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 the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(B\). Then \(B = 180^{\circ}-(105^{\circ}+45^{\circ})=30^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{c}{\sin C}\). Here, \(c = 7\), \(A = 45^{\circ}\), \(C = 30^{\circ}\). So \(\frac{a}{\sin45^{\circ}}=\frac{7}{\sin30^{\circ}}\).
Since \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\) and \(\sin30^{\circ}=\frac{1}{2}\), we have \(a=\frac{7\times\sin45^{\circ}}{\sin30^{\circ}}\).
Substitute the values: \(a=\frac{7\times\frac{\sqrt{2}}{2}}{\frac{1}{2}}\).

Step3: Simplify the expression

\(a = 7\sqrt{2}\) (because \(\frac{7\times\frac{\sqrt{2}}{2}}{\frac{1}{2}}=7\sqrt{2}\)).

Answer:

\(7\sqrt{2}\)