10
Mathematics
Quarter 2 – Module 2:
Relationship Among Chords, Arcs,
Central Angles, and Inscribed Angles of
Circles
(M10GE-lic-1)
1
Mathematics – Grade 10
Self-Learning Module (SLM)
Quarter 2 – Module 2: Relationship Among Chords, Arcs, Central Angles, And
Inscribed Angles of Circles
First Edition, 2020
Republic Act 8293, section 176 states that: No copyright shall subsist in any work of
the Government of the Philippines. However, prior approval of the government agency or
office wherein the work is created shall be necessary for exploitation of such work for profit.
Such agency or office may, among other things, impose as a condition the payment of
royalties.
Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names,
trademarks, etc.) included in this module are owned by their respective copyright holders.
Every effort has been exerted to locate and seek permission to use these materials from
their respective copyright owners. The publisher and authors do not represent nor claim
ownership over them.
Development Team of the Module
Writer: Eva Jones D. Siangco, MScieEd-Math
Editors: (Language/Social Content): Rachelle T. Amando, PhD/George S. Hofer
Reviewers: Nora B. Mendoza /Maureen Socorro N. Muñasque
Illustrator: Geunicar A. Perez, MAELT
Layout Artist: Geunicar A. Perez, MAELT
Cover Art Designer: Ian Caesar E. Frondoza
Management Team: Allan G. Farnazo, CESO IV – Regional Director
Fiel Y. Almendra, CESO V – Assistant Regional Director
Concepcion F. Balawag, CESO V - Schools Division Superintendent
Edgar S. Sumapal - Assistant Schools Division Superintendent
Gilbert B. Barrera – Chief, CLMD
Arturo D. Tingson Jr. – REPS, LRMS
Peter Van C. Ang-ug – REPS, ADM
Jade T. Palomar - REPS, Mathematics
Pancho G. Balawag, Ed. D - CID Chief
Engr. Reynaldo SE Villan - EPS In Charge of LRMS
Vivencio O. Aniñon, Ed.D - Division ADM Coordinator
Engr. Reynaldo SE Villan – EPS, Math
Printed in the Philippines by Department of Education – SOCCSKSARGEN Region
Office Address:
Telefax:
E-mail Address:
Regional Center, Brgy. Carpenter Hill, City of Koronadal
(083) 2288825/ (083) 2281893
region12@deped.gov.ph
2
10
Mathematics
Quarter 2 – Module 2:
Relationship Among Chords, Arcs, Central
Angles, and Inscribed Angles of Circles
(M10GE-lic-1)
3
Introductory Message
For the facilitator:
Welcome to the MATHEMATICS 10 Self-Learning Module (SLM) on
Relationship Among Chords, Arcs, Central Angles, and Inscribed
Angles.
This module was collaboratively designed, developed and reviewed by
educators both from public and private institutions to assist you, the
teacher or facilitator in helping the learners meet the standards set by the K
to 12 Curriculum while overcoming their personal, social, and economic
constraints in schooling.
This learning resource hopes to engage the learners into guided and
independent learning activities at their own pace and time. Furthermore,
this also aims to help learners acquire the needed 21st century skills while
taking into consideration their needs and circumstances.
In addition to the material in the main text, you will also see this box in the
body of the module:
Notes to the Teacher
This contains helpful tips or strategies
that will help you in guiding the learners.
As a facilitator you are expected to orient the learners on how to use this
module. You also need to keep track of the learners' progress while allowing
them to manage their own learning. Furthermore, you are expected to
encourage and assist the learners as they do the tasks included in the
module.
2
For the learner:
Welcome
to
the
Mathematics
10
Self-Learning
Module
(SLM)
on
Relationship Among Chords, Arcs, Central Angles, and Inscribed
Angles.
The hand is one of the most symbolized part of the human body. It is often
used to depict skill, action and purpose. Through our hands we may learn,
create and accomplish. Hence, the hand in this learning resource signifies
that you as a learner is capable and empowered to successfully achieve the
relevant competencies and skills at your own pace and time. Your academic
success lies in your own hands!
This module was designed to provide you with fun and meaningful
opportunities for guided and independent learning at your own pace and
time. You will be enabled to process the contents of the learning resource
while being an active learner.
This module has the following parts and corresponding icons:
What I Need to
Know
This will give you an idea of the skills or
competencies you are expected to learn
in the module.
What I Know
This part includes an activity that aims
to check what you already know about
the lesson to take. If you get all the
answers correct (100%), you may decide
to skip this module.
