Question

solving for all unknown measures in a triangle
determine the measures of all unknown angles and side lengths of (\triangle pqr). round side lengths to the nearest hundredth.
(mangle r=)
(pr=)
(pq=)

Explanation:

Step1: Find \(m\angle R\)

The sum of angles in a triangle is \(180^{\circ}\). So \(m\angle R=180^{\circ}-(98^{\circ} + 31^{\circ})\)

$$m\angle R=180^{\circ}-129^{\circ}=51^{\circ}$$

Step2: Use the Law of Sines to find \(PR\)

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let \(a = PR\), \(A = 31^{\circ}\), \(b = 11\), \(B=98^{\circ}\)
\(\frac{PR}{\sin31^{\circ}}=\frac{11}{\sin98^{\circ}}\)

$$PR=\frac{11\times\sin31^{\circ}}{\sin98^{\circ}}$$

$$PR=\frac{11\times0.5150}{0.9903}\approx5.73$$

Step3: Use the Law of Sines to find \(PQ\)

Let \(a = PQ\), \(A = 51^{\circ}\), \(b = 11\), \(B = 98^{\circ}\)
\(\frac{PQ}{\sin51^{\circ}}=\frac{11}{\sin98^{\circ}}\)

$$PQ=\frac{11\times\sin51^{\circ}}{\sin98^{\circ}}$$

$$PQ=\frac{11\times0.7771}{0.9903}\approx8.65$$

Answer:

\(m\angle R = 51^{\circ}\), \(PR\approx5.73\), \(PQ\approx8.65\)