Question

consider circle o below. the length of arc ba is 8.4 cm and the length of the radius is 8 cm. the measure of angle aoc is 45°. rounded to the nearest whole degree, what is the measure of angle boa? rounded to the nearest tenth of a centimeter, what is the length of arc bac?

Explanation:

Step1: Use the arc - length formula \(s = r\theta\) (where \(s\) is arc length, \(r\) is radius, \(\theta\) is in radians) to find \(\theta_{BOA}\) in radians

Given \(s = 8.4\) cm and \(r = 8\) cm. From \(s=r\theta\), we have \(\theta=\frac{s}{r}\). Substituting the values: \(\theta=\frac{8.4}{8}=1.05\) radians.

Step2: Convert radians to degrees

Use the conversion formula \(\theta_{degrees}=\theta_{radians}\times\frac{180^{\circ}}{\pi}\). So \(\theta_{BOA}=1.05\times\frac{180^{\circ}}{\pi}\approx 60^{\circ}\) (since \(\pi\approx3.14\)).

Step3: Find the length of arc \(BAC\)

The formula for arc length is \(s = r\theta\). The measure of \(\angle BOC=\angle BOA+\angle AOC\). \(\angle BOA\approx60^{\circ}\) and \(\angle AOC = 45^{\circ}\), so \(\angle BOC=60^{\circ}+45^{\circ}=105^{\circ}\). Convert \(105^{\circ}\) to radians: \(\theta = 105\times\frac{\pi}{180}=\frac{7\pi}{12}\) radians.
Using \(s = r\theta\) with \(r = 8\) cm, \(s=8\times\frac{7\pi}{12}=\frac{14\pi}{3}\approx14.7\) cm (since \(\pi\approx3.14\)).

Answer:

The measure of angle \(BOA\) is \(60^{\circ}\). The length of arc \(BAC\) is \(14.7\) cm.