Question

solve the triangle.
write each answer as an integer or as a decimal rounded to the nearest tenth.
m∠p =
m∠q =
m∠r =

Explanation:

Step1: Use the Law of Cosines for angle P

The Law of Cosines is \(q^{2}=p^{2}+r^{2}-2pr\cos Q\). For \(\cos P=\frac{q^{2}+r^{2}-p^{2}}{2qr}\), where \(p = 12\), \(q = 5\), \(r = 11\).
\(\cos P=\frac{5^{2}+11^{2}-12^{2}}{2\times5\times11}=\frac{25 + 121-144}{110}=\frac{2}{110}=\frac{1}{55}\)
\(P=\cos^{-1}(\frac{1}{55})\approx 88.9^{\circ}\)

Step2: Use the Law of Cosines for angle Q

\(\cos Q=\frac{p^{2}+r^{2}-q^{2}}{2pr}\)
\(\cos Q=\frac{12^{2}+11^{2}-5^{2}}{2\times12\times11}=\frac{144 + 121 - 25}{264}=\frac{240}{264}=\frac{10}{11}\)
\(Q=\cos^{-1}(\frac{10}{11})\approx 24.6^{\circ}\)

Step3: Use the angle - sum property of a triangle

Since the sum of the interior angles of a triangle is \(180^{\circ}\), \(R=180^{\circ}-P - Q\)
\(R = 180^{\circ}-88.9^{\circ}-24.6^{\circ}=66.5^{\circ}\)

Answer:

\(m\angle P\approx88.9^{\circ}\)
\(m\angle Q\approx24.6^{\circ}\)
\(m\angle R\approx66.5^{\circ}\)