Question

solve for $x$. round to the nearest tenth, if necessary.

Explanation:

Step1: Find the measure of angle \(N\)

In a right - triangle \( \triangle NOP\), the sum of angles is \(180^{\circ}\). Since \(\angle O = 90^{\circ}\) and \(\angle P=69^{\circ}\), then \(\angle N=180^{\circ}-(90^{\circ} + 69^{\circ})=21^{\circ}\)

Step2: Use the tangent function

The tangent of an angle in a right - triangle is defined as \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \(\angle N\), \(\tan N=\frac{OP}{ON}\). We know \(OP = 80\) and \(ON=x\), and \(\angle N = 21^{\circ}\). So \(\tan(21^{\circ})=\frac{80}{x}\)

Step3: Solve for \(x\)

Cross - multiply to get \(x=\frac{80}{\tan(21^{\circ})}\). Since \(\tan(21^{\circ})\approx0.384\), then \(x=\frac{80}{0.384}\approx208.3\)

Answer:

\(x\approx208.3\)