Question

in the figure shown, \\( \overline { w z } \\) and \\( \overline { x y } \\) intersect at point q. yq, wq, wx, and qx are 28, 35, 30, and 60, respectively. what is the length of \\( \overline { y z } \\)?

Explanation:

Step1: Check similarity of triangles

Since \(\angle WQX=\angle YQZ\) (vertically - opposite angles) and \(\frac{WQ}{YQ}=\frac{35}{28}=\frac{5}{4}\), \(\frac{QX}{QZ}=\frac{60}{QZ}\). Also, if \(\frac{WQ}{YQ}=\frac{QX}{QZ}\), then \(\triangle WQX\sim\triangle YQZ\) (by the Side - Angle - Side similarity criterion).

First, find \(QZ\). We know that \(WX = 30\). Let \(QZ=x\).

Since \(\triangle WQX\sim\triangle YQZ\), \(\frac{WQ}{YQ}=\frac{QX}{QZ}=\frac{WX}{YZ}\)

We have \(\frac{WQ}{YQ}=\frac{35}{28}=\frac{5}{4}\), and \(QX = 60\). Let's use \(\frac{WQ}{YQ}=\frac{QX}{QZ}\)

\(\frac{35}{28}=\frac{60}{QZ}\)

Cross - multiply: \(35\times QZ=28\times60\)

\(QZ=\frac{28\times60}{35}\)

\(QZ = 48\)

Step2: Use the similarity ratio to find \(YZ\)

Since \(\frac{WQ}{YQ}=\frac{WX}{YZ}\) and \(\frac{WQ}{YQ}=\frac{5}{4}\), \(WX = 30\)

Let \(YZ = y\), then \(\frac{5}{4}=\frac{30}{y}\)

Cross - multiply: \(5y=4\times30\)

\(5y = 120\)

\(y=\frac{120}{5}=24\)

Answer:

\(24\)