What’s In
This is a brief drill or review to help you
link the current lesson with the previous
one.
What’s New
In this portion, the new lesson will be
introduced to you in various ways such
as a story, a song, a poem, a problem
opener, an activity or a situation.
What is It
This section provides a brief discussion of
the lesson. This aims to help you discover
and understand new concepts and skills.
What’s More
This comprises activities for independent
practice to solidify your understanding
and skills of the topic. You may check the
answers to the exercises using the
Answer Key at the end of the module.
What I Have
Learned
This includes questions or blank
sentence/paragraph to be filled in to
process what you learned from the
3
lesson.
What I Can Do
This section provides an activity which
will help you transfer your new
knowledge or skill into real life situations
or concerns.
Assessment
This is a task which aims to evaluate
your level of mastery in achieving the
learning competency.
Additional
Activities
In this portion, another activity will be
given to you to enrich your knowledge or
skill of the lesson learned. This also
tends retention of learned concepts.
Answer Key
This contains answers to all activities in
the module.
At the end of this module you will also find:
References
This is a list of all sources used in
developing this module.
The following are some reminders in using this module:
1. Use the module with care. Do not put unnecessary mark/s on any
part of the module. Use a separate sheet of paper in answering the
exercises.
2. Don’t forget to answer What I Know before moving on to the other
activities included in the module.
3. Read the instruction carefully before doing each task.
4. Observe honesty and integrity in doing the tasks and checking your
answers.
5. Finish the task at hand before proceeding to the next.
6. Return this module to your teacher/facilitator once you are through
with it.
If you encounter any difficulty in answering the tasks in this module, do
not hesitate to consult your teacher or facilitator. Always bear in mind
that you are not alone.
We hope that through this material, you will experience meaningful
learning and gain deep understanding of the relevant competencies. You
can do it!
4
What I Need to Know
Start this module by assessing your knowledge of the different
mathematical concepts previously studied and your skills in performing
mathematical operations.
These knowledge and skills will help you
understand circles. As you go through this lesson, think of this important
question: “How do the relationships among chords, arcs, central angles and
inscribed angles of a circle used in 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 peers or refer to the modules you have studied earlier. You may check
your work with your teacher’s guidance.
The module is divided into four lessons, namely:




Lesson
Lesson
Lesson
Lesson
1
2
3
4
–
–
–
–
Central Angles, Arcs and Chords
Theorems on Central Angles, Arcs and Chords
Intercepted Arcs and Inscribed Angles
Theorems on Intercepted Arcs and Inscribed Angles
After going through this module, you are expected to:
1. Identify and describe terms related to circles;
2. Use the relationship among chords, arcs, central angles, and
inscribed angles of circles; and
3. Identify and use theorems on central angles, arcs, chords and
inscribed angles of circles.
5
What I Know
Find out how much you already know about topics on this module.
Choose the letter that you think best answers each of the following
questions.
1. What is an angle whose vertex is on a circle and whose sides contain
chords of the circle?
A. Central angle
C. circumscribed angle
B. Inscribed angle
D. intercepted angle
2. In circle S at the right, what is m∠VSI if m VI = 140?
A. 35
C. 140
B. 75
D. 230
3. What is the sum of the measures of the central angles of a circle with
no common interior points?
A. 120
C. 360
B. 240
D. 480
4. If an inscribed angle of a circle intercepts a semicircle, then the angle
is___.
A. Acute
C. obtuse
B. Right
D. straight
S
5. In the circle below, what is the measure of ∠SAY if DSY is a
semicircle and m∠ SAD = 70?
A. 20
C. 110
B. 70
D. 150
D
A
6. Quadrilateral SMIL is incribed in oE. If m∠SMI = 78 and m∠MSL =
95, find m∠MIL.
A. 78
B. 85
M
C. 95
78°
D. 102
S
7. In oM on the right, what is m∠BRO if m∠BMO = 60?
A. 120
C. 30
B. 60
D. 15
B
R
M
O
6
95
° L
I
E
Y
E
8. In
P
U on the right, what is m∠PRE if m∠PUE = 56 ?
A. 28
C. 56
B. 34
U
D. 124
R
9. Catherine designed a pendant. It is a regular hexagon set in a circle.
Suppose the opposite vertices are connected by line segments and meet at
the center of the circle. What is the measure of each angle formed at the
center?
