Select The Correct Answer.Points $A$ And $B$ Lie On A Circle Centered At Point $O$. If $OA = 5$ And $\frac{\text{length Of } \hat{AB}}{\text{circumference}} = \frac{1}{4}$, What Is The Area Of Sector

Feb 26, 2025 by ADMIN 200 views

**Understanding the Problem and Solution Approach**

Introduction

In this problem, we are given a circle centered at point $O$ with points $A$ and $B$ lying on its circumference. We are also given that $OA = 5$ and the ratio of the length of arc $\hat{AB}$ to the circumference of the circle is $\frac{1}{4}$ . Our goal is to find the area of the sector formed by points $A$ and $B$ .

Recalling Circle Properties

Before we dive into the solution, let's recall some important properties of circles. The circumference of a circle is given by the formula $C = 2\pi r$ , where $r$ is the radius of the circle. The length of an arc on a circle can be calculated using the formula $s = \frac{\theta}{360} \times 2\pi r$ , where $\theta$ is the central angle subtended by the arc.

Calculating the Circumference

Given that $OA = 5$ , we can calculate the circumference of the circle using the formula $C = 2\pi r$ . Substituting $r = 5$ , we get:

C = 2\pi \times 5 = 10\pi

Calculating the Length of Arc $\hat{AB}$

We are given that the ratio of the length of arc $\hat{AB}$ to the circumference of the circle is $\frac{1}{4}$ . Let's denote the length of arc $\hat{AB}$ as $s$ . Then, we can write:

\frac{s}{10\pi} = \frac{1}{4}

Solving for $s$ , we get:

s = \frac{10\pi}{4} = \frac{5\pi}{2}

Calculating the Central Angle

The length of arc $\hat{AB}$ is given by the formula $s = \frac{\theta}{360} \times 2\pi r$ . We can rearrange this formula to solve for $\theta$ :

\theta = \frac{360 \times s}{2\pi r}

Substituting $s = \frac{5\pi}{2}$ and $r = 5$ , we get:

\theta = \frac{360 \times \frac{5\pi}{2}}{2\pi \times 5} = 90

Calculating the Area of Sector $AOB$

The area of a sector of a circle can be calculated using the formula $A = \frac{\theta}{360} \times \pi r^2$ . Substituting $\theta = 90$ and $r = 5$ , we get:

A = \frac{90}{360} \times \pi \times 5^2 = \frac{1}{4} \times \pi \times 25 = \frac{25\pi}{4}

Conclusion

In this problem, we were given a circle centered at point $O$ with points $A$ and $B$ lying on its circumference. We were also given that $OA = 5$ and the ratio of the length of arc $\hat{AB}$ to the circumference of the circle is $\frac{1}{4}$ . Our goal was to find the area of the sector formed by points $A$ and $B$ . We calculated the circumference of the circle, the length of arc $\hat{AB}$ , the central angle, and finally the area of sector $AOB$ . The final answer is $\boxed{\frac{25\pi}{4}}$ .

Discussion

This problem requires a good understanding of circle properties and formulas. The key concept used in this problem is the relationship between the length of an arc and the central angle subtended by it. The problem also requires the use of formulas to calculate the circumference, length of arc, central angle, and area of sector. The solution approach involves a step-by-step calculation of each of these quantities.

Solutions to Related Problems

A circle has a radius of 6 cm. If the length of an arc is 12 cm, what is the central angle subtended by the arc?

The length of an arc is given by the formula $s = \frac{\theta}{360} \times 2\pi r$ . We can rearrange this formula to solve for $\theta$ :
$\theta = \frac{360 \times s}{2\pi r}$
Substituting $s = 12$ and $r = 6$ , we get:
$\theta = \frac{360 \times 12}{2\pi \times 6} = 120$
Therefore, the central angle subtended by the arc is 120°.
A circle has a circumference of 20 cm. If the length of an arc is 5 cm, what is the central angle subtended by the arc?

The length of an arc is given by the formula $s = \frac{\theta}{360} \times 2\pi r$ . We can rearrange this formula to solve for $\theta$ :
$\theta = \frac{360 \times s}{2\pi r}$
Substituting $s = 5$ and $C = 20$ , we get:
$\theta = \frac{360 \times 5}{2\pi \times 10} = 18$
Therefore, the central angle subtended by the arc is 18°.
A circle has a radius of 8 cm. If the central angle subtended by an arc is 60°, what is the length of the arc?

