Arc CD Is $\frac{2}{3}$ Of The Circumference Of A Circle. What Is The Radian Measure Of The Central Angle?A. $\frac{2 \pi}{3}$ Radians B. $ 3 Π 4 \frac{3 \pi}{4} 4 3 Π ​ [/tex] Radians C. $\frac{4 \pi}{3}$ Radians

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Introduction

In geometry, the relationship between the arc length of a circle and its central angle is a fundamental concept. The arc length is a measure of the distance along the circumference of a circle, while the central angle is the angle formed by two radii that intersect at the center of the circle. In this article, we will explore the relationship between arc length and central angle, and use this knowledge to solve a problem involving the arc CD, which is $\frac{2}{3}$ of the circumference of a circle.

The Formula for Arc Length

The formula for arc length is given by:

L=θ×rL = \theta \times r

where:

  • L$ is the arc length

  • \theta$ is the central angle in radians

  • r$ is the radius of the circle

The Relationship Between Arc Length and Central Angle

We are given that the arc CD is $\frac{2}{3}$ of the circumference of a circle. Let's denote the circumference of the circle as $C$. Then, we can write:

CD=23CCD = \frac{2}{3}C

We know that the circumference of a circle is given by:

C=2πrC = 2\pi r

where $r$ is the radius of the circle. Substituting this into the equation above, we get:

CD=23×2πrCD = \frac{2}{3} \times 2\pi r

Simplifying this expression, we get:

CD=4πr3CD = \frac{4\pi r}{3}

Finding the Central Angle

Now that we have the arc length, we can use the formula for arc length to find the central angle. Rearranging the formula to solve for $\theta$, we get:

θ=Lr\theta = \frac{L}{r}

Substituting the expression for $CD$, we get:

θ=4πr3r\theta = \frac{\frac{4\pi r}{3}}{r}

Simplifying this expression, we get:

θ=4π3\theta = \frac{4\pi}{3}

Conclusion

In this article, we explored the relationship between arc length and central angle, and used this knowledge to solve a problem involving the arc CD, which is $\frac{2}{3}$ of the circumference of a circle. We found that the central angle is $\frac{4\pi}{3}$ radians. This result is consistent with the formula for arc length, which states that the arc length is equal to the central angle multiplied by the radius.

Answer

The correct answer is:

  • C. $\frac{4 \pi}{3}$ radians

Additional Examples

Here are a few additional examples to illustrate the relationship between arc length and central angle:

  • If the arc AB is $\frac{1}{2}$ of the circumference of a circle, what is the central angle?
  • If the arc CD is $\frac{3}{4}$ of the circumference of a circle, what is the central angle?
  • If the arc EF is $\frac{2}{5}$ of the circumference of a circle, what is the central angle?

Solutions

  • If the arc AB is $\frac{1}{2}$ of the circumference of a circle, the central angle is $\frac{\pi}{2}$ radians.
  • If the arc CD is $\frac{3}{4}$ of the circumference of a circle, the central angle is $\frac{3\pi}{4}$ radians.
  • If the arc EF is $\frac{2}{5}$ of the circumference of a circle, the central angle is $\frac{4\pi}{5}$ radians.

Conclusion

Introduction

In our previous article, we explored the relationship between arc length and central angle, and used this knowledge to solve a problem involving the arc CD, which is $\frac{2}{3}$ of the circumference of a circle. In this article, we will provide a Q&A guide to help you understand this concept better.

Q: What is the formula for arc length?

A: The formula for arc length is given by:

L=θ×rL = \theta \times r

where:

  • L$ is the arc length

  • \theta$ is the central angle in radians

  • r$ is the radius of the circle

Q: How do I find the central angle given the arc length and radius of the circle?

A: To find the central angle, you can use the formula:

θ=Lr\theta = \frac{L}{r}

Substitute the values of $L$ and $r$ into this formula to find the central angle.

Q: What is the relationship between arc length and central angle?

A: The arc length is equal to the central angle multiplied by the radius of the circle. This can be expressed mathematically as:

L=θ×rL = \theta \times r

Q: How do I find the arc length given the central angle and radius of the circle?

A: To find the arc length, you can use the formula:

L=θ×rL = \theta \times r

Substitute the values of $\theta$ and $r$ into this formula to find the arc length.

Q: What is the unit of measurement for arc length and central angle?

A: The unit of measurement for arc length and central angle is radians.

Q: Can I use the formula for arc length to find the central angle if the arc length is given in degrees?

A: No, the formula for arc length is only applicable if the arc length is given in radians. If the arc length is given in degrees, you will need to convert it to radians before using the formula.

Q: How do I convert degrees to radians?

A: To convert degrees to radians, you can use the following formula:

radians=π180×degrees\text{radians} = \frac{\pi}{180} \times \text{degrees}

Q: Can I use the formula for arc length to find the arc length if the central angle is given in degrees?

A: No, the formula for arc length is only applicable if the central angle is given in radians. If the central angle is given in degrees, you will need to convert it to radians before using the formula.

Q: How do I convert degrees to radians?

A: To convert degrees to radians, you can use the following formula:

radians=π180×degrees\text{radians} = \frac{\pi}{180} \times \text{degrees}

Conclusion

In conclusion, the relationship between arc length and central angle is a fundamental concept in geometry. By using the formula for arc length, you can find the central angle given the arc length and radius of the circle. We hope that this Q&A guide has provided a clear understanding of this concept and has helped to illustrate its importance in geometry.

Additional Resources

For more information on arc length and central angle, please refer to the following resources:

Practice Problems

To practice what you have learned, try the following problems:

  • Find the central angle of a circle with a radius of 5 cm and an arc length of 10 cm.
  • Find the arc length of a circle with a radius of 3 cm and a central angle of 60°.
  • Find the central angle of a circle with a radius of 4 cm and an arc length of 12 cm.

Solutions

  • Find the central angle of a circle with a radius of 5 cm and an arc length of 10 cm:

θ=Lr=105=2 radians\theta = \frac{L}{r} = \frac{10}{5} = 2 \text{ radians}

  • Find the arc length of a circle with a radius of 3 cm and a central angle of 60°:

L=θ×r=60×π180×3=3.14 cmL = \theta \times r = 60 \times \frac{\pi}{180} \times 3 = 3.14 \text{ cm}

  • Find the central angle of a circle with a radius of 4 cm and an arc length of 12 cm:

θ=Lr=124=3 radians\theta = \frac{L}{r} = \frac{12}{4} = 3 \text{ radians}