Question

in triangle t r s, side t s is 14, angle at r is 77°, angle at s is 58°, find the length of side t r (denoted as s). write your answer as an integer or as a decimal rounded to the nearest tenth. s = blank submit

Explanation:

Step1: Find angle at T

The sum of angles in a triangle is \(180^\circ\). So, \(\angle T = 180^\circ - 77^\circ - 58^\circ = 45^\circ\).

Step2: Apply Law of Sines

Law of Sines states \(\frac{s}{\sin S}=\frac{14}{\sin R}\). We know \(\angle S = 58^\circ\), \(\angle R = 77^\circ\), and the side opposite \(\angle R\) is \(14\), side opposite \(\angle S\) is \(s\). So, \(\frac{s}{\sin 58^\circ}=\frac{14}{\sin 77^\circ}\).

Step3: Solve for s

First, calculate \(\sin 58^\circ\approx0.8480\) and \(\sin 77^\circ\approx0.9744\). Then, \(s=\frac{14\times\sin 58^\circ}{\sin 77^\circ}=\frac{14\times0.8480}{0.9744}\approx\frac{11.872}{0.9744}\approx12.2\).

Answer:

\(12.2\)