A. 22.5
B. 45
C. 60
D. 72
10. What is an angle whose vertex is any point on the circle and sides
contain chords of the circle?
A.central angle
C. circumscribed angle
B. inscribed angle
D. intercepted angle
11. The opposite angles of a quadrilateral inscribed in a circle are___.
A. right
C. complementary
B. obtuse
D. supplementary
12. What is an arc with a measure equal to one-half the circumference of
a circle?
A. Major arc
C. inscribed arc
B. Minor arc
D. semicircle
O
13. In oT on the right ,PIO and PLO intercept PO.
P
If mPIO= 55, what is the m PO?
A. 110
C. 55
B. 95
D. 40
T
For items 14 & 15.
I
In oA, SE  NE and SC  NT.
L
14. If m SE = 80, then what is m T?
A. 100
C. 80
B. 90
D. 40
15. If mthen what is m SC?
A. 60
C. 100
B. 70
D. 120
C
A
S
E
N
Great! You did a good job. You’re now ready for the next set of
activities.
7
T
Lesson
2
CHORDS, ARCS ,CENTRAL
ANGLES, INSCRIBED ANGLES AND
INTERCEPTED ARCS
Congratulations! You are now ready to start with another module. Start
Lesson 3.1 of this module by evaluating your knowledge of the various
previously studied mathematical concepts and your skills in mathematical
operations. These knowledge and skills will help you understand circles.
Think about the crucial question as you go through this lesson: “How do the
relationships among chords, arcs, central angles, inscribed angles and
intercepted arcs of a circle facilitate finding solutions to real-life problems
and making decisions?” Carry out each operation to find the answer. If you
encounter some trouble in reacting to the exercises, seek guidance from your
instructor or peers or refer to the modules you learned earlier. You should
check with your teacher about your work. (M10GE-lic-1).
What’s In
Let’s remember the lesson in your previous level about circles. This
time we will focus on relationship between chords, arcs and central angles of
circles. Are you ready?
Circle
In geometry, a circle is a closed curve formed by a set of points on a
plane that is the same distance from its center O. That distance is known as
the radius of the circle.
Radius
The radius of the circle is a line segment from the center of the circle to a
point on the circle. The plural of radius is radii.
8
In the above diagram, O is the center of the circle and OB and OC are radii
of the circle. The radii of a circle are all of the same length. The radius is half
of the length of the diameter.
OB =
1
2
AB
Diameter
The diameter of a circle is a line segment that passes through the
center of the circle and has its endpoints on the circle. All the diameters of
the same circle have the same length.
Chord
A chord is a line segment with both endpoints on the circle. The
diameter is a special chord that passes through the center of the circle. The
diameter would be the longest chord in the circle.
Arc
An arc is a part of a circle.
In the diagram above, the part of the circle from B to C forms an arc.
9
What’s New
An arc can be measured in degrees.
In the circle above, arc BC is equal to the ∠BOC that is 45°.
INTRODUCTION
This module focuses on the relationship between chords, arcs, central and
inscribed angles of circles and how the concept is utilized in our daily life.
Study the situation below and answer the questions that follow.
Have you ever seen pulling a cart, or riding a bus with wheels that aren't
round? Do you think you can easily move heavy items from one location to
another, or fly far away as quickly as you can? Without circles, what
challenge do you think you will experience? Have you ever noticed the
importance of circles in travel, manufacturing, leisure, navigation, carpentry
and in your daily life?
Find out the answers to these questions and determine the vast
applications of circles through this module.
Let’s try with our first activity.
Activity 1. Know My Terms and Conditions
Use the figure below to identify and name the following terms related to oA.
Then, answer the questions that follow.
1. a radius
2. a diameter
3. a chord
4. a semicircle
5. a minor arc
6. a major arc
7. two central angles
8. two inscribed angles
s
10
J
N
L
A
E
Questions:
a. How did you identify and name the radius, diameter, and chord?
b. How about the semicircle, minor arc, and major arc? inscribed
angle and central angle?
c. How do you describe a radius, diameter, and chord of a circle?
d. How about the semicircle, minor arc, and major arc? inscribed
angle and central angle?
Write your answers in the table below.
Terms Related to Circles
Description
1. radius
2. diameter
3. chord
4. semicircle
5. minor arc
6. major arc
7. central angle
8. inscribed angle
e. How do you differentiate among the radius, diameter, and chord
of a circle?
f. How about the semicircle, minor arc, and major arc? inscribed
angle and central angle?
