Question
In the figure below, the radius of circle O is 3 centimeters.
image d21f3160931c471f81df2762a7a8b6a8
What is the length of the arc bound by central angle POQ?
A.
π
2
cm
B.
π
cm
C.
3
π
cm
D.
3
π
2
cm
image d21f3160931c471f81df2762a7a8b6a8
What is the length of the arc bound by central angle POQ?
A.
π
2
cm
B.
π
cm
C.
3
π
cm
D.
3
π
2
cm
Answers
Answered by
GPT-4o mini
To find the length of an arc bound by a central angle in a circle, we can use the formula for the arc length \( L \):
\[
L = r \theta
\]
where:
- \( L \) is the length of the arc,
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.
From the question, we know the radius \( r = 3 \) cm. However, we need to determine the value of \( \theta \) (the central angle POQ) in radians.
Without additional information about the angle or the image (e.g., whether it is a right angle, etc.), I can infer that if this question is from a standard geometry problem, the angle might be one commonly encountered.
For example:
- If \( \theta = \frac{\pi}{2} \) radians (90 degrees), then:
\[
L = 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \, \text{cm}
\]
- If \( \theta = \pi \) radians (180 degrees), then:
\[
L = 3 \cdot \pi = 3\pi \, \text{cm}
\]
- If \( \theta = \frac{3\pi}{2} \) radians (270 degrees), then:
\[
L = 3 \cdot \frac{3\pi}{2} = \frac{9\pi}{2} \, \text{cm}
\]
- If \( \theta = 2\pi \) radians (360 degrees), then:
\[
L = 3 \cdot 2\pi = 6\pi \, \text{cm}
\]
Given the choices provided, it looks like we should refer back to a standard angle found in geometry. It's quite common in problems like this to see either \( \frac{\pi}{2} \) (for quarter-circles) or \( \pi \) (for half-circles), but since the most reasonable interpretation without estimating the angle based on the diagram would default to **Option D,** \( \frac{3\pi}{2} \) cm.
Thus, if we assume that the angle POQ is \( \frac{\pi}{2} \):
The length of the arc bound by central angle POQ is \( \frac{3\pi}{2} \) cm.
So, the answer is:
**D. \( \frac{3\pi}{2} \) cm**
\[
L = r \theta
\]
where:
- \( L \) is the length of the arc,
- \( r \) is the radius of the circle,
- \( \theta \) is the central angle in radians.
From the question, we know the radius \( r = 3 \) cm. However, we need to determine the value of \( \theta \) (the central angle POQ) in radians.
Without additional information about the angle or the image (e.g., whether it is a right angle, etc.), I can infer that if this question is from a standard geometry problem, the angle might be one commonly encountered.
For example:
- If \( \theta = \frac{\pi}{2} \) radians (90 degrees), then:
\[
L = 3 \cdot \frac{\pi}{2} = \frac{3\pi}{2} \, \text{cm}
\]
- If \( \theta = \pi \) radians (180 degrees), then:
\[
L = 3 \cdot \pi = 3\pi \, \text{cm}
\]
- If \( \theta = \frac{3\pi}{2} \) radians (270 degrees), then:
\[
L = 3 \cdot \frac{3\pi}{2} = \frac{9\pi}{2} \, \text{cm}
\]
- If \( \theta = 2\pi \) radians (360 degrees), then:
\[
L = 3 \cdot 2\pi = 6\pi \, \text{cm}
\]
Given the choices provided, it looks like we should refer back to a standard angle found in geometry. It's quite common in problems like this to see either \( \frac{\pi}{2} \) (for quarter-circles) or \( \pi \) (for half-circles), but since the most reasonable interpretation without estimating the angle based on the diagram would default to **Option D,** \( \frac{3\pi}{2} \) cm.
Thus, if we assume that the angle POQ is \( \frac{\pi}{2} \):
The length of the arc bound by central angle POQ is \( \frac{3\pi}{2} \) cm.
So, the answer is:
**D. \( \frac{3\pi}{2} \) cm**