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 \(A = 180-(41 + 62)=77^{\circ}\).

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 \(41^{\circ}\) be \(a\), opposite \(62^{\circ}\) be \(b\), and opposite \(77^{\circ}\) (the side of length \(29\)) be \(c\).
First, find \(a\): \(\frac{a}{\sin41^{\circ}}=\frac{29}{\sin77^{\circ}}\), so \(a=\frac{29\times\sin41^{\circ}}{\sin77^{\circ}}\approx\frac{29\times0.6561}{0.9744}\approx19.5\).
Then, find \(b\): \(\frac{b}{\sin62^{\circ}}=\frac{29}{\sin77^{\circ}}\), so \(b = \frac{29\times\sin62^{\circ}}{\sin77^{\circ}}\approx\frac{29\times0.8829}{0.9744}\approx26.3\).

Step3: Calculate the perimeter

Perimeter \(P=a + b+ c\). Substituting \(a\approx19.5\), \(b\approx26.3\), and \(c = 29\), we get \(P\approx19.5+26.3 + 29=74.8\).

Answer:

\(74.8\)