Question

wx || vy. find xz.
18
6
3
xz =
submit

Explanation:

Step1: Identify Similar Triangles

Since \( \overline{WX} \parallel \overline{VY} \), by the Basic Proportionality Theorem (Thales' theorem), triangles \( \triangle WXZ \) and \( \triangle VYZ \) are similar. So, the ratios of corresponding sides are equal.

Step2: Set Up Proportion

Let \( XZ = x \). Then \( YZ = x - 3 \) (since \( XY = 3 \)). The ratio of \( WZ \) to \( VZ \) should equal the ratio of \( XZ \) to \( YZ \). \( WZ = 18 \), \( VZ = 6 \), so \( \frac{18}{6}=\frac{x}{x - 3} \).

Step3: Solve the Proportion

Simplify \( \frac{18}{6}=3 \), so \( 3=\frac{x}{x - 3} \). Cross - multiply: \( 3(x - 3)=x \). Expand: \( 3x-9 = x \). Subtract \( x \) from both sides: \( 2x-9 = 0 \). Add 9 to both sides: \( 2x = 9 \)? Wait, no, wait. Wait, maybe I mixed up the sides. Wait, actually, the sides: \( WZ = 18 \), \( VZ = 6 \), so the ratio of similarity is \( \frac{WZ}{VZ}=\frac{18}{6}=3 \). So the ratio of \( XZ \) to \( YZ \) should be 3? Wait, no, maybe the segments: \( WZ = WV+VZ=18 \), \( WV = 18 - 6=12 \)? No, wait the diagram: \( W \) to \( V \) is 6, \( V \) to \( Z \) is 18? Wait, no, the black segment is \( WZ = 18 \), and \( WV = 6 \), so \( VZ=18 - 6 = 12 \)? Wait, I think I misidentified the segments. Let's re - examine. The blue triangle: \( WX \parallel VY \), so \( \triangle VYZ\sim\triangle WXZ \). So \( \frac{VZ}{WZ}=\frac{YZ}{XZ} \). \( VZ = 6 \), \( WZ=6 + 12=18 \)? Wait, no, the length from \( W \) to \( Z \) is 18, from \( W \) to \( V \) is 6, so from \( V \) to \( Z \) is \( 18 - 6=12 \). And \( XY = 3 \), let \( XZ=x \), then \( YZ=x - 3 \). So the ratio of \( VZ \) to \( WZ \) is \( \frac{6}{18}=\frac{1}{3} \), and the ratio of \( YZ \) to \( XZ \) is \( \frac{x - 3}{x} \). So \( \frac{x - 3}{x}=\frac{6}{18}=\frac{1}{3} \). Cross - multiply: \( 3(x - 3)=x \). \( 3x-9=x \). \( 2x = 9 \)? No, that's wrong. Wait, maybe the correct proportion is \( \frac{WV}{WZ}=\frac{XY}{XZ} \). \( WV = 6 \), \( WZ = 18 \), \( XY = 3 \). So \( \frac{6}{18}=\frac{3}{XZ} \). Then cross - multiply: \( 6\times XZ=18\times3 \). \( 6XZ = 54 \). Divide both sides by 6: \( XZ = 9 \). Wait, that makes more sense. Because \( \frac{WV}{WZ}=\frac{6}{18}=\frac{1}{3} \), and \( \frac{XY}{XZ}=\frac{3}{XZ} \), so \( \frac{1}{3}=\frac{3}{XZ} \), so \( XZ = 9 \).

Answer:

\( 9 \)