Questions: Determine the remaining sides and angles of the triangle ABC. c=9 mi, B=35.49°, C=30.07° Find the measure of angle A. A=114.44° (Type an integer or a decimal.) Find the length of side a. a= mi (Round to the nearest mile as needed.)

Solution Steps

To solve the triangle, we can use the following approach:

1. Find the measure of angle \( A \): Use the fact that the sum of angles in a triangle is \( 180^\circ \). Subtract the given angles \( B \) and \( C \) from \( 180^\circ \) to find angle \( A \).

2. Find the length of side \( a \): Use the Law of Sines, which states that \(\frac{a}{\sin A} = \frac{c}{\sin C}\). Rearrange this formula to solve for \( a \).

To find angle \( A \), we use the fact that the sum of the angles in a triangle is \( 180^\circ \): \[ A = 180^\circ - B - C = 180^\circ - 35.49^\circ - 30.07^\circ = 114.44^\circ \]

Using the Law of Sines, we can find the length of side \( a \): \[ \frac{a}{\sin A} = \frac{c}{\sin C} \] Rearranging gives: \[ a = \frac{c \cdot \sin A}{\sin C} \] Substituting the known values: \[ a = \frac{9 \cdot \sin(114.44^\circ)}{\sin(30.07^\circ)} \approx 16.3525 \] Rounding to the nearest mile, we find: \[ a \approx 16 \]

Final Answer