To find the measure of the central angle in radians and determine its range, follow these steps:
1. Convert the angle from degrees to radians:
The given measure of the arc is [tex]\( 85^\circ \)[/tex]. One radian is equivalent to [tex]\( \frac{\pi}{180} \)[/tex] degrees. Therefore, the measure of the central angle in radians can be calculated using the conversion formula:
[tex]\[
\text{Radians} = \text{Degrees} \times \left( \frac{\pi}{180} \right)
\][/tex]
Plugging in [tex]\( 85^\circ \)[/tex]:
[tex]\[
85^\circ \times \left( \frac{\pi}{180} \right) = \frac{85\pi}{180} \approx 1.4835298641951802 \text{ radians}
\][/tex]
2. Determine the range of the central angle:
Next, we need to identify which range this radian measure falls into among the given options:
- [tex]\( 0 \)[/tex] to [tex]\( \frac{\pi}{2} \approx 1.570796 \)[/tex]
- [tex]\( \frac{\pi}{2} \approx 1.570796 \)[/tex] to [tex]\( \pi \approx 3.141593 \)[/tex]
- [tex]\( \pi \approx 3.141593 \)[/tex] to [tex]\( \frac{3\pi}{2} \approx 4.712389 \)[/tex]
- [tex]\( \frac{3\pi}{2} \approx 4.712389 \)[/tex] to [tex]\( 2\pi \approx 6.283185 \)[/tex]
We see that [tex]\( 1.4835298641951802 \)[/tex] radians is less than [tex]\( \frac{\pi}{2} \)[/tex] and greater than [tex]\( 0 \)[/tex]. Therefore, it falls within the range [tex]\( 0 \)[/tex] to [tex]\( \frac{\pi}{2} \)[/tex].
Hence, the central angle of the arc measuring [tex]\( 85^\circ \)[/tex] is approximately [tex]\( 1.4835298641951802 \)[/tex] radians, and it falls within the range:
[tex]\[
0 \text{ to } \frac{\pi}{2} \text{ radians}
\][/tex]
