Question

if ( us = 54 ), ( st = 82 ), and ( vw = 45 ), find the length of ( overline{xv} ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Find the third angle of each triangle

In \(\triangle UST\), using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)). Let \(\angle U = 180^{\circ}-\angle S-\angle T\). Given \(\angle S = 60^{\circ}\), \(\angle T = 40^{\circ}\), then \(\angle U=180^{\circ}-60^{\circ}-40^{\circ}=80^{\circ}\).
In \(\triangle WXV\), using the angle - sum property of a triangle. Let \(\angle W = 180^{\circ}-\angle X-\angle V\). Given \(\angle X = 80^{\circ}\), \(\angle V = 40^{\circ}\), then \(\angle W = 60^{\circ}\).

Step2: Prove the similarity of triangles

Since \(\angle U=\angle X = 80^{\circ}\), \(\angle S=\angle W = 60^{\circ}\), \(\angle T=\angle V = 40^{\circ}\), by the AA (Angle - Angle) similarity criterion, \(\triangle UST\sim\triangle WXV\).

Step3: Set up the proportion

For similar triangles \(\triangle UST\) and \(\triangle WXV\), the ratios of corresponding sides are equal. That is \(\frac{US}{WX}=\frac{ST}{XV}\). We know \(US = 54\), \(ST = 82\), \(WX = 45\). Substituting these values into the proportion \(\frac{54}{45}=\frac{82}{XV}\).

Step4: Solve for \(XV\)

Cross - multiply: \(54\times XV=45\times82\). Then \(XV=\frac{45\times82}{54}\).

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Answer:

\(68.3\)