Select The Correct Answer.A Circle Has A Radius Of 22 Centimeters. Arc XY Has A Length Of 66 5 Π \frac{66}{5} \pi 5 66 ​ Π Centimeters. What Is The Radian Measure Of The Corresponding Central Angle?A. 4 5 Π \frac{4}{5} \pi 5 4 ​ Π B. $\frac{3}{5}

Introduction

In geometry, the relationship between the length of an arc and the central angle it subtends is a fundamental concept. Given the length of an arc and the radius of the circle, we can determine the radian measure of the corresponding central angle. In this article, we will explore how to solve for the radian measure of a central angle using the formula: arc length = radius * central angle.

Understanding the Problem

A circle has a radius of 22 centimeters. Arc XY has a length of $\frac{66}{5} \pi$ centimeters. We need to find the radian measure of the corresponding central angle.

Recalling the Formula

The formula to find the arc length is given by:

arc length = radius * central angle

We can rearrange this formula to solve for the central angle:

central angle = arc length / radius

Applying the Formula

Substituting the given values into the formula, we get:

central angle = $\frac{66}{5} \pi$ / 22

To simplify the expression, we can divide the numerator and denominator by their greatest common divisor, which is 11:

central angle = $\frac{6}{5} \pi$ / 2

Now, we can simplify the expression by dividing the numerator and denominator by 2:

central angle = $\frac{3}{5} \pi$

Conclusion

Therefore, the radian measure of the corresponding central angle is $\frac{3}{5} \pi$.

Answer

The correct answer is B. $\frac{3}{5} \pi$.

Additional Information

To further understand the concept, let's consider an example. Suppose we have a circle with a radius of 10 centimeters and an arc length of 15 centimeters. Using the formula, we can find the central angle:

central angle = 15 / 10 = 1.5 radians

This means that the central angle corresponding to the arc length of 15 centimeters is 1.5 radians.

Real-World Applications

The concept of finding the radian measure of a central angle has numerous real-world applications. For instance, in engineering, architects use this concept to design buildings and bridges. In physics, the radian measure of a central angle is used to calculate the angular displacement of an object.

Practice Problems

1. A circle has a radius of 15 centimeters. Arc XY has a length of $\frac{99}{7} \pi$ centimeters. What is the radian measure of the corresponding central angle?
2. A circle has a radius of 20 centimeters. Arc XY has a length of $\frac{88}{5} \pi$ centimeters. What is the radian measure of the corresponding central angle?

Solutions

1. central angle = $\frac{99}{7} \pi$ / 15 = $\frac{33}{7} \pi$ / 5 = $\frac{11}{5} \pi$
2. central angle = $\frac{88}{5} \pi$ / 20 = $\frac{44}{5} \pi$ / 10 = $\frac{22}{5} \pi$

Conclusion

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

A: The radian measure of a central angle is the ratio of the arc length to the radius of the circle.

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

A: To find the radian measure of a central angle, you can use the formula: arc length = radius * central angle. Rearranging this formula, you can solve for the central angle: central angle = arc length / radius.

Q: What is the relationship between the radian measure of a central angle and the arc length?

A: The radian measure of a central angle is directly proportional to the arc length. As the arc length increases, the radian measure of the central angle also increases.

Q: Can I use the radian measure of a central angle to find the arc length?

A: Yes, you can use the radian measure of a central angle to find the arc length. By rearranging the formula: arc length = radius * central angle, you can solve for the arc length: arc length = radius * central angle.

Q: What is the difference between the radian measure of a central angle and the degree measure?

A: The radian measure of a central angle is a unit of measurement that is based on the ratio of the arc length to the radius of the circle. The degree measure, on the other hand, is a unit of measurement that is based on a 360-degree circle.

Q: Can I convert the radian measure of a central angle to degrees?

A: Yes, you can convert the radian measure of a central angle to degrees. To do this, you can use the conversion factor: 1 radian = $\frac{180}{\pi}$ degrees.

Q: What are some real-world applications of the radian measure of a central angle?

A: The radian measure of a central angle has numerous real-world applications, including:

• Engineering: Architects and engineers use the radian measure of a central angle to design buildings and bridges.
• Physics: Physicists use the radian measure of a central angle to calculate the angular displacement of an object.
• Navigation: Pilots and sailors use the radian measure of a central angle to navigate.

Q: Can I use the radian measure of a central angle to solve problems involving circular motion?

A: Yes, you can use the radian measure of a central angle to solve problems involving circular motion. By using the formula: arc length = radius * central angle, you can calculate the distance traveled by an object moving in a circular path.

Q: What are some common mistakes to avoid when working with the radian measure of a central angle?

A: Some common mistakes to avoid when working with the radian measure of a central angle include:

• Confusing the radian measure with the degree measure.
• Failing to convert the radian measure to degrees when necessary.
• Not using the correct formula to calculate the arc length.

Q: Can I use the radian measure of a central angle to solve problems involving circular functions?

A: Yes, you can use the radian measure of a central angle to solve problems involving circular functions. By using the formula: arc length = radius * central angle, you can calculate the value of circular functions such as sine, cosine, and tangent.

Conclusion

In conclusion, the radian measure of a central angle is a fundamental concept in geometry that has numerous real-world applications. By understanding the relationship between the radian measure of a central angle and the arc length, you can solve problems involving circular motion and circular functions.