# Unit 10 Circles Homework 7 Answer Key: The Ultimate Guide

## Introduction

If you're a high school student studying geometry, you're probably familiar with circles. In Unit 10, you'll dive deeper into this topic and explore various properties of circles. Homework 7 will test your understanding of these concepts, and in this article, we'll provide you with the answer key to help you check your work.

## What is Unit 10 Circles Homework 7?

Homework 7 is a set of problems that your geometry teacher will assign you to complete independently. The questions will cover different aspects of circles, such as arcs, chords, tangents, and sectors. You'll need to apply various formulas and theorems to solve each problem.

## Why is it Important to Understand Circles?

Circles are not just a theoretical concept in geometry. They have practical applications in real life, such as in architecture, engineering, and physics. Understanding circles and their properties will help you to solve real-world problems and make better decisions based on data.

## How to Approach Homework 7

To tackle Homework 7, you need to have a solid grasp of the concepts covered in Unit 10. Start by reviewing your notes and textbook, and make sure you understand the definitions, formulas, and theorems related to circles. Then, read each problem carefully and identify what is being asked. Use diagrams and equations to organize your thoughts and develop a plan to solve the problem. Finally, double-check your work and make sure you've answered the question correctly.

## Answer Key for Homework 7

### Problem 1

Find the length of arc AB, given that the radius of circle O is 6 cm and the central angle AOB measures 60 degrees.

Solution: The formula for the length of an arc is L = rθ, where r is the radius and θ is the central angle in radians. Since the angle is given in degrees, we need to convert it to radians by multiplying it by π/180. Thus, θ = 60π/180 = π/3. Substituting r = 6 cm and θ = π/3, we get L = 6π/3 = 2π cm. Therefore, the length of arc AB is 2π cm.

### Problem 2

Find the measure of angle AOB, given that the length of arc AB is 4π cm and the radius of circle O is 2 cm.

Solution: The formula for the length of an arc is L = rθ, where r is the radius and θ is the central angle in radians. We are given L = 4π cm and r = 2 cm. Solving for θ, we get θ = L/r = (4π)/(2) = 2π radians. To convert this to degrees, we multiply by 180/π. Thus, θ = (2π)(180/π) = 360 degrees. Therefore, the measure of angle AOB is 360 degrees.

### Problem 3

Find the length of chord AB, given that the radius of circle O is 5 cm and the distance from the center of the circle to chord AB is 4 cm.

Solution: The formula for the length of a chord in terms of the radius and the distance from the center to the chord is c = 2sqrt(r^2 - d^2), where r is the radius and d is the distance from the center to the chord. Substituting r = 5 cm and d = 4 cm, we get c = 2sqrt(5^2 - 4^2) = 6 cm. Therefore, the length of chord AB is 6 cm.

### Problem 4

Find the length of tangent BC, given that the radius of circle O is 7 cm and the distance from point B to the center of the circle is 8 cm.

Solution: The formula for the length of a tangent in terms of the radius and the distance from the point of contact to the center of the circle is t = sqrt(r^2 - d^2), where r is the radius and d is the distance from the point of contact to the center. Substituting r = 7 cm and d = 8 cm - 7 cm = 1 cm, we get t = sqrt(7^2 - 1^2) = sqrt(48) = 4sqrt(3) cm. Therefore, the length of tangent BC is 4sqrt(3) cm.

### Problem 5

Find the area of sector AOB, given that the radius of circle O is 9 cm and the central angle AOB measures 120 degrees.

Solution: The formula for the area of a sector is A = (1/2)r^2θ, where r is the radius and θ is the central angle in radians. Since the angle is given in degrees, we need to convert it to radians by multiplying it by π/180. Thus, θ = 120π/180 = 2π/3. Substituting r = 9 cm and θ = 2π/3, we get A = (1/2)(9^2)(2π/3) = 27π/2 sq.cm. Therefore, the area of sector AOB is 27π/2 sq.cm.

## Conclusion

Congratulations! You've reached the end of this article and hopefully have a better understanding of Unit 10 Circles Homework 7. Remember, mastering circles requires practice and patience, but it's worth the effort. Keep studying and exploring this fascinating topic, and you'll be amazed at how circles are all around us in the world.