Question

an arc on a circle measures 125°. the measure of the central angle, in radians, is within which range?
○ 0 to \\(\frac{\pi}{2}\\) radians
○ \\(\frac{\pi}{2}\\) to \\(\pi\\) radians
○ \\(\pi\\) to \\(\frac{3\pi}{2}\\) radians
○ \\(\frac{3\pi}{2}\\) to \\(2\pi\\) radians

Explanation:

Step1: Recall the conversion formula

To convert degrees to radians, we use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \).

Step2: Convert \( 125^\circ \) to radians

Substitute \( 125^\circ \) into the formula: \( 125 \times \frac{\pi}{180} = \frac{25\pi}{36} \approx 2.18 \) radians.

Step3: Analyze the ranges

- \( 0 \) to \( \frac{\pi}{2} \approx 1.57 \): \( 2.18>1.57 \), so not in this range.
- \( \frac{\pi}{2} \approx 1.57 \) to \( \pi \approx 3.14 \): \( 1.57 < 2.18 < 3.14 \), so this range is possible. Wait, no, wait: Wait, \( \pi\approx3.14 \), \( \frac{3\pi}{2}\approx4.71 \), \( 2\pi\approx6.28 \). Wait, let's recalculate the conversion correctly. Wait, \( 125^\circ\times\frac{\pi}{180}=\frac{25\pi}{36}\approx\frac{25\times3.14}{36}\approx2.18 \) radians. Now, let's check the ranges:
- \( 0 \) to \( \frac{\pi}{2}\approx1.57 \): \( 2.18>1.57 \), so no.
- \( \frac{\pi}{2}\approx1.57 \) to \( \pi\approx3.14 \): \( 1.57 < 2.18 < 3.14 \), so this is true? Wait, but wait, the options are: Wait, the third option is \( \pi \) to \( \frac{3\pi}{2} \) (≈3.14 to 4.71), fourth is \( \frac{3\pi}{2} \) to \( 2\pi \) (≈4.71 to 6.28). Wait, no, I made a mistake. Wait, \( 125^\circ \) is less than \( 180^\circ \) (which is \( \pi \) radians, ≈3.14). Wait, \( 90^\circ=\frac{\pi}{2}\approx1.57 \), \( 180^\circ=\pi\approx3.14 \), \( 270^\circ=\frac{3\pi}{2}\approx4.71 \), \( 360^\circ = 2\pi\approx6.28 \). So \( 125^\circ \) is between \( 90^\circ \) ( \( \frac{\pi}{2} \)) and \( 180^\circ \) ( \( \pi \))? Wait, no, \( 125^\circ \) is more than \( 90^\circ \) ( \( \frac{\pi}{2}\approx1.57 \)) and less than \( 180^\circ \) ( \( \pi\approx3.14 \))? Wait, no, \( 125^\circ \) is 125 degrees, which is between 90 (π/2) and 180 (π)? Wait, no, 90 is π/2, 180 is π, 270 is 3π/2, 360 is 2π. So 125 degrees is between 90 (π/2) and 180 (π)? Wait, no, 125 is less than 180, so between π/2 and π? But wait, when we convert 125 degrees to radians, it's approximately 2.18, and π is approximately 3.14, so 2.18 is between π/2 (1.57) and π (3.14). But wait, the third option is π to 3π/2 (3.14 to 4.71), which is 180 to 270 degrees. The fourth is 3π/2 to 2π (4.71 to 6.28), 270 to 360. Wait, maybe I made a mistake in the conversion. Wait, let's recalculate \( 125\times\frac{\pi}{180} \). Let's compute \( 125\div180=\frac{25}{36}\approx0.694 \), so \( \frac{25}{36}\pi\approx0.694\times3.14\approx2.18 \) radians. Now, π is approximately 3.14, 3π/2 is approximately 4.71, 2π is approximately 6.28. So 2.18 is between π/2 (1.57) and π (3.14)? Wait, no, π/2 is 1.57, π is 3.14, so 2.18 is between them. But wait, the options are:

1. 0 to π/2 (0 to 1.57)
2. π/2 to π (1.57 to 3.14)
3. π to 3π/2 (3.14 to 4.71)
4. 3π/2 to 2π (4.71 to 6.28)

So 2.18 is in the range of π/2 to π? But wait, 125 degrees is more than 90 (π/2) and less than 180 (π), so the central angle (which is equal to the arc measure in degrees) converted to radians is 125*(π/180)≈2.18, which is between π/2 (1.57) and π (3.14). Wait, but that would be option 2. But wait, maybe I messed up. Wait, no, let's check the options again. Wait, the problem says "the measure of the central angle, in radians, is within which range". The central angle is equal to the arc measure, so 125 degrees. Convert to radians: 125*(π/180)=25π/36≈2.18. Now, π≈3.14, 3π/2≈4.71, 2π≈6.28. So 2.18 is between π/2 (1.57) and π (3.14)? Wait, no, π is 3.14, so 2.18 is less than π. So the range π/2 to π (1.57 to 3.14) includes 2.18. But wait, let's check the options again. Wait, maybe I made a mistake in the conversion. Wait, 180 degrees is π radians, so 90 degrees is π/2, 270 is 3π/2, 360 is 2π. So 125 degrees is between 90 and 180, so radians between π/2 and π. So the answer should be the second option: \( \frac{\pi}{2} \) to \( \pi \) radians. Wait, but let's confirm with the calculation. 25π/36≈2.18, π/2≈1.57, π≈3.14. So 1.57 < 2.18 < 3.14, so it's in the range \( \frac{\pi}{2} \) to \( \pi \) radians. Wait, but the third option is π to 3π/2 (3.14 to 4.71), which is 180 to 270 degrees, which is more than 125. So the correct range is \( \frac{\pi}{2} \) to \( \pi \) radians.

Answer:

\( \frac{\pi}{2} \) to \( \pi \) radians (the second option)