Question

find the length of side a. do not use a calculator.
a = \square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find angle \( C \)

By the angle - sum property of a triangle (\( A + B + C=180^{\circ} \)), \( C = 180^{\circ}-(60^{\circ}+75^{\circ})=45^{\circ} \).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here, \( b = 9\sqrt{2}\), \(A = 60^{\circ}\), \(B = 45^{\circ}\). So \(\frac{a}{\sin60^{\circ}}=\frac{9\sqrt{2}}{\sin45^{\circ}}\).
Since \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\) and \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\), we have \(a=\frac{9\sqrt{2}\times\sin60^{\circ}}{\sin45^{\circ}}\).
Substitute the sine values: \(a=\frac{9\sqrt{2}\times\frac{\sqrt{3}}{2}}{\frac{\sqrt{2}}{2}}\).
Cancel out \(\sqrt{2}\) and \(2\): \(a = 9\sqrt{3}\).

Answer:

\(9\sqrt{3}\)