Chords, Arcs, Central Angles and Inscribed Angles PDF

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A coursework about Chords, Arcs, Central Angles and Inscribed Angles...

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CHORDS, ARCS, CENTRAL ANGLES AND INSCRIBED ANGLES In this lesson, you will learn the following: 1. Describes the relationship between the central angle and the intercepted arc. 2. Describes the relationship between the inscribed angle and the intercepted arc. 3. Derives inductively the relations among chords, arcs, central angles and inscribed angles. 4. Proves theorems related to chords, arcs, central angles, and inscribed angles.

EXPLORE You have been introduced to circles in the previous grade levels. You also encountered some problems and terms involving circles which are important in learning Geometry. In this lesson, you will look at how chords, angles and arcs are related in a circle. You will also learn the relationships among chords, arcs ad angles which are useful in solving real world problems. As you go through this lesson, think of this important question: “How do the relationships among chords, arcs, and angles of a circle facilitate finding solutions to real-life problems and making decisions?” To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or refer to your textbook.

ACTIVITY 1: Spin a Prize In a shopping mart, a customer gets a chance to spin a roulette and a prize for every Php 500 purchase. The supervisor gave the roulette designer the following instructions. Prize A face towel A detergent soap bar An eco-bag A gift certificate worth Php 1000

Color Blue Yellow Red

Percent 20% 50% 25%

Green

5%

How should the roulette be created to accurately reflect the indicated area in percent of the given color and label for the prize? Questions: 1. What geometric concept can you associate with the problem? Why?

2. Can you cite any problems or situations associated with this geometric concept? Please specify.

1

3. How will you design the roulette based from the geometric concept you have stated above?

End of EXPLORE You have just tried to find out how geometric concepts let you analyzed and solved some real-life situations. Let us now strengthen your insights by doing the succeeding activities which will help you later on to deal with more challenging problems. Now move on to the next activity to learn the knowledge and skills needed for you to become a good problem solver.

FIRM UP In this section, your goal is to learn and understand key concepts of chords, arcs, central angles and inscribed angles which are important in solving problems involving circles. There are activities in this section which will help you discover and understand the terms and conditions which are useful in dealing with real life problems related to chords, arcs and angles of a circle. Now, take a look at the next activity.

ACTIVITY 2: Measure Me and You Use the figures below to answer the questions that follow: M B

C

A

F

O R

E

N

Q D

Figure 1

P Figure 2

1. What is the measure of each of the following angles in Figure 1? Use a protractor. a. ∠BAC c. ∠EAF e. ∠DAE b. ∠BAF d. ∠CAD

2

´ , NO ´ , PO ´ , QO ´ ´ are radii of circle O. What is 2. In Figure 2, MO and RO the measure of each of the following angles? Use a protractor. a. ∠MON c. ∠NOP e. ∠POQ b. ∠MOR d. ∠QOR 3. How do you describe the angles in each figure? 4. In Figure 2, what is the intercepted arc of ∠MON? How about ∠NOP? ∠POQ? ∠QOR? ∠MOR? Complete the table below. Central Angle Measure Intercepted Arc a. ∠MON b. ∠NOP c. ∠POQ d. ∠QOR e. ∠MOR

Were you able to measure the angles accurately and find the sum of their measures? Were you able to determine the relationship between the measures of the central angle and its intercepted arc? For sure you were able to do it.

ACTIVITY 3: Travel Safely Use the situation below to answer the questions that follow.

Rowel is designing a mag wheel like the one shown below. He decided to put 6 spokes which divide the rim into 6 equal parts. Questions:

a. What is the degree measure of each arc along the rim? How about each angle formed by the spokes at the hub?

How did you find the preceding activities? b.Are If you were to design a wheel, how many spokes will you use to divide the rim? Why? you ready to learn about the relations among chords, arcs, and central angles of a circle? I am sure you are!!! From the activities done, you were able to recall and describe the terms related to circles. You were able to find out how circles are illustrated CENTRAL ANGLES AND ARCS in real-life situations. But how do the relationships A central angle of a circle is an angle among chords, arcs, and formed by two rays whose vertex is the center central angles of a circle facilitate finding solutions of the circle. Each ray intersects the circle at a to real-life problems and point, dividing it into arcs. making decisions? You will find these out in the activities in the next section. Before doing these activities, read and

Arc

B

3

Centr Ang

understand first some important notes on this lesson and the examples

In the figure on the right, is a central angle. Its sides divide ʘ A into arcs. One arc is the curve containing points B and C. The other arc is the curve containing points B, D, and C. D

Arc Arcs of a Circle An arc is a part of a circle. The symbol for arc is . A semicircle is an arc with a measure equal to one-half the circumference of a circle. It is named by using the two endpoints and another point on the arc.

C

Example: The curve from point N to point Z is an arc. It is part of ʘ O and is named as arc NZ or NZ. Other arcs of ʘ O are CN, CZ, CZN, CNZ, and NCZ. If m CNZ is one-half the circumference of ʘ O, then it is a semicircle.

O N

Z

A minor arc is an arc of the circle that measures less than a semicircle. It is named usually by using the two endpoints of the arc.

N

Examples: JN, NE, and JE O

A major arc is an arc of a circle that measures greater than a semicircle. It is named by using the two endpoints and another point on the arc.

J E

Examples: JEN, JNE, and EJN Degree Measure of an Arc 1. The degree measure of a minor arc is the measure of the central angle which intercepts the arc.

O

Example:∠GEO is a central angle. It intercepts ʘ E at points G and O. The measure of GO is equal to the measure of ∠GEO. If m∠GEO =118°, then mGO = 118°.

G

E

M

2. The degree measure of a major arc is equal to 360 minus the measure of the minor arc with the same endpoints. Example: If mGO = 118°, then mOMG = 360° – mGO. That is, mOMG = 360° – 118° = 242°. Answer: mOMG = 242° 3. The degree measure of a semicircle is 180°.

ACTIVITY 4: Identify and Name Me Use A below to identify and name the following. Then, answer the questions that follow. 1. 2 semicircles in the figure

2. 4 minor arcs and their corresponding major arcs

F

E

G

B

4

A

3. 4 central angles

D

Questions: a. How did you identify and name the semicircles?

C

How about the minor arcs and the major arcs? Central angles? b. Do you think the circle has more semicircles, arcs, and central angles? Show.

ACTIVITY 5: Find My Degree In ʘA below, m ∠LAM =42, m ∠HAG= 30, and ∠KAH is a right angle. Find the following measure of an angle or an arc, and explain how you arrived at your answer. 1. m ∠LAK

6. m LK

2. m ∠JAK

7. m JK

3. m ∠LAJ

8. m LMG

4. m ∠JAH

9. m JH

5. m ∠KAM

10. m KLM

K

L

M

J A H

Were you able to identify and name the arcs and central angles in the

G

Name the angles and their intercepted arcs in the figure below. Answer the questions that follow. L K

ACTIVITY 6: My Angles and Intercepted Arcs M

A

J

Angles

Arc that the angle intercepts

Questions: 1. How did you identify and name the angles in the figure? How about the arcs that these angles intercept?

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