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. A degree is a unit of measurement for angles, while a radian is a unit of measurement for the ratio of the arc length to the radius of a circle. The conversion between degrees and radians is given by the formula:

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

Converting $250^{\circ}$ to Radians

To find the radian measure of the central angle, we need to convert $250^{\circ}$ to radians using the formula above.

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

$$\text{radians} = \frac{250\pi}{180}$$

$$\text{radians} = \frac{25\pi}{18}$$

Understanding the Range of Radian Measures

Now that we have the radian measure of the central angle, we need to determine within which range it falls. The range of radian measures can be understood by considering the unit circle.

• A radian measure of 0 corresponds to a central angle of 0 degrees, which is the starting point of the unit circle.
• A radian measure of $\frac{\pi}{2}$ corresponds to a central angle of 90 degrees, which is the point where the unit circle intersects the y-axis.
• A radian measure of $\pi$ corresponds to a central angle of 180 degrees, which is the point where the unit circle intersects the x-axis.

Determining the Range of the Radian Measure

Based on the conversion above, we have:

$$\text{radians} = \frac{25\pi}{18}$$

Since $\frac{25\pi}{18}$ is greater than $\frac{\pi}{2}$ and less than $\pi$, the radian measure of the central angle falls within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Conclusion

In conclusion, the radian measure of the central angle corresponding to an arc on a circle measuring $250^{\circ}$ falls within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Final Answer

The final answer is B. $\frac{\pi}{2}$ to $\pi$ radians.

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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. A degree is a unit of measurement for angles, while a radian is a unit of measurement for the ratio of the arc length to the radius of a circle. The conversion between degrees and radians is given by the formula:

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

Converting $250^{\circ}$ to Radians

To find the radian measure of the central angle, we need to convert $250^{\circ}$ to radians using the formula above.

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

$$\text{radians} = \frac{250\pi}{180}$$

$$\text{radians} = \frac{25\pi}{18}$$

Understanding the Range of Radian Measures

Now that we have the radian measure of the central angle, we need to determine within which range it falls. The range of radian measures can be understood by considering the unit circle.

• A radian measure of 0 corresponds to a central angle of 0 degrees, which is the starting point of the unit circle.
• A radian measure of $\frac{\pi}{2}$ corresponds to a central angle of 90 degrees, which is the point where the unit circle intersects the y-axis.
• A radian measure of $\pi$ corresponds to a central angle of 180 degrees, which is the point where the unit circle intersects the x-axis.

Determining the Range of the Radian Measure

Based on the conversion above, we have:

$$\text{radians} = \frac{25\pi}{18}$$

Since $\frac{25\pi}{18}$ is greater than $\frac{\pi}{2}$ and less than $\pi$, the radian measure of the central angle falls within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Conclusion

In conclusion, the radian measure of the central angle corresponding to an arc on a circle measuring $250^{\circ}$ falls within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Final Answer

The final answer is B. $\frac{\pi}{2}$ to $\pi$ radians.

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Q&A: Understanding the Relationship Between Degrees and Radians

Q: What is the relationship between degrees and radians?

A: The relationship between degrees and radians is given by the formula:

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

Q: How do I convert degrees to radians?

A: To convert degrees to radians, you can use the formula above. For example, to convert $250^{\circ}$ to radians, you would use:

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

Q: What is the range of radian measures?

A: The range of radian measures can be understood by considering the unit circle. A radian measure of 0 corresponds to a central angle of 0 degrees, which is the starting point of the unit circle. A radian measure of $\frac{\pi}{2}$ corresponds to a central angle of 90 degrees, which is the point where the unit circle intersects the y-axis. A radian measure of $\pi$ corresponds to a central angle of 180 degrees, which is the point where the unit circle intersects the x-axis.

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

A: To determine the range of the radian measure, you can compare the radian measure to the values of $\frac{\pi}{2}$ and $\pi$. If the radian measure is greater than $\frac{\pi}{2}$ and less than $\pi$, it falls within the range of $\frac{\pi}{2}$ to $\pi$ radians.

Q: What is the final answer?

A: The final answer is B. $\frac{\pi}{2}$ to $\pi$ radians.

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Additional Resources

• For more information on the relationship between degrees and radians, see the formula above.
• For more information on converting degrees to radians, see the example above.
• For more information on the range of radian measures, see the explanation above.
• For more information on determining the range of the radian measure, see the explanation above.

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Conclusion

In conclusion, the radian measure of the central angle corresponding to an arc on a circle measuring $250^{\circ}$ falls within the range of $\frac{\pi}{2}$ to $\pi$ radians. We hope this article has been helpful in understanding the relationship between degrees and radians, converting degrees to radians, and determining the range of radian measures.