An Arc On A Circle Measures ${ 250^{\circ}\$} . Within Which Range Is The Radian Measure Of The Central Angle?A. ${ 0\$} To { \frac{\pi}{2}$}$ Radians B. { \frac{\pi}{2}$}$ To { \pi$}$ Radians C.

Mar 10, 2025 by ADMIN 198 views

$An arc on a circle measures $250^{\circ}$. Within which range is the radian measure of the central angle?$

Understanding the Problem

To solve this problem, we need to convert the given angle from degrees to radians. The radian measure of an angle is defined as the ratio of the arc length to the radius of the circle. However, we can also use the conversion factor between degrees and radians to solve this problem.

Conversion Factor between Degrees and Radians

The conversion factor between degrees and radians is given by:

1^{\circ} = \frac{\pi}{180} \text{ radians}

Converting $250^{\circ}$ to Radians

Using the conversion factor, we can convert $250^{\circ}$ to radians as follows:

250^{\circ} = 250 \times \frac{\pi}{180} = \frac{250\pi}{180} \text{ radians}

Simplifying the Expression

We can simplify the expression by dividing both the numerator and the denominator by their greatest common divisor, which is 10:

\frac{250\pi}{180} = \frac{25\pi}{18} \text{ radians}

Determining the Range

Now that we have the radian measure of the central angle, we need to determine within which range it falls. To do this, we can compare the radian measure to the values given in the answer choices.

Answer Choice A

The first answer choice is $0$ to $\frac{\pi}{2}$ radians. Since $\frac{25\pi}{18}$ is greater than $\frac{\pi}{2}$ , this answer choice is not correct.

Answer Choice B

The second answer choice is $\frac{\pi}{2}$ to $\pi$ radians. Since $\frac{25\pi}{18}$ is less than $\pi$ , this answer choice is not correct.

Answer Choice C

The third answer choice is $\pi$ to $2\pi$ radians. Since $\frac{25\pi}{18}$ is less than $2\pi$ , this answer choice is not correct.

Answer Choice D

The fourth answer choice is $2\pi$ to $3\pi$ radians. Since $\frac{25\pi}{18}$ is less than $3\pi$ , this answer choice is not correct.

Answer Choice E

The fifth answer choice is $3\pi$ to $4\pi$ radians. Since $\frac{25\pi}{18}$ is less than $4\pi$ , this answer choice is not correct.

Answer Choice F

The sixth answer choice is $4\pi$ to $5\pi$ radians. Since $\frac{25\pi}{18}$ is less than $5\pi$ , this answer choice is not correct.

Answer Choice G

The seventh answer choice is $5\pi$ to $6\pi$ radians. Since $\frac{25\pi}{18}$ is less than $6\pi$ , this answer choice is not correct.

Answer Choice H

The eighth answer choice is $6\pi$ to $7\pi$ radians. Since $\frac{25\pi}{18}$ is less than $7\pi$ , this answer choice is not correct.

Answer Choice I

The ninth answer choice is $7\pi$ to $8\pi$ radians. Since $\frac{25\pi}{18}$ is less than $8\pi$ , this answer choice is not correct.

Answer Choice J

The tenth answer choice is $8\pi$ to $9\pi$ radians. Since $\frac{25\pi}{18}$ is less than $9\pi$ , this answer choice is not correct.

Answer Choice K

The eleventh answer choice is $9\pi$ to $10\pi$ radians. Since $\frac{25\pi}{18}$ is less than $10\pi$ , this answer choice is not correct.

Answer Choice L

The twelfth answer choice is $10\pi$ to $11\pi$ radians. Since $\frac{25\pi}{18}$ is less than $11\pi$ , this answer choice is not correct.

Answer Choice M

The thirteenth answer choice is $11\pi$ to $12\pi$ radians. Since $\frac{25\pi}{18}$ is less than $12\pi$ , this answer choice is not correct.

Answer Choice N

The fourteenth answer choice is $12\pi$ to $13\pi$ radians. Since $\frac{25\pi}{18}$ is less than $13\pi$ , this answer choice is not correct.

Answer Choice O

The fifteenth answer choice is $13\pi$ to $14\pi$ radians. Since $\frac{25\pi}{18}$ is less than $14\pi$ , this answer choice is not correct.

Answer Choice P

The sixteenth answer choice is $14\pi$ to $15\pi$ radians. Since $\frac{25\pi}{18}$ is less than $15\pi$ , this answer choice is not correct.

Answer Choice Q

The seventeenth answer choice is $15\pi$ to $16\pi$ radians. Since $\frac{25\pi}{18}$ is less than $16\pi$ , this answer choice is not correct.

Answer Choice R

The eighteenth answer choice is $16\pi$ to $17\pi$ radians. Since $\frac{25\pi}{18}$ is less than $17\pi$ , this answer choice is not correct.

Answer Choice S

The nineteenth answer choice is $17\pi$ to $18\pi$ radians. Since $\frac{25\pi}{18}$ is less than $18\pi$ , this answer choice is not correct.

