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? 45 60 67

Explanation:

Part 1: Measure of angle \( BOA \)

Step 1: Recall the arc length formula

The formula for the length of an arc \( s \) is \( s = r\theta \), where \( r \) is the radius and \( \theta \) is the central angle in radians. First, we need to find \( \theta \) (angle \( BOA \)) in radians and then convert it to degrees.
Given \( s = 8.4 \) cm and \( r = 8 \) cm, from \( s = r\theta \), we can solve for \( \theta \): \( \theta=\frac{s}{r} \)
Substitute the values: \( \theta=\frac{8.4}{8}=1.05 \) radians.

Step 2: Convert radians to degrees

To convert radians to degrees, we use the conversion factor \( 1 \) radian \( =\frac{180^\circ}{\pi} \). So, \( \theta_{degrees}=\theta_{radians}\times\frac{180^\circ}{\pi} \)
Substitute \( \theta_{radians} = 1.05 \): \( \theta_{degrees}=1.05\times\frac{180^\circ}{\pi}\approx1.05\times57.2958^\circ\approx60.16^\circ\approx60^\circ \) (rounded to the nearest whole degree). Wait, but let's check again. Wait, maybe I made a mistake. Wait, let's recalculate: \( \frac{8.4}{8}=1.05 \) radians. \( 1.05\times\frac{180}{\pi}\approx1.05\times57.2957795\approx60.1605685^\circ\approx60^\circ \)? Wait, but the options have 67? Wait, maybe I messed up. Wait, no, wait the arc length formula when \( \theta \) is in degrees is \( s=\frac{\theta}{360^\circ}\times2\pi r \). Let's use that formula. Let's redo it with the degree formula.

Step 1 (Alternative): Use arc length formula with degrees

The arc length formula in degrees is \( s=\frac{\theta}{360^\circ}\times2\pi r \), where \( \theta \) is the central angle in degrees. We need to solve for \( \theta \) (angle \( BOA \)).
Given \( s = 8.4 \) cm, \( r = 8 \) cm.
Rearrange the formula: \( \theta=\frac{s\times360^\circ}{2\pi r} \)
Substitute the values: \( \theta=\frac{8.4\times360^\circ}{2\pi\times8} \)
Calculate numerator: \( 8.4\times360 = 3024 \)
Denominator: \( 2\pi\times8 = 16\pi\approx50.2655 \)
Then \( \theta=\frac{3024}{50.2655}\approx60.16^\circ\approx60^\circ \)? But the options have 67. Wait, maybe I misread the arc length. Wait, the arc BA is 8.4 cm. Wait, maybe the radius is 8 cm, let's check again. Wait, maybe the first approach with radians: \( \theta=\frac{s}{r}=\frac{8.4}{8}=1.05 \) radians. Convert to degrees: \( 1.05\times\frac{180}{\pi}\approx60.16^\circ\approx60^\circ \). But the options include 67. Wait, maybe there's a mistake in my calculation. Wait, no, let's check the second part about arc BAC. Maybe I need to check that first. Wait, no, let's focus on angle BOA. Wait, maybe the arc length is 8.4, radius 8. Let's recalculate \( \theta \) in degrees:

\( s = \frac{\theta}{360} \times 2\pi r \)

\( 8.4 = \frac{\theta}{360} \times 2\pi \times 8 \)

\( 8.4 = \frac{\theta}{360} \times 16\pi \)

\( \theta = \frac{8.4 \times 360}{16\pi} \)

Calculate \( 8.4 \times 360 = 3024 \)

\( 16\pi \approx 50.26548 \)

\( \theta = \frac{3024}{50.26548} \approx 60.16^\circ \approx 60^\circ \). But the options have 67. Wait, maybe the arc length is not 8.4? Wait, the problem says "the length of arc BA is 8.4 cm". Wait, maybe I made a mistake. Wait, no, let's check the second part.

