Questions: In triangle RST, m angle R=14°, ST=83 feet, and RT=125 feet. Find the measure of angle S.

Solution Steps

To find the measure of \(\angle S\) in \(\triangle RST\), we can use the Law of Sines. The Law of Sines states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in a triangle. We can set up the equation using the known values and solve for \(\angle S\).

In triangle \(RST\), we have the following information:

• \(m \angle R = 14^\circ\)
• \(ST = 83\) feet
• \(RT = 125\) feet

Using the Law of Sines, we can express the relationship between the sides and angles as follows: \[ \frac{ST}{\sin R} = \frac{RT}{\sin S} \] Substituting the known values: \[ \frac{83}{\sin(14^\circ)} = \frac{125}{\sin S} \]

Rearranging the equation to solve for \(\sin S\): \[ \sin S = \frac{125 \cdot \sin(14^\circ)}{83} \] Calculating \(\sin(14^\circ)\) gives approximately \(0.2419\). Thus: \[ \sin S \approx \frac{125 \cdot 0.2419}{83} \approx 0.1606 \]

To find \(\angle S\), we take the inverse sine: \[ \angle S = \arcsin(0.1606) \approx 9.244^\circ \]

Final Answer