Question

find the exact value of each part labeled with a variable in the following figure.

(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the hypotenuse of the right - triangle with angle 30°

In a 30 - 60 - 90 triangle, if the side opposite the 30° angle is \(x\), the hypotenuse is \(2x\). Let the side opposite the 30° angle in the large right - triangle be \(x\). We know that \(\sin30^{\circ}=\frac{f}{57}\), so \(f = 57\times\sin30^{\circ}=\frac{57}{2}\). Also, \(\cos30^{\circ}=\frac{d}{57}\).

Step2: Calculate the value of \(d\)

Since \(\cos30^{\circ}=\frac{\sqrt{3}}{2}\) and \(\cos30^{\circ}=\frac{d}{57}\), we can solve for \(d\) using the formula \(d = 57\times\cos30^{\circ}\).
\[d = 57\times\frac{\sqrt{3}}{2}=\frac{57\sqrt{3}}{2}\]

Answer:

\(\frac{57\sqrt{3}}{2}\)