Question

$\overline{wx} \parallel \overline{vy}$. find $vw$.

$vw = \square$

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Explanation:

Step1: Identify Similar Triangles

Since $\overline{WX} \parallel \overline{VY}$, by the Basic Proportionality Theorem (Thales' theorem), $\triangle WXZ \sim \triangle VYZ$. So the ratios of corresponding sides are equal: $\frac{VW + VZ}{VZ} = \frac{XY + YZ}{YZ}$. Wait, actually, let's define the sides. Let $VW = x$, $VZ = 14$, $XY = 6$, $YZ = 13 - 6 = 7$? Wait, no, the total length from X to Z is 13? Wait, the diagram shows X to Y is 6, and the vertical segment is 13, so Y to Z is $13 - 6 = 7$? Wait, no, maybe the sides are $XY = 6$, $YZ = 13$? Wait, no, the two segments on the right: X to Y is 6, and the vertical line (maybe XZ) is 13? Wait, no, the triangles: $\triangle WXZ$ and $\triangle VYZ$ are similar because $WX \parallel VY$. So corresponding sides: $WX$ corresponds to $VY$, $XZ$ corresponds to $YZ$, and $WZ$ corresponds to $VZ$. Wait, $WZ = VW + VZ = x + 14$, $VZ = 14$, $XZ = XY + YZ = 6 + 13 = 19$? No, maybe I misread. Wait, the right side: X to Y is 6, Y to Z is 13? Wait, the diagram has X---Y with length 6, and Y---Z with length 13? Then XZ is 6 + 13 = 19? No, the vertical segment (maybe XZ) is 13, and XY is 6, so YZ is 13 - 6 = 7? Wait, the tick marks on the vertical line suggest that XZ and the other vertical line are equal? No, the tick marks are on the right vertical line, maybe indicating that XZ is 13, and XY is 6, so YZ is 13 - 6 = 7. Then, since $WX \parallel VY$, $\triangle WXY \sim \triangle VZY$? No, $\triangle WXZ \sim \triangle VYZ$ by AA similarity (since $WX \parallel VY$, corresponding angles are equal). So the ratio of sides: $\frac{VW}{VZ} = \frac{XY}{YZ}$. Wait, $XY = 6$, $YZ = 13 - 6 = 7$? No, maybe $YZ = 13$, $XY = 6$, so $XZ = 6 + 13 = 19$? No, this is confusing. Wait, let's start over. Let’s denote:

- $VW = x$ (what we need to find)
- $VZ = 14$
- $XY = 6$
- $YZ = 13$ (maybe the length from Y to Z is 13, and X to Y is 6, so XZ is 6 + 13 = 19? No, the vertical segment (XZ) is 13, and XY is 6, so YZ is 13 - 6 = 7. Wait, the two segments on the right: X to Y is 6, Y to Z is 7 (since 6 + 7 = 13). Then, since $WX \parallel VY$, $\triangle WXZ \sim \triangle VYZ$ (AA similarity, because $\angle X = \angle Y$ (alternate interior angles) and $\angle Z$ is common). So the ratio of corresponding sides: $\frac{WX}{VY} = \frac{XZ}{YZ} = \frac{WZ}{VZ}$. Wait, $WZ = VW + VZ = x + 14$, $VZ = 14$, $XZ = XY + YZ = 6 + 7 = 13$? No, XZ is 13, XY is 6, YZ is 7. Then $\frac{VW + VZ}{VZ} = \frac{XZ}{YZ}$? No, $\frac{WZ}{VZ} = \frac{XZ}{YZ}$. So $\frac{x + 14}{14} = \frac{13}{7}$? Wait, no, $XZ$ is 13, $YZ$ is 7? Then $\frac{x + 14}{14} = \frac{13}{7}$. Solving: $x + 14 = 14 \times \frac{13}{7} = 26$, so $x = 26 - 14 = 12$. Wait, that makes sense. Let's check: If $VW = 12$, $VZ = 14$, so $WZ = 26$. $XZ = 13$, $YZ = 7$. Then $\frac{26}{14} = \frac{13}{7}$, which simplifies to $\frac{13}{7} = \frac{13}{7}$. Correct. And $\frac{XY}{YZ} = \frac{6}{7}$? No, wait, maybe the ratio is $\frac{VW}{VZ} = \frac{XY}{YZ}$. Wait, $XY = 6$, $YZ = 7$, $VZ = 14$, so $\frac{x}{14} = \frac{6}{7}$, then $x = 14 \times \frac{6}{7} = 12$. Yes, that's better. So by the Basic Proportionality Theorem (Thales' theorem), since $WX \parallel VY$, the line $VY$ divides $WZ$ and $XZ$ proportionally. So $\frac{VW}{VZ} = \frac{XY}{YZ}$. Here, $XY = 6$, $YZ = 13 - 6 = 7$ (since XZ is 13, XY is 6, so YZ is 13 - 6 = 7), $VZ = 14$. So $\frac{VW}{14} = \frac{6}{7}$. Solving for $VW$: $VW = 14 \times \frac{6}{7} = 12$.

Step2: Solve for VW

Using the proportion from similar triangles: $\frac{VW}{VZ} = \frac{XY}{YZ}$. Substitute $VZ = 14$, $XY = 6$, $YZ = 13 - 6 = 7$. So $\frac{VW}{14} = \frac{6}{7}$. Multiply both sides by 14: $VW = 14 \times \frac{6}{7} = 12$.

Answer:

12