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.

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Understanding the Relationship Between Degrees and Radians

When dealing with angles, it's essential to understand the relationship between degrees and radians. The radian measure of an angle is defined as the ratio of the arc length to the radius of the circle. To convert degrees to radians, we use the formula:

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Converting $250^{\circ}$ to Radians

Using the formula above, we can convert $250^{\circ}$ to radians:

$$\text{radians} = \frac{\pi}{180} \times 250 = \frac{5\pi}{18}$$

Understanding the Range of Radian Measures

To determine the range within which the radian measure of the central angle lies, we need to consider the possible values of the radian measure. The radian measure of an angle can range from $0$ to $\pi$ radians.

Analyzing the Options

Let's analyze the options given:

A. $0$ to $\frac{\pi}{2}$ radians B. $\frac{\pi}{2}$ to $\pi$ radians C. $\pi$ to $2\pi$ radians

Evaluating Option A

Option A suggests that the radian measure of the central angle lies between $0$ and $\frac{\pi}{2}$ radians. However, we have calculated that the radian measure of $250^{\circ}$ is $\frac{5\pi}{18}$, which is greater than $\frac{\pi}{2}$ radians. Therefore, option A is not correct.

Evaluating Option B

Option B suggests that the radian measure of the central angle lies between $\frac{\pi}{2}$ and $\pi$ radians. Since $\frac{5\pi}{18}$ is greater than $\frac{\pi}{2}$ but less than $\pi$, option B is a possible range for the radian measure of the central angle.

Evaluating Option C

Option C suggests that the radian measure of the central angle lies between $\pi$ and $2\pi$ radians. However, we have calculated that the radian measure of $250^{\circ}$ is $\frac{5\pi}{18}$, which is less than $\pi$. Therefore, option C is not correct.

Conclusion

Based on our analysis, the radian measure of the central angle lies within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Final Answer

The final answer is B.

Understanding the Relationship Between Degrees and Radians

When dealing with angles, it's essential to understand the relationship between degrees and radians. The radian measure of an angle is defined as the ratio of the arc length to the radius of the circle. To convert degrees to radians, we use the formula:

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Q&A Session

Q: What is the radian measure of an angle?

A: The radian measure of an angle is defined as the ratio of the arc length to the radius of the circle.

Q: How do I convert degrees to radians?

A: To convert degrees to radians, we use the formula:

$$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$

Q: What is the range of radian measures?

A: The radian measure of an angle can range from $0$ to $\pi$ radians.

Q: How do I determine the range of the radian measure of a central angle?

A: To determine the range of the radian measure of a central angle, we need to consider the possible values of the radian measure. We can use the formula above to convert the angle from degrees to radians and then determine the range.

Q: What is the radian measure of $250^{\circ}$?

A: Using the formula above, we can convert $250^{\circ}$ to radians:

$$\text{radians} = \frac{\pi}{180} \times 250 = \frac{5\pi}{18}$$

Q: Within which range does the radian measure of the central angle lie?

A: Based on our analysis, the radian measure of the central angle lies within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Q: What are some common mistakes to avoid when converting degrees to radians?

A: Some common mistakes to avoid when converting degrees to radians include:

• Forgetting to multiply by $\frac{\pi}{180}$
• Not considering the range of radian measures
• Not using the correct formula

Q: How can I practice converting degrees to radians?

A: You can practice converting degrees to radians by using online calculators or worksheets. You can also try converting different angles from degrees to radians and vice versa.

Conclusion

Converting degrees to radians is an essential skill in mathematics, particularly in trigonometry and calculus. By understanding the relationship between degrees and radians and using the correct formula, you can accurately convert angles from degrees to radians and vice versa.

Final Tips

• Always use the correct formula to convert degrees to radians
• Consider the range of radian measures when determining the range of the radian measure of a central angle
• Practice converting different angles from degrees to radians and vice versa to build your skills and confidence

Additional Resources

• Online calculators and worksheets for converting degrees to radians
• Textbooks and online resources for trigonometry and calculus
• Practice problems and exercises for converting degrees to radians

Final Answer

The final answer is B.