Question

find the perimeter of the triangle to the right. the perimeter is (type an integer or decimal rounded to the nearest tenth as needed.)

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\). Then \(A = 180-(42 + 69)=69^{\circ}\).

Step2: Use the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). Let the side opposite \(42^{\circ}\) be \(a\), opposite \(69^{\circ}\) (one of them) be \(b = 22\), and the other side (opposite \(69^{\circ}\)) be \(c\).
Since two angles (\(69^{\circ}\) and \(69^{\circ}\)) are equal, the sides opposite them are equal. So \(c = 22\).
For \(a\): \(\frac{a}{\sin42^{\circ}}=\frac{22}{\sin69^{\circ}}\), then \(a=\frac{22\times\sin42^{\circ}}{\sin69^{\circ}}\).
\(\sin42^{\circ}\approx0.6691\), \(\sin69^{\circ}\approx0.9336\).
\(a=\frac{22\times0.6691}{0.9336}\approx15.8\).

Step3: Calculate the perimeter

Perimeter \(P=a + b+ c\). Substitute \(a\approx15.8\), \(b = 22\), \(c = 22\).
\(P=15.8+22 + 22=59.8\).

Answer:

\(59.8\)