Question

question
find the perimeter of △rst. round your answer to the nearest tenth if necessary. figures are not
necessarily drawn to scale.

Explanation:

Step1: Determine similarity of triangles

Since the angles of \(\triangle OPQ\) and \(\triangle RST\) are equal (\(41^{\circ},81^{\circ},58^{\circ}\)), the two triangles are similar by the AA (Angle - Angle) similarity criterion.

Step2: Find the scale factor

The ratio of corresponding sides. Let's take the side \(PQ = 83\) and \(SR=57.6\). The scale factor \(k=\frac{SR}{PQ}=\frac{57.6}{83}\approx0.694\).

Step3: Calculate the lengths of other sides of \(\triangle RST\)

For side \(OT = 55\), the corresponding side \(ST=x\). Using the scale factor \(x = 55\times k=55\times\frac{57.6}{83}\approx38.2\).
For side \(OP = 72\), the corresponding side \(RT = 44\) (already given).

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

Perimeter \(P=ST + SR+RT\). Substitute \(ST\approx38.2\), \(SR = 57.6\), \(RT = 44\).
\(P\approx38.2+57.6 + 44=139.8\)

Answer:

\(139.8\)