 5 \). For \( \triangle XYZ \), \( XY = 12 \), and we need to find \( ZX \). The ratio of \( XY \) to \( UV \) is \( \frac{12}{3}=4 \).

Step2: Apply the similarity ratio

Since the triangles are similar, the ratio of corresponding sides is equal. So, \( ZX \) corresponds to \( UW \). Multiply the length of \( UW \) by the ratio. \( UW = 5 \), ratio is 4, so \( ZX = 5\times4 = 20 \)? Wait, no, wait. Wait, let's check the angles. Wait, \( \triangle UVW \): right angle at \( V \), so \( UV = 3 \), \( VW = 4 \), \( UW = 5 \) (3-4-5 triangle). \( \triangle XYZ \): right angle at \( Y \), \( XY = 12 \), so the corresponding side to \( UV \) is \( XY \), and corresponding side to \( UW \) is \( ZX \), and corresponding side to \( VW \) is \( YZ \). So the ratio of similarity is \( \frac{XY}{UV}=\frac{12}{3} = 4 \). Then \( ZX \) (corresponding to \( UW \)) is \( UW\times4 = 5\times4 = 20 \)? Wait, but let's check again. Wait, maybe I mixed up the correspondence. Let's see the angles: \( \angle W \) corresponds to \( \angle Z \), \( \angle V \) (right angle) corresponds to \( \angle Y \) (right angle), \( \angle U \) corresponds to \( \angle X \). So sides: \( VW \) (4) corresponds to \( YZ \), \( UV \) (3) corresponds to \( XY \) (12), \( UW \) (5) corresponds to \( ZX \). So ratio is \( \frac{XY}{UV}=\frac{12}{3}=4 \). Then \( ZX = UW\times4 = 5\times4 = 20 \). Wait, but the options have D as 20. Wait, but let me check again. Wait, maybe I made a mistake. Wait, \( XY \) is 12, \( UV \) is 3, so scale factor is 4. Then \( UW \) is 5, so \( ZX = 5\times4 = 20 \). Yes.

Answer:

D. 20