An arc on a circle measures [tex]$125^{\circ}$[/tex]. The measure of the central angle, in radians, is within which range?

A. 0 to [tex]$\frac{\pi}{2}$[/tex] radians
B. [tex]$\frac{\pi}{2}$[/tex] to [tex]$\pi$[/tex] radians
C. [tex]$\pi$[/tex] to [tex]$\frac{3\pi}{2}$[/tex] radians
D. [tex]$\frac{3\pi}{2}$[/tex] to [tex]$2\pi$[/tex] radians



Answer :

To determine within which range the measure of a central angle, measured at 125 degrees, falls in radians, follow these steps:

1. Convert the central angle from degrees to radians:

The conversion factor from degrees to radians is [tex]\( \frac{\pi}{180} \)[/tex]. Therefore, the central angle in radians can be found using:

[tex]\[ \text{Central Angle in Radians} = 125^\circ \times \frac{\pi}{180} \][/tex]

2. Perform the calculation:

[tex]\[ 125^\circ \times \frac{\pi}{180} = \frac{125 \pi}{180} \][/tex]

Simplify this fraction:

[tex]\[ \frac{125 \pi}{180} = \frac{25 \pi}{36} \][/tex]

This gives us the measure of the central angle in radians.

3. Determine the numerical value of the central angle in radians:

[tex]\[ \frac{25 \pi}{36} \approx 2.181661564992912 \, \text{radians} \][/tex]

4. Identify which range the angle falls into:

Compare this value to the given ranges:

- [tex]\(0 \)[/tex] to [tex]\( \frac{\pi}{2} \approx 1.5708 \)[/tex]
- [tex]\(\frac{\pi}{2} \approx 1.5708 \)[/tex] to [tex]\( \pi \approx 3.1416 \)[/tex]
- [tex]\(\pi \approx 3.1416 \)[/tex] to [tex]\( \frac{3\pi}{2} \approx 4.7124 \)[/tex]
- [tex]\(\frac{3\pi}{2} \approx 4.7124 \)[/tex] to [tex]\( 2\pi \approx 6.2832 \)[/tex]

So, [tex]\(2.181661564992912\)[/tex] falls between [tex]\( \frac{\pi}{2} \approx 1.5708 \)[/tex] and [tex]\( \pi \approx 3.1416 \)[/tex].

Hence, the measure of the central angle, in radians, falls in the range [tex]\(\frac{\pi}{2}\)[/tex] to [tex]\(\pi\)[/tex] radians.