The length of an arc is given by the formula $s = \frac{\theta}{360} \times 2\pi r$ . Substituting $\theta = 60$ and $r = 8$ , we get:
$s = \frac{60}{360} \times 2\pi \times 8 = \frac{1}{6} \times 16\pi = \frac{8\pi}{3}$
Therefore, the length of the arc is $\frac{8\pi}{3}$ cm.
Q&A: Understanding Circle Properties and Formulas =====================================================

Q: What is the circumference of a circle?

A: The circumference of a circle is given by the formula $C = 2\pi r$ , where $r$ is the radius of the circle.

Q: How do I calculate the length of an arc on a circle?

A: The length of an arc on a circle can be calculated using the formula $s = \frac{\theta}{360} \times 2\pi r$ , where $\theta$ is the central angle subtended by the arc.

Q: What is the relationship between the length of an arc and the central angle subtended by it?

A: The length of an arc is directly proportional to the central angle subtended by it. The formula $s = \frac{\theta}{360} \times 2\pi r$ shows that the length of an arc is equal to the fraction of the circumference that the central angle subtends.

Q: How do I calculate the central angle subtended by an arc?

A: The central angle subtended by an arc can be calculated using the formula $\theta = \frac{360 \times s}{2\pi r}$ , where $s$ is the length of the arc.

Q: What is the area of a sector of a circle?

A: The area of a sector of a circle can be calculated using the formula $A = \frac{\theta}{360} \times \pi r^2$ , where $\theta$ is the central angle subtended by the sector.

Q: How do I calculate the area of a sector of a circle?

A: To calculate the area of a sector of a circle, you need to know the central angle subtended by the sector and the radius of the circle. You can then use the formula $A = \frac{\theta}{360} \times \pi r^2$ to calculate the area.

Q: What is the relationship between the area of a sector and the central angle subtended by it?

A: The area of a sector is directly proportional to the central angle subtended by it. The formula $A = \frac{\theta}{360} \times \pi r^2$ shows that the area of a sector is equal to the fraction of the area of the circle that the central angle subtends.

Q: How do I use the formulas to solve problems involving circles?

A: To use the formulas to solve problems involving circles, you need to identify the given information and the unknown quantity. You can then use the formulas to calculate the unknown quantity.

Q: What are some common mistakes to avoid when working with circle formulas?

A: Some common mistakes to avoid when working with circle formulas include:

Not converting the central angle from degrees to radians
Not using the correct formula for the length of an arc or the area of a sector
Not substituting the correct values for the variables in the formulas
Not checking the units of the answer to ensure that they are correct

Q: How do I check my answers to ensure that they are correct?

A: To check your answers, you can use the following steps:

Read the problem carefully to ensure that you understand what is being asked
Identify the given information and the unknown quantity
Use the formulas to calculate the unknown quantity
Check the units of the answer to ensure that they are correct
Check the answer to ensure that it makes sense in the context of the problem

Q: What are some real-world applications of circle formulas?

A: Circle formulas have many real-world applications, including:

Calculating the circumference and area of a circle to determine the size of a circular object
Calculating the length of an arc to determine the distance traveled by an object moving along a circular path
Calculating the area of a sector to determine the area of a portion of a circle
Calculating the central angle subtended by an arc to determine the angle of a circular object

Q: How do I use circle formulas to solve problems involving circular motion?

A: To use circle formulas to solve problems involving circular motion, you need to identify the given information and the unknown quantity. You can then use the formulas to calculate the unknown quantity.

Q: What are some common problems that involve circular motion?

A: Some common problems that involve circular motion include:

Calculating the distance traveled by an object moving along a circular path
Calculating the time it takes for an object to complete one revolution around a circular path
Calculating the speed of an object moving along a circular path
Calculating the acceleration of an object moving along a circular path

Q: How do I use circle formulas to solve problems involving circular shapes?

A: To use circle formulas to solve problems involving circular shapes, you need to identify the given information and the unknown quantity. You can then use the formulas to calculate the unknown quantity.

Q: What are some common problems that involve circular shapes?

A: Some common problems that involve circular shapes include:

Calculating the circumference and area of a circle to determine the size of a circular object
Calculating the length of an arc to determine the distance traveled by an object moving along a circular path
Calculating the area of a sector to determine the area of a portion of a circle
Calculating the central angle subtended by an arc to determine the angle of a circular object