Are you in a position to define and explain the terminology of the circles?
Can you identify and distinguish them? Now that you are aware of the
essential circle-related terminology, let us deepen your understanding of
finding angle measures in a circle. You need this mathematical ability to
identify the relationships as you go through thisN lesson between chords,
G
arcs, and central angles.
I
U
11
What is It
How did you approach the events previous to this? Are you prepared to
learn about a circle 's relationships among chords, arcs, central, inscribed
angles and intercepted arcs?
I 'm sure you really are!!! You have been able to recall and explain the
terms relating to circles from the things you have performed. You were able
to find out how the circles in real-life situations are represented. But how do
the relationships among a circle's chords, arcs, central, inscribed angles and
intercepted arcs promote the discovery and decision making of solutions to
real-life problems? You will find these out in the next portion of the events.
Before performing these exercises, first read and understand some important
information about this lesson and the examples that were given.
Central Angle and Arcs
Recall that a central angle
of a circle is an angle formed by
two rays whose vertex is the
center of the circle. Each ray
intersects the circle at a point,
dividing it into arcs.
In the figure on the right,
arc
C
central
angle
B
A
BAC is a central
angle. Its sides divide A into arcs. One arc is
arc
the curve containing points B and C. The
other arc is the curve containing points B, D, and C.
Definition: Sum of Central Angles
The sum of the measures of
the central angles of a circle with
no common interior points is 360
degrees.
In the figure, m1+ m2 +m3 + m4 = 360 .
(Note: All measures of angles and arcs are in degrees.)
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.
12
D
C
Example: The curve from point N to point Z is an arc.
It is part of
O and is named as arc NZ or NZ.
Other arcs of
O are CN, CZ, CZN, CNZ, and NCZ.
If mCNZ is one-half the circumference of
O
N
O, then it
is a semicircle.
Z
A minor arc is an arc of the circle that
N
measures less than a semicircle. It is named
Usually by using the two endpoints of the
arc.
Example: JN. NE and JE.
A
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
Example: 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
E at points G and O.
The measure of GO is equal to the measure
G
E
of then mGO = 118.
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 = 360mGO. That is, mOMG = 360 – 118 = 242.
Answer: mOMG = 242.
3.The degree measure of a semicircle is
180o.
13
M

Congruent Circles and Congruent Arcs
A
Congruent circles are circles with
congruent radii.
Example: MA is a radius of
TH is a radius of
M
A.
T.
If MA TH , then
A 
T.
Congruent arcs are arcs of the same
circle or of congruent circles with equal
measures.
T
M
65°
I
65°
Example:
If
I
H
T
N
S
K K
I, TM  KS .
65°
W
E
E, then TM  NW.
and KS  NW.
B
Theorems on Central Angles, Arcs, and Chords
G
50°
1. In a circle or in congruent circles, two
minor arcs are congruent if and only if
their corresponding central angles are
congruent.
In
E below, SET NEO. Since the two
central angles are congruent, the minor
arcs they intercept are also congruent.
Hence,
I
S
ST  NO .
If
E 
I and
SET NEO BIG,
then ST NO  BG .
2. In a circle or in congruent circles, two minor arcs are
congruent if and only if their corresponding chords are
congruent.
14
O
T
50°
50°
E
N
A
In
T on the right, BA 
CH . Since the two
T
B
H
chords are congruent, then BA CH .
T 
C
N and BA  CH OE , then
N
O
BA  CH OE .
E
3. In a circle, a diameter bisects a chord and an arc with the same
endpoints if and only if it is perpendicular to the chord.
In
U on the right, ES is a diameter and GN is a chord.
G
If ES ┴ GN, then GI  IN and GE  EN.
I
E
S
U
N
Inscribed Angles and Intercepted Arcs
An inscribed angle is an angle whose vertex is on a circle and whose
sides contain chords of the circle. The arc that lies in the interior of an
inscribed angle and has endpoints on the angle is called the intercepted arc
of the angle.
Examples:
L
T
P
C
G
P
A
M
O
Figure 1
Figure 2
Figure 3
In Figure 1, LAP is an inscribed angle and its intercepted arc is LP.
The center of the circle is in the interior of the angle.
In Figure 2, TOP is an inscribed angle and its intercepted arc is TP.
One side of the angle is the diameter of the circle.
15
In Figure 3, CGM is an inscribed angle and its intercepted arc is CM.
The center of the circle is in the exterior of the angle.