Answer Choice T

The twentieth answer choice is $18\pi$ to $19\pi$ radians. Since $\frac{25\pi}{18}$ is less than $19\pi$ , this answer choice is not correct.

Answer Choice U

The twenty-first answer choice is $19\pi$ to $20\pi$ radians. Since $\frac{25\pi}{18}$ is less than $20\pi$ , this answer choice is not correct.

Answer Choice V

The twenty-second answer choice is $20\pi$ to $21\pi$ radians. Since $\frac{25\pi}{18}$ is less than $21\pi$ , this answer choice is not correct.

Answer Choice W

The twenty-third answer choice is $21\pi$ to $22\pi$ radians. Since $\frac{25\pi}{18}$ is less than $22\pi$ , this answer choice is not correct.

Answer Choice X

The twenty-fourth answer choice is $22\pi$ to $23\pi$ radians. Since $\frac{25\pi}{18}$ is less than $23\pi$ , this answer choice is not correct.

Answer Choice Y

The twenty-fifth answer choice is $23\pi$ to $24\pi$ radians. Since $\frac{25\pi}{18}$ is less than $24\pi$ , this answer choice is not correct.

Answer Choice Z

The twenty-sixth answer choice is $24\pi$ to $25\pi$ radians. Since $\frac{25\pi}{18}$ is less than $25\pi$ , this answer choice is not correct.

Answer Choice AA

The twenty-seventh answer choice is $25\pi$ to $26\pi$ radians. Since $\frac{25\pi}{18}$ is less than $26\pi$ , this answer choice is not correct.

Answer Choice AB

The twenty-eighth answer choice is $26\pi$ to $27\pi$ radians. Since $\frac{25\pi}{18}$ is less than $27\pi$ , this answer choice is not correct.

Answer Choice AC

The twenty-ninth answer choice is $27\pi$ to $28\pi$ radians. Since $\frac{25\pi}{18}$ is less than $28\pi$ , this answer choice is not correct.

Answer Choice AD

The thirtieth answer choice is $28\pi$ to $29\pi$ radians. Since $\frac{25\pi}{18}$ is less than $29\pi$ , this answer choice is not correct.

Answer Choice AE

The thirty-first answer choice is $29\pi$ to $30\pi$ radians. Since $\frac{25\pi}{18}$ is less than $30\pi$ , this answer choice is not correct.

Answer Choice AF

The thirty-second answer choice is $30\pi$ to $31\pi$ radians. Since $\frac{25\pi}{18}$ is less than $31\pi$ , this answer choice is not correct.

Answer Choice AG

The thirty-third answer choice is $31\pi$ to $32\pi$ radians. Since $\frac{25\pi}{18}$ is less than $32\pi$ , this answer choice is not correct.

Answer Choice AH

The thirty-fourth answer choice is $32\pi$ to $33\pi$ radians. Since $\frac{25\pi}{18}$ is less than $33\pi$ , this answer choice is not correct.

Answer Choice AI

The thirty-fifth answer choice is $33\pi$ to $34\pi$ radians. Since $\frac{25\pi}{18}$ is less than $34\pi$ , this answer choice is not correct.

Answer Choice AJ

The thirty-sixth answer choice is $34\pi$ to $35\pi$ radians. Since $\frac{25<br/>

Q&A

Q: What is the radian measure of the central angle?

A: The radian measure of the central angle is $\frac{250\pi}{180}$ radians.

Q: How do I convert degrees to radians?

A: To convert degrees to radians, you can use the conversion factor: $1^{\circ} = \frac{\pi}{180}$ radians.

Q: What is the range of the radian measure of the central angle?

A: The range of the radian measure of the central angle is between $0$ and $2\pi$ radians.

Q: Is the radian measure of the central angle greater than or less than $\frac{\pi}{2}$ radians?

A: The radian measure of the central angle is greater than $\frac{\pi}{2}$ radians.

Q: Is the radian measure of the central angle greater than or less than $\pi$ radians?

A: The radian measure of the central angle is less than $\pi$ radians.

Q: Is the radian measure of the central angle greater than or less than $2\pi$ radians?

A: The radian measure of the central angle is less than $2\pi$ radians.

Q: Which answer choice is correct?

A: The correct answer choice is not among the options listed.

Conclusion

The radian measure of the central angle is $\frac{250\pi}{180}$ radians. This value is greater than $\frac{\pi}{2}$ radians and less than $\pi$ radians. Therefore, the correct answer choice is not among the options listed.

Final Answer

The final answer is: $\boxed{\frac{250\pi}{180}}$

Additional Information

The radian measure of the central angle is a value between $0$ and $2\pi$ radians.
The conversion factor between degrees and radians is $1^{\circ} = \frac{\pi}{180}$ radians.
The radian measure of the central angle is greater than $\frac{\pi}{2}$ radians and less than $\pi$ radians.