Question

find the length of side a.
round to the nearest tenth.

Explanation:

Step1: Find angle \( A \)

The sum of angles in a triangle is \( 180^{\circ} \). So \( A=180^{\circ}-100^{\circ}-35^{\circ}=45^{\circ} \)

Step2: Apply the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \). We want to find \( a \), and we know \( B = 100^{\circ}\), \( C=35^{\circ}\), \(A = 45^{\circ}\), and \(b = 15\). Using \( \frac{a}{\sin A}=\frac{b}{\sin B}\), we substitute the values: \( a=\frac{b\times\sin A}{\sin B}\)
Substitute \( b = 15\), \(A = 45^{\circ}\), \(B=100^{\circ}\)
\(a=\frac{15\times\sin(45^{\circ})}{\sin(100^{\circ})}\)
We know that \( \sin(45^{\circ})=\frac{\sqrt{2}}{2}\approx0.707\), \( \sin(100^{\circ})\approx0.985\)
\(a=\frac{15\times0.707}{0.985}=\frac{10.605}{0.985}\approx10.8\)

Answer:

\(10.8\)