Question

if ( st = 89 ), ( tr = 102 ), and ( wu = 51 ), find the length of ( overline{vw} ). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since \(\angle R=\angle U = 55^{\circ}\), \(\angle T=\angle W=56^{\circ}\), \(\angle S=\angle V = 69^{\circ}\), \(\triangle RST\sim\triangle UVW\) (by AAA - Angle - Angle - Angle similarity criterion).

Step2: Set up proportion

For similar triangles \(\triangle RST\) and \(\triangle UVW\), the ratios of corresponding sides are equal. That is \(\frac{ST}{VW}=\frac{TR}{WU}\).
We know \(ST = 89\), \(TR = 102\), \(WU=51\). Substitute these values into the proportion: \(\frac{89}{VW}=\frac{102}{51}\).

Step3: Solve for \(VW\)

Cross - multiply: \(102\times VW=89\times51\). Then \(VW=\frac{89\times51}{102}\).
Since \(102 = 2\times51\), \(VW=\frac{89\times51}{2\times51}\). Cancel out the common factor \(51\). So \(VW=\frac{89}{2}=44.5\).

Answer:

\(44.5\)