Part 2: Length of arc \( BAC \)

Step 1: Find the measure of angle \( BOC \)

First, we know angle \( AOC = 45^\circ \), and we found angle \( BOA \approx 60^\circ \) (from part 1). So angle \( BOC = \angle BOA + \angle AOC = 60^\circ + 45^\circ = 105^\circ \)? Wait, no, arc BAC is from B to A to C, so the central angle is \( \angle BOA + \angle AOC \). Wait, but first, let's confirm angle \( BOA \). Wait, maybe my first calculation was wrong. Wait, let's use the arc length formula again. Wait, \( s = r\theta \) (theta in radians). So \( \theta = s/r = 8.4/8 = 1.05 \) radians. Convert to degrees: \( 1.05 \times (180/\pi) \approx 60.16^\circ \approx 60^\circ \). Then angle \( BOC = 60^\circ + 45^\circ = 105^\circ \). Now, the length of arc BAC is the length of arc BA plus arc AC. Arc AC: central angle \( 45^\circ \), radius 8 cm. Arc length of AC: \( s_{AC} = \frac{45^\circ}{360^\circ} \times 2\pi \times 8 = \frac{1}{8} \times 16\pi = 2\pi \approx 6.283 \) cm. Arc BA is 8.4 cm. So total arc BAC: \( 8.4 + 6.283 \approx 14.683 \approx 14.7 \) cm? Wait, but maybe the central angle for BAC is \( \angle BOC \), which is \( \angle BOA + \angle AOC \). Wait, if \( \angle BOA \) is \( \theta \), then \( \angle BOC = \theta + 45^\circ \). Let's use the arc length formula for BAC: \( s_{BAC} = \frac{\theta + 45^\circ}{360^\circ} \times 2\pi r \). We know \( \theta = 8.4/(2\pi r/360) \)? Wait, no, let's do it properly.

Wait, maybe the first part's angle \( BOA \) is 67 degrees (from the options). Let's check: if \( \theta = 67^\circ \), convert to radians: \( 67 \times (\pi/180) \approx 1.17 \) radians. Then arc length \( s = r\theta = 8 \times 1.17 \approx 9.36 \) cm, which is not 8.4. So that's not right. Wait, maybe the arc length formula was misapplied. Wait, no, \( s = r\theta \) (theta in radians) is correct. So \( \theta = 8.4/8 = 1.05 \) radians \( \approx 60^\circ \), so arc length of BA is 8.4, which matches. Then arc AC: central angle 45 degrees, so arc length \( s_{AC} = (45/360) \times 2\pi \times 8 = (1/8) \times 16\pi = 2\pi \approx 6.28 \) cm. Then total arc BAC: \( 8.4 + 6.28 = 14.68 \approx 14.7 \) cm.

But let's go back to the first part. The options for angle BOA are 45, 60, 67. Let's check 67 degrees: arc length would be \( s = r\theta = 8 \times (67 \times \pi/180) \approx 8 \times 1.17 \approx 9.36 \) cm, which is not 8.4. 60 degrees: \( s = 8 \times (60 \times \pi/180) = 8 \times (\pi/3) \approx 8.377 \) cm, which is approximately 8.4 cm (rounded). Ah! So \( 8 \times (\pi/3) \approx 8.377 \approx 8.4 \) cm. So angle \( BOA = 60^\circ \) (since \( \theta = 60^\circ = \pi/3 \) radians, \( s = 8 \times \pi/3 \approx 8.377 \approx 8.4 \) cm). That makes sense. So angle \( BOA = 60^\circ \).

Now, for arc BAC: central angle is \( \angle BOA + \angle AOC = 60^\circ + 45^\circ = 105^\circ \). Now, calculate the arc length:

Step 1: Convert angle to radians

\( 105^\circ = 105 \times \frac{\pi}{180} = \frac{7\pi}{12} \) radians.

Step 2: Calculate arc length

Using \( s = r\theta \), where \( r = 8 \) cm and \( \theta = \frac{7\pi}{12} \) radians:

\( s = 8 \times \frac{7\pi}{12} = \frac{56\pi}{12} = \frac{14\pi}{3} \approx \frac{14 \times 3.1416}{3} \approx \frac{43.9824}{3} \approx 14.6608 \approx 14.7 \) cm.

Answer:

s:

Part 1: Measure of angle \( BOA \)

The measure of angle \( BOA \) is \( \boldsymbol{60^\circ} \) (since \( 8 \times \frac{\pi}{3} \approx 8.4 \) cm, which matches the arc length, so \( \theta = 60^\circ \)).

Part 2: Length of arc \( BAC \)

The length of arc \( BAC \) is approximately \( \boldsymbol{14.7} \) cm.