Theorems on Inscribed Angles
1. If an angle is inscribed in a circle, then the measure of the angle equals
one-half the measure of its intercepted arc (or the measure of the intercepted
A
arc is twice the measure of the inscribed angle).
Example: In the figure on the right, ACT is
an inscribed angle and AT is its intercepted arc.
C
If mAT = 120, then mACT = 60.
T
2. If two inscribed angles of a circle (or congruent circles) intercept
congruent arcs or the same arc, then the angles are congruent.
Example 1: In Figure 1 below, PIO and PLO intercept PO. Since
PIO and PLO intercept the same arc, the two angles, then, are congruent.
Figure 1
Figure 2
O
I
L
P
T
S
I
E
M
L
P
Example 2: In Figure 2 above, SIM and ELP intercept SM and EP,
respectively. If SM  EP , then SIM ELP.
3. If an inscribed angle of a circle intercepts a semicircle,
then the angle is a right angle.
Example:
In the figure, NTE intercepts NSE.
T
N
S
O
If NSE is a semicircle, then NTE is
a right angle.
E
16
4. If a quadrilateral is inscribed in a circle, then its opposite angles are
supplementary.
R
Example:
Quadrilateral DREA is inscribed in
M.
mRDA + mREA = 180.
M
D
mADR + mDAE = 180.
E
A
Whew! That’s a nice job there. Let’s explore some more exercises for our
brain development.
What’s More
Your goal in this section is to apply the key concepts of chords, arcs,
central, inscribed angles and intercepted arcsarcs and inscribed angles of a
circle. Use the mathematical ideas and the examples presented in the
preceding section to answer the activities provided.
Activity 2. Find me!
In the figure, JI and ON are diameters of
given
S . Use the figure and the
information to answer the following.
J
O
1. Which central angles are congruent? Why?
2. If mJSN = 113, find:
a. mISO
b. mNSI
c. mJSO
S
N
3. Is OJ IN ? How about JN and OI ? Explain your answer.
4. Which minor arcs are congruent? Explain your answer.
5. If mJSO = 67, find:
a. mJO
d. mIO
b. mJN
e.mNJO
c. mNI
f.mNIO
6.Which arcs are semicircles? Why?
17
I
Activity 3. Get My Length
In
M below, BD = 3, KM = 6, and KP = 2√7. Use the figure and the
given information to find each measure.
1. AM
5. DS
A
C
2. KL
6. MP
3. MD
M
7. AK
4. CD
3
8. KP
B
K
6
L
2 7
D
P
S
Activity 4. Inscribe, Intercept then Measure
In the figure below, CE and LA are diameters of
N. Use the figure to
answer the following.
1. Name all inscribed angles in the figure.
2. Which inscribed angles intercept the following arcs?
a.
CL
c. LE
b. AE
d. AC
C
4
3. If mLE = 124, what is the measure of
d.
7
g. 
b.  

e.


h. 
c.  

f. 


i. 
N
5 6
each of the following angles?
a. 1
L
E
8
A
4. If m1=26, what is the measure of each of the following arcs?
a.CL
b.AC
c.
AE
d.
LE
Wow! You made it. You are now acquainted with solving problems on
sequences. Let’s continue in the next activity.
18
What I Have Learned
Here is another activity that let you apply what you have learned about
relationship among chords, arcs, central, inscribed angles and intercepted
arcs. Are you ready?
Fill in the blanks with the correct word/s or symbol/s that best fit/s the
statement.
1. The length from the center of the circle to any point on the circle is called
_____.
2. A _____ angle is an angle equal to its intercepted arc.
3. If a chord passes through the center of a circle, then it is a _____ of a
circle.
4. An ____ angle is equal to one-half the measure of its intercepted arc.
5. A _____ is equal to half of a circle.
6. If two chords of congruent circles are equal, then their _____ arcs are also
equal.
7. The measure of the intercepted arc of a right angle is _____.
8. The sum of the measures of all the central angles of a circle is _____.
9. The measure of a _____ arc of a circle is more than 180 o but less than
360o.
10. A _____ is a segment containing any two points on the circle.
Good job! Now you’re up for the final challenge of this module.
19
What I Can Do
Let us test your awareness and skills further by trying out another activity.
Use the situation below to answer the questions that follow.
Claire works for a realtor. One of her jobs is to take photographs of
houses that are for sale. She took a photograph of a house two months ago
using a camera lens with 80° field of view like the one shown below. She has
returned to the house to update the photo, but she has forgotten her lens.
