Question

all 2025 geometry b wwva solving for side lengths of right triangles what is the length of $overline{ab}$? round to the nearest tenth. 10 m 75° x 9.7 m 37.3 m 10.4 m 38.6 m

Explanation:

Step1: Identify trigonometric relation

In right - triangle \(ABC\) with right - angle at \(C\), we know \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). Here, \(\theta = 75^{\circ}\), the opposite side to \(\angle A\) is \(BC = 10\) m and the hypotenuse is \(AB=x\). So, \(\sin A=\sin75^{\circ}=\frac{BC}{AB}\).

Step2: Solve for \(AB\)

We know that \(\sin75^{\circ}=\sin(45^{\circ}+ 30^{\circ})=\sin45^{\circ}\cos30^{\circ}+\cos45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659\). Since \(\sin75^{\circ}=\frac{10}{x}\), then \(x=\frac{10}{\sin75^{\circ}}\).

Step3: Calculate the value of \(x\)

\(x=\frac{10}{0.9659}\approx10.4\) m.

Answer:

\(10.4\) m