Question

find the length of side x. round to the nearest tenth.
x =

Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\). Then \(A = 180^{\circ}-45^{\circ}-75^{\circ}=60^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here, if \(x\) is opposite the \(75^{\circ}\) angle and the side of length \(23\) is opposite the \(60^{\circ}\) angle. So \(\frac{x}{\sin75^{\circ}}=\frac{23}{\sin60^{\circ}}\).

Step3: Solve for \(x\)

We know that \(\sin75^{\circ}=\sin(45^{\circ} + 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.966\), and \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\approx0.866\).
From \(\frac{x}{\sin75^{\circ}}=\frac{23}{\sin60^{\circ}}\), we get \(x=\frac{23\times\sin75^{\circ}}{\sin60^{\circ}}\).
Substitute the values: \(x=\frac{23\times0.966}{0.866}\approx25.4\).

Answer:

\(25.4\)