Question

1. \\( \triangle xzw \sim \triangle xyv \\), find the perimeter of \\( \triangle xzw \\).

Explanation:

Step1: Find the ratio of similarity

Since \(\triangle XZW\sim\triangle XYV\), the ratio of similarity \(k=\frac{XW}{XV}\). Given \(XW = 32 + 40=72\) and \(XV = 40\), so \(k=\frac{72}{40}=\frac{9}{5}\).

Step2: Find the length of \(ZY\)

Let \(ZY = x\). Using the ratio of similarity \(\frac{ZW}{YV}=\frac{XW}{XV}\). We know \(YV = 30\), so \(\frac{x + 28}{28}=\frac{9}{5}\). Cross - multiply: \(5(x + 28)=9\times28\), \(5x+140 = 252\), \(5x=252 - 140=112\), \(x=\frac{112}{5} = 22.4\). Then \(ZW=22.4 + 28=50.4\).

Step3: Calculate the perimeter of \(\triangle XZW\)

The perimeter of \(\triangle XYV\) is \(P_{XYV}=30 + 28+40 = 98\). Using the ratio of similarity, if the ratio of similarity \(k = \frac{9}{5}\), and let the perimeter of \(\triangle XZW\) be \(P_{XZW}\). Since for similar triangles \(\frac{P_{XZW}}{P_{XYV}}=\frac{XW}{XV}\), then \(P_{XZW}=\frac{9}{5}\times98=\frac{882}{5}=176.4\).

Another way:

Step1: Find the ratio of similarity

Since \(\triangle XZW\sim\triangle XYV\), the ratio of similarity \(k=\frac{XW}{XV}\). \(XW=32 + 40 = 72\), \(XV = 40\), so \(k=\frac{72}{40}=\frac{9}{5}\).

Step2: Calculate the perimeter of \(\triangle XZW\)

The sides of \(\triangle XYV\) are \(a = 28\), \(b = 30\), \(c = 40\). The perimeter of \(\triangle XYV\) is \(P_{XYV}=28 + 30+40=98\). For similar triangles, the ratio of perimeters is equal to the ratio of corresponding side lengths. Let the perimeter of \(\triangle XZW\) be \(P_{XZW}\). Then \(P_{XZW}=P_{XYV}\times\frac{XW}{XV}\). Substitute \(P_{XYV} = 98\), \(\frac{XW}{XV}=\frac{72}{40}=\frac{9}{5}\), so \(P_{XZW}=98\times\frac{9}{5}=\frac{98\times9}{5}=\frac{882}{5}=176.4\)

Answer:

The perimeter of \(\triangle XZW\) is \(176.4\)