Question

if ( pq = 37 ), ( qo = 32 ), ( po = 42 ), ( tr = 16 ), and ( sr = 21 ), find the perimeter of ( \triangle rst ). 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 \(\triangle PQO\), using the angle - sum property of a triangle (\(\angle Q+\angle P+\angle O = 180^{\circ}\)), we have \(\angle O=180^{\circ}-(74^{\circ}+48^{\circ}) = 58^{\circ}\).
In \(\triangle RST\), \(\angle T=180^{\circ}-(48^{\circ}+58^{\circ}) = 74^{\circ}\).
Since \(\angle Q=\angle T = 74^{\circ}\), \(\angle P=\angle S = 48^{\circ}\), \(\angle O=\angle R = 58^{\circ}\), \(\triangle PQO\sim\triangle TSR\) (by AAA similarity criterion).

Step2: Find the ratio of similarity

The ratio of similarity \(k=\frac{TR}{QO}\). Given \(QO = 32\) and \(TR = 16\), so \(k=\frac{16}{32}=\frac{1}{2}\).

Step3: Find the lengths of \(ST\) and \(TS\)

If \(\triangle PQO\sim\triangle TSR\), then \(\frac{SR}{PO}=\frac{TR}{QO}=\frac{ST}{PQ}\).
We know \(PQ = 37\), \(PO = 42\), \(QO = 32\), \(TR = 16\), \(SR = 21\).
Since \(\frac{ST}{PQ}=\frac{1}{2}\), then \(ST=\frac{1}{2}PQ=\frac{37}{2}=18.5\).
Since \(\frac{TS}{PO}=\frac{1}{2}\), then \(TS=\frac{1}{2}PO = 21\).

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

The perimeter of \(\triangle RST\) is \(P=ST + SR+TR\).
Substitute \(ST = 18.5\), \(SR = 21\), \(TR = 16\) into the formula: \(P=18.5+21 + 16\).
\(P=55.5\).

Answer:

\(55.5\)