Question

the measure of angle a is 15°, and the length of side bc is 8. what are the lengths of the other two sides, rounded to the nearest tenth? ac = 29.9 ab = dropdown with options 2.1, 8.3, 20.7, 30.9

Explanation:

Step1: Identify triangle type

This is a right - triangle with \(\angle C = 90^{\circ}\), \(\angle A=15^{\circ}\) and \(BC = 8\). We can use trigonometric ratios. In a right - triangle, \(\sin A=\frac{BC}{AB}\) and \(\tan A=\frac{BC}{AC}\) (we already know \(AC = 29.9\), let's verify with \(\tan A\): \(\tan15^{\circ}=\frac{8}{AC}\), \(AC=\frac{8}{\tan15^{\circ}}\approx\frac{8}{0.2679}\approx29.9\), which matches). Now for \(AB\), using \(\sin A=\frac{BC}{AB}\), so \(AB=\frac{BC}{\sin A}\)

Step2: Calculate \(AB\)

We know that \(A = 15^{\circ}\) and \(BC = 8\). \(\sin15^{\circ}\approx0.2588\)
\(AB=\frac{8}{\sin15^{\circ}}=\frac{8}{0.2588}\approx30.9\) (rounded to the nearest tenth)

Answer:

\(AB = 30.9\)