Arc CD Is 2/3 Of The Circumference Of A Circle. What Is The Radian Measure Of The Central Angle?

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Understanding the relationship between arc length and central angles is a fundamental concept in mathematics, particularly in trigonometry and geometry. When dealing with circles, the arc length, which is a portion of the circle's circumference, is directly proportional to the central angle that subtends it. This article aims to explore this relationship in detail and specifically address the problem where arc CD is $\frac{2}{3}$ of the circumference of a circle, and we need to determine the radian measure of the central angle.

Fundamentals of Arc Length and Central Angles

Before diving into the specific problem, it is crucial to grasp the underlying principles of arc length and central angles. A central angle is an angle whose vertex is at the center of a circle, and its sides intersect the circle at two distinct points. The arc formed between these two points is called the intercepted arc. The measure of the central angle is directly related to the length of the intercepted arc.

The circumference of a circle, denoted by C, is the total distance around the circle. It is given by the formula:

C=2πrC = 2 \pi r

where r is the radius of the circle. An arc length, denoted by s, is a portion of the circumference. The relationship between the arc length, the central angle, and the radius is given by the formula:

s=rθs = r\theta

where s is the arc length, r is the radius, and θ\theta is the central angle measured in radians. This formula is the cornerstone of understanding the connection between arc length and central angles.

To fully appreciate this relationship, consider the following:

  • A complete circle corresponds to a central angle of 2$\pi$ radians (or 360 degrees).
  • Half a circle corresponds to a central angle of $\pi$ radians (or 180 degrees).
  • A quarter of a circle corresponds to a central angle of $ rac{\pi}{2}$ radians (or 90 degrees).

The central angle essentially carves out a fraction of the total circumference, and the arc length represents that fraction. Therefore, if we know the fraction of the circumference an arc represents, we can determine the corresponding fraction of the total central angle (2$\pi$ radians).

Solving the Problem Arc CD is 2/3 of the Circumference

Now, let's address the specific problem: Arc CD is $\frac{2}{3}$ of the circumference of a circle. Our goal is to find the radian measure of the central angle that subtends arc CD. We can approach this problem using the principles outlined above. Let $\theta$ be the central angle we are trying to find.

We are given that the arc length s of arc CD is $\frac{2}{3}$ of the circumference C. Mathematically, this can be written as:

s=23Cs = \frac{2}{3} C

We know that the circumference C is given by:

C=2πrC = 2 \pi r

Substituting this into the equation for s, we get:

s=23(2πr)=4πr3s = \frac{2}{3} (2 \pi r) = \frac{4 \pi r}{3}

We also know the relationship between arc length, radius, and central angle:

s=rθs = r \theta

Now, we can substitute the expression for s that we found earlier:

4πr3=rθ\frac{4 \pi r}{3} = r \theta

To solve for $\theta$, we can divide both sides of the equation by r (assuming r is not zero):

4π3=θ\frac{4 \pi}{3} = \theta

Therefore, the central angle $\theta$ is $\frac{4 \pi}{3}$ radians. This corresponds to option C in the provided choices.

Detailed Explanation of the Solution

The solution involves a few key steps that are worth elaborating on to ensure a thorough understanding. First, we established the relationship between the arc length and the circumference. We were given that the arc CD is $\frac{2}{3}$ of the circumference. This means that the length of the arc is $\frac{2}{3}$ of the total distance around the circle.

Next, we expressed the circumference in terms of the radius using the formula $C = 2 \pi r$. This allowed us to write the arc length s as a fraction of the radius:

s=23(2πr)=4πr3s = \frac{2}{3} (2 \pi r) = \frac{4 \pi r}{3}

The crucial step is recognizing the formula that connects arc length, radius, and central angle: $s = r \theta$. This formula tells us that the arc length is equal to the product of the radius and the central angle in radians. By substituting the expression we found for s into this formula, we could relate the central angle directly to the radius:

4πr3=rθ\frac{4 \pi r}{3} = r \theta

Finally, we solved for $\theta$ by dividing both sides by r. This step is valid as long as the radius r is not zero, which is a reasonable assumption for any circle. The result is:

θ=4π3 radians\theta = \frac{4 \pi}{3} \text{ radians}

This result signifies that the central angle that subtends arc CD is $\frac{4}{3}$ times the radian measure of a straight angle ($\pi$ radians). In terms of degrees, this angle would be $\frac{4}{3} \times 180 = 240$ degrees.

Practical Applications and Extensions

The concept of arc length and central angles has numerous practical applications in various fields, including:

  • Navigation: Calculating distances along curved paths, such as airplane routes or ship trajectories.
  • Engineering: Designing curved structures, such as bridges, arches, and gears.
  • Astronomy: Determining the angular size of celestial objects and their distances.
  • Computer Graphics: Creating and manipulating circular shapes and arcs in graphical interfaces.

Furthermore, this concept can be extended to more advanced topics in mathematics, such as:

  • Trigonometry: The unit circle definition of trigonometric functions relies heavily on the relationship between arc length and angles.
  • Calculus: Arc length is a fundamental concept in integral calculus, used to calculate the length of curves.
  • Differential Geometry: The curvature of a curve is related to the rate of change of the central angle with respect to arc length.

Conclusion

In conclusion, understanding the relationship between arc length and central angles is crucial for solving various problems in mathematics and its applications. The problem of finding the radian measure of the central angle when the arc length is a fraction of the circumference highlights the importance of this relationship. By applying the formula $s = r \theta$ and the concept of proportionality, we can effectively determine the central angle. The solution to the problem, where arc CD is $\frac{2}{3}$ of the circumference, is $\frac{4 \pi}{3}$ radians. This concept is not only fundamental in geometry and trigonometry but also has far-reaching applications in various fields, making it an essential topic for students and professionals alike.

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Arc CD is $\frac{2}{3}$ of the circumference of a circle. What is the measurement in radians of the central angle that corresponds to arc CD?