Question

if ( st = 34 ), ( tr = 30 ), and ( wu = 6 ), find the length of ( overline{vw} ). 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 \(\triangle STR\), using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), the third angle \(\angle R=180^{\circ}-(97^{\circ}+38^{\circ}) = 45^{\circ}\).
In \(\triangle VWU\), using the angle - sum property of a triangle (\(A + B + C = 180^{\circ}\)), the third angle \(\angle W=180^{\circ}-(38^{\circ}+45^{\circ})=97^{\circ}\).

Step2: Prove the similarity of triangles

Since \(\angle S=\angle V = 38^{\circ}\), \(\angle R=\angle U = 45^{\circ}\), and \(\angle T=\angle W = 97^{\circ}\), by the AA (angle - angle) similarity criterion, \(\triangle STR\sim\triangle VWU\).

Step3: Set up the proportion for similar triangles

For similar triangles \(\triangle STR\) and \(\triangle VWU\), the ratios of corresponding sides are equal. That is \(\frac{ST}{VW}=\frac{TR}{WU}\).

Step4: Substitute the given values into the proportion

We know that \(ST = 34\), \(TR = 30\), and \(WU = 6\). Substituting these values into \(\frac{ST}{VW}=\frac{TR}{WU}\), we get \(\frac{34}{VW}=\frac{30}{6}\).

Step5: Solve for \(VW\)

Cross - multiply: \(30\times VW=34\times6\). Then \(VW=\frac{34\times6}{30}\).

$$VW=\frac{204}{30}=6.8$$

Answer:

\(6.8\)