Question

wx || vy. find uv.
x 12 y 9 u
diagram of a triangle with a line segment parallel to the base
wv = 28 (assumed from the diagram, as the number 28 is visible near the segment wv)

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{WX} \parallel \overline{VY} \), by the Basic Proportionality Theorem (Thales' theorem), triangles \( \triangle WXU \) and \( \triangle VYU \) are similar. So, the ratios of corresponding sides are equal: \( \frac{UY}{UX} = \frac{UV}{UW} \).

Step2: Calculate \( UX \)

\( UX = UY + YX = 9 + 12 = 21 \). Let \( UV = x \), then \( UW = UV + VW = x + 28 \).

Step3: Set Up Proportion

From similarity, \( \frac{9}{21} = \frac{x}{x + 28} \). Cross - multiply: \( 9(x + 28)=21x \).

Step4: Solve for \( x \)

Expand: \( 9x + 252 = 21x \). Subtract \( 9x \): \( 252 = 12x \). Divide by 12: \( x=\frac{252}{12}=21 \). Wait, no, wait. Wait, maybe the segments are \( WU \) and \( WV \)? Wait, maybe I mixed up. Let's re - examine. If \( WX \parallel VY \), then \( \triangle UVY \sim \triangle UWX \). So \( \frac{UV}{UW}=\frac{UY}{UX} \). Let \( UV = x \), \( UW=x + 28 \), \( UY = 9 \), \( UX=9 + 12=21 \). So \( \frac{x}{x + 28}=\frac{9}{21}=\frac{3}{7} \). Cross - multiply: \( 7x=3(x + 28) \), \( 7x = 3x+84 \), \( 4x = 84 \), \( x = 21 \)? Wait, no, that can't be. Wait, maybe the length of \( VW \) is 28? Wait, maybe the correct proportion is \( \frac{UY}{UX}=\frac{UV}{UW} \), but \( UW=UV + VW \). Wait, maybe I made a mistake in the segment labels. Let's assume that \( VW = 28 \), and we need to find \( UV \). Let \( UV = x \), then \( UW=x + 28 \). The ratio of \( UY \) to \( UX \) is \( \frac{9}{9 + 12}=\frac{9}{21}=\frac{3}{7} \). So \( \frac{UV}{UW}=\frac{3}{7}\), so \( \frac{x}{x + 28}=\frac{3}{7} \). Cross - multiply: \( 7x=3x + 84 \), \( 4x = 84 \), \( x = 21 \). Wait, but let's check again. If \( UV = 21 \), then \( UW=21 + 28 = 49 \). Then \( \frac{21}{49}=\frac{3}{7}\), and \( \frac{9}{21}=\frac{3}{7} \), so it works.

Answer:

\( UV = 21 \)