Question

several of the exercises on right triangle applications involved a figure similar to the one shown here, in which angles α and β and the length of line segment ab are known, and the length of side cd is to be determined. use the law of sines to obtain x in terms of α, β, and d. choose the correct answer below. a. ( x = \frac{d sin alpha sin \beta}{sin (\beta - alpha)} ) b. ( x = \frac{d sin alpha sin \beta}{sin (alpha + \beta)} ) c. ( x = \frac{sin (alpha + \beta)}{d sin alpha sin \beta} ) d. ( x = \frac{sin (\beta - alpha)}{d sin alpha sin \beta} )

Explanation:

Step1: Find the length of BD in \(\triangle ABD\)

In \(\triangle ABD\), \(\angle ABD = 180^{\circ}-\beta\), \(\angle ADB=\beta - \alpha\). By the law of sines \(\frac{BD}{\sin\alpha}=\frac{AB}{\sin(\beta - \alpha)}\), since \(AB = d\), then \(BD=\frac{d\sin\alpha}{\sin(\beta - \alpha)}\).

Step2: Find the length of \(x\) in \(\triangle BCD\)

In right - triangle \(BCD\), \(\sin\beta=\frac{x}{BD}\), substituting \(BD\) from step1, we get \(x = BD\sin\beta=\frac{d\sin\alpha\sin\beta}{\sin(\beta - \alpha)}\).

Answer:

A. \(x=\frac{d\sin\alpha\sin\beta}{\sin(\beta - \alpha)}\)