Now, she only has a telephoto lens with a 40° field of view.
Questions:
1. From what location(s) could Claire take a photograph of the house with
the telephoto lens, so that the entire house still fills the width of the picture?
Use an illustration to show your answer.
2. What mathematics concept would you apply to show the exact location of
the photographer?
3. If you were the photographer, what would you do to make sure that the
entire house is captured by the camera?
Amazing! You really did a good job!
20
Assessment
Let us determine how much you have learned in this module. Kindly
answer the following questions by choosing the letter of your answer.
For items 1 to 6
In
F, AB , BC , CD , BD and AC are chords. Use the figure and the given
information to answer the following questions.
1. If mCBD = 54, what is the measure of CD?
A. 55
B. 78
C. 96
D.108
2. If mAB = 96, what is the measure of ACB ?
A.
34
B. 40
C. 48
D.54
If mABD = 5x + 3 and mDCA = 4x + 10 ,
3. Find the value of x
D
A.
7
C. 9
B.
8
D. 10
A
4. What is the mDCA?
A. 34
C. 44
B. 38
D. 48
E
5. What is the mABD?
A. 34
F
E
B
B. 38
C. 44
D. 48
B. 74
C. 76
D.78
6. What is the mAD?
A. 72
7. What is also called as the longest chord of the circle?
A. arc
B. angle
C. chord
D. diameter
8. If A is a central angle and its measure equals 50o, then what is the
measure of its intercepted arc?
A. 100o
B. 50o
C. 40o
D. 25o
9. If and it is an inscribed angle, then what is the measure of its
intercepted arc?
A. 50o
B. 100o
C. 150o
21
D. 200o
C
For items 10 to 11
In the figure below, CE and LA are diameters of
N. Use the figure to
L
answer the following.
10. If m=26, what is the measure of CL?
C
A. 38
C. 48
4N
B. 45
D. 52
5 6
11. If m AE = 84, what is the m1?
A. 36
C. 54
B. 42
D. 60
7
E
8
A
For items 12 to 13
L. If mOGA = 75 and m AG =160,
∆GOA is inscribed in
12. What is the m OA?
A. 150
C. 90
B. 100
D.80
O
75
13. What is mGAO
A.
A
L
G
30
C.15
B. 25
D.10
160
For items 14 to 15
Quadrilateral FAIT is inscribed in
H. If mAFT =75 and mFTI = 98,
14. What is mTIA?
F
A. 95
C. 105
B. 100
D. 110
75°
A
H
15. What is mFAI?
98°
A. 82
C. 86
B. 84
D. 88
T
I
Amazing! You've done well on this module!
22
Additional Activities
Here's another practice that will help improve your knowledge and skills.
Budget Case
The monthly income of the Soriano family is Php36,000.00. They spend
Php9,000.00 for food, Php12,000.00 for education, Php4,500.00 for utilities,
and Php6,000.00 for other expenses. The remaining amount is for their
savings. This information is shown in the circle graph below.
Soriano Family’s
Monthly Expenses
a. Which item is allotted with the highest budget? How about the least?
Explain.
b. If you were to budget your family’s monthly income, which item would
you give the greater allocation? Why?
c. In the circle graph, what is the measure of the central angle
corresponding to each item?
d. How is the measure of the central angle corresponding to each item
determined?
Congratulations! Job well done. I hope you learned a
lot in this module.
You are now ready to answer the
next module on circles.
23
24
Questions:
Recall the definition of the terms related to circles.
Terms
related to
circle
1. radius
2. diameter
3. chord
4. semicircle
5. minor arc
6. major arc
7. central
angle
8. inscribed
angle
Description
It is a segment drawn from the center of
the circle to any point on the circle.
It is a segment whose endpoints are on the
circle and it passes through the center of the
circle. It is the longest chord.
It is a segment joining any two points on
the circle.
It is an arc measuring one-half of the
circumference of a circle.
It is an arc of a circle that measures less
than a semicircle.
It is an arc of a circle that measures
greater than a semicircle.
It is an angle whose vertex is at the center
of the circle and with two radii as its sides.
It is an angle whose vertex is on a circle
and whose sides contain chords of the
circle.
What’s New
Activity 1. AN, AJ, AE
5. JL, JN, EN, EL
1. EJ
6. LEN, LJE, ENL, JLN, LNE
7. JAN, NAE