Question

solving for all unknown measures in a triangle
determine the measures of all unknown angles and side lengths of △pqr. round side lengths to the nearest hundredth.
m∠r = □°
pr ≈ □
pq ≈ □

Explanation:

Step1: Find \( m\angle R \)

The sum of angles in a triangle is \( 180^\circ \). Given \( m\angle P = 99^\circ \) and \( m\angle Q = 31^\circ \), we calculate \( m\angle R \) as:
\[
m\angle R = 180^\circ - 99^\circ - 31^\circ = 50^\circ
\]

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 \( QR = 11 \) (opposite \( \angle P \)), \( PR \) opposite \( \angle Q \), and \( PQ \) opposite \( \angle R \). For \( PR \):
\[
\frac{PR}{\sin Q} = \frac{QR}{\sin P} \implies PR = \frac{QR \cdot \sin Q}{\sin P}
\]
Substitute \( QR = 11 \), \( \sin Q = \sin 31^\circ \approx 0.5150 \), \( \sin P = \sin 99^\circ \approx 0.9877 \):
\[
PR = \frac{11 \cdot 0.5150}{0.9877} \approx \frac{5.665}{0.9877} \approx 5.74
\]

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

For \( PQ \):
\[
\frac{PQ}{\sin R} = \frac{QR}{\sin P} \implies PQ = \frac{QR \cdot \sin R}{\sin P}
\]
Substitute \( QR = 11 \), \( \sin R = \sin 50^\circ \approx 0.7660 \), \( \sin P = \sin 99^\circ \approx 0.9877 \):
\[
PQ = \frac{11 \cdot 0.7660}{0.9877} \approx \frac{8.426}{0.9877} \approx 8.53
\]

Answer:

\( m\angle R = 50^\circ \), \( PR \approx 5.74 \), \( PQ \approx 8.53 \)