Question

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Explanation:

Step1: Identify Similar Triangles

Since \(\overline{XY} \parallel \overline{UW}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle VUW \sim \triangle VXY\) (similar triangles) because corresponding angles are equal (alternate interior angles due to parallel lines).

Step2: Set Up Proportion

For similar triangles, the ratios of corresponding sides are equal. So, \(\frac{VW}{VX} = \frac{VU}{VY}\). First, find \(VX\): \(VX = VW + WX = 18 + 36 = 54\). Let \(VY = x\), then \(VU = x - 14\) (since \(YU = 14\), so \(VY = VU + UY\) implies \(VU = VY - UY\)). Wait, actually, from the diagram, \(VU = 14\)? Wait, no, the side \(VU\) is part of \(VY\). Wait, correct proportion: \(\frac{VW}{VX} = \frac{VU}{VY}\). Wait, \(VW = 18\), \(VX = VW + WX = 18 + 36 = 54\), \(VU = 14\)? No, wait, the side \(VU\) is 14? Wait, no, the length from \(V\) to \(U\) is 14, and from \(U\) to \(Y\) is... Wait, no, the triangle: \(V\) to \(U\) is 14, \(U\) to \(Y\) is unknown, and \(V\) to \(Y\) is what we need. Wait, actually, the correct proportion is \(\frac{VW}{VX} = \frac{VU}{VY}\). Wait, \(VW = 18\), \(VX = 18 + 36 = 54\), \(VU = 14\)? No, that can't be. Wait, maybe I mixed up the sides. Let's re-express: Since \(\overline{UW} \parallel \overline{XY}\), the triangles \(\triangle VUW\) and \(\triangle VXY\) are similar. So, \(\frac{VW}{VX} = \frac{VU}{VY}\). Wait, \(VW = 18\), \(VX = VW + WX = 18 + 36 = 54\), so \(\frac{18}{54} = \frac{VU}{VY}\). Simplify \(\frac{18}{54} = \frac{1}{3}\). Wait, but \(VU\) is 14? No, that's not right. Wait, maybe the side \(VU\) is 14, and \(VY\) is \(VU + UY\), but actually, the correct proportion is \(\frac{VW}{VX} = \frac{VU}{VY}\). Wait, no, maybe it's \(\frac{VW}{WX} = \frac{VU}{UY}\)? Wait, no, Thales' theorem: If a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. So, in \(\triangle VXY\), line \(UW\) is parallel to \(XY\), intersecting \(VX\) at \(W\) and \(VY\) at \(U\). So, \(\frac{VW}{WX} = \frac{VU}{UY}\). Ah, that's the correct application! So, \(VW = 18\), \(WX = 36\), \(VU = 14\), and \(UY = VY - VU = VY - 14\). So, \(\frac{VW}{WX} = \frac{VU}{UY}\) becomes \(\frac{18}{36} = \frac{14}{VY - 14}\). Simplify \(\frac{18}{36} = \frac{1}{2}\), so \(\frac{1}{2} = \frac{14}{VY - 14}\). Cross-multiplying: \(VY - 14 = 28\), so \(VY = 28\).

Answer:

28