Question

$\overline{yz} \parallel \overline{vx}$. find $wy$.

(image of a geometric figure with segments: zv = 14, vw = 7, xy = 30, and we need to find wy. the figure has points z, y, w, v, x with $\overline{yz} \parallel \overline{vx}$.)

$wy = \square$

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Explanation:

Step1: Identify Similar Triangles

Since \(\overline{YZ} \parallel \overline{VX}\), by the Basic Proportionality Theorem (Thales' theorem), \(\triangle WVX \sim \triangle WYZ\) (similar triangles). So the ratios of corresponding sides are equal.

Step2: Set Up Proportion

Let \(WY = x\). Then \(WX = x - 30\) (wait, no, actually, looking at the sides: \(WV = 7\), \(WZ = WV + VZ = 7 + 14 = 21\), \(WX\) and \(WY\) are corresponding, \(WV\) and \(WZ\) are corresponding. So the proportion is \(\frac{WV}{WZ}=\frac{WX}{WY}\)? Wait, no, better: \(\frac{WV}{WZ}=\frac{WX}{WY}\)? Wait, actually, \(VZ = 14\), \(WV = 7\), so \(WZ = WV + VZ = 21\). And \(XY = 30\), let \(WX = y\), then \(WY = WX + XY = y + 30\). Since \(\triangle WVX \sim \triangle WYZ\), \(\frac{WV}{WZ}=\frac{WX}{WY}\), so \(\frac{7}{21}=\frac{y}{y + 30}\). Wait, no, maybe I mixed up. Wait, actually, the sides: \(WV = 7\), \(WZ = 7 + 14 = 21\), \(WX\) and \(WY\) are corresponding, \(VX\) and \(YZ\) are corresponding, but maybe the correct proportion is \(\frac{WV}{WZ}=\frac{WX}{WY}\). Wait, let's re - express. Since \(\overline{VX}\parallel\overline{YZ}\), \(\angle WVX=\angle WYZ\) (corresponding angles) and \(\angle W\) is common, so \(\triangle WVX\sim\triangle WYZ\) by AA similarity. Therefore, \(\frac{WV}{WY}=\frac{WZ}{WX}\)? No, wait, corresponding sides: \(WV\) corresponds to \(WY\)? No, \(WV\) is part of \(WZ\), and \(WX\) is part of \(WY\). Wait, \(WZ = WV + VZ=7 + 14 = 21\), \(WY = WX + XY\), and \(XY = 30\). So the ratio of similarity is \(\frac{WV}{WZ}=\frac{WX}{WY}\), so \(\frac{7}{21}=\frac{WX}{WX + 30}\). Solving \(\frac{1}{3}=\frac{WX}{WX + 30}\), cross - multiply: \(WX + 30 = 3WX\), \(30 = 2WX\), \(WX = 15\). Then \(WY = WX + XY = 15 + 30 = 45\)? Wait, no, that can't be. Wait, maybe the proportion is \(\frac{WV}{VZ}=\frac{WX}{XY}\)? Wait, no, let's look again. The correct proportion: since \(VX\parallel YZ\), the triangles \(\triangle WVX\) and \(\triangle WYZ\) are similar, so \(\frac{WV}{WZ}=\frac{WX}{WY}\). \(WV = 7\), \(WZ = 7 + 14 = 21\), so \(\frac{7}{21}=\frac{1}{3}\). Let \(WX = k\), then \(WY = k + 30\). So \(\frac{1}{3}=\frac{k}{k + 30}\), \(k + 30 = 3k\), \(2k = 30\), \(k = 15\). Then \(WY = 15+30 = 45\). Wait, but let's check again. Alternatively, \(\frac{WV}{WZ}=\frac{WX}{WY}\), \(WV = 7\), \(WZ = 21\), so the ratio is \(\frac{1}{3}\). So \(WX=\frac{1}{3}WY\). But \(WY=WX + 30\), so \(WY=\frac{1}{3}WY+30\), \(\frac{2}{3}WY = 30\), \(WY = 30\times\frac{3}{2}=45\). Wait, that makes sense.

Answer:

\(45\)