Question

consider circle o. the length of arc ba is 8.4 cm and the length of the radius is 8 cm. the measure of angle aoc is 46°. 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: Recall arc - length formula

The arc - length formula is $s = r\theta$, where $s$ is the arc - length, $r$ is the radius of the circle, and $\theta$ is the central angle in radians. First, find the measure of $\angle BOA$ in radians. Given $s_{BA}=8.4$ cm and $r = 8$ cm. From $s = r\theta$, we have $\theta=\frac{s}{r}$. So, $\theta_{BOA}=\frac{8.4}{8}=1.05$ radians.

Step2: Convert radians to degrees

To convert radians to degrees, use the formula $\theta_{degrees}=\frac{180^{\circ}}{\pi}\times\theta_{radians}$. So, $\angle BOA=\frac{180^{\circ}}{\pi}\times1.05\approx60^{\circ}$.

Step3: Find the measure of $\angle BOC$

Since $\angle AOC = 45^{\circ}$, and we know $\angle BOA\approx60^{\circ}$, then $\angle BOC=\angle BOA+\angle AOC=60^{\circ}+ 45^{\circ}=105^{\circ}$. Convert $105^{\circ}$ to radians: $\theta_{BOC}=105\times\frac{\pi}{180}=\frac{7\pi}{12}$ radians.

Step4: Find the length of arc $BAC$

Using the arc - length formula $s = r\theta$ with $r = 8$ cm and $\theta=\frac{7\pi}{12}$ radians. $s_{BAC}=8\times\frac{7\pi}{12}=\frac{14\pi}{3}\approx14.7$ cm.

Answer:

Measure of $\angle BOA$: $60^{\circ}$
Length of arc $BAC$: $14.7$ cm