Question

find the perimeter of \\( \triangle u v w \\). round your answer to the nearest tenth if necessary. figures are not necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since \(\angle R = 34^{\circ}\), \(\angle T=49^{\circ}\), \(\angle S = 97^{\circ}\) in \(\triangle RST\) and \(\angle W = 49^{\circ}\), \(\angle V = 97^{\circ}\), \(\angle U=34^{\circ}\) in \(\triangle UVW\), by AA (angle - angle) similarity criterion \(\triangle RST\sim\triangle UVW\).

Step2: Find the scale factor

The ratio of corresponding sides. Let's take the pair of sides \(RS = 20\) and \(WV=37.5\). The scale factor \(k=\frac{WV}{RS}=\frac{37.5}{20}=\frac{15}{8}\)

Step3: Find the other sides of \(\triangle UVW\)

For side \(UW\): Since \(RT = 27\) in \(\triangle RST\), then \(UW=27\times\frac{15}{8}=\frac{405}{8}=50.625\)
For side \(UV\): Since \(ST = 15\) in \(\triangle RST\), then \(UV = 15\times\frac{15}{8}=\frac{225}{8}=28.125\)

Step4: Calculate the perimeter of \(\triangle UVW\)

Perimeter \(P=37.5 + 50.625+28.125\)
\(P=37.5+(50.625 + 28.125)\)
\(P=37.5 + 78.75\)
\(P = 116.25\)

Answer:

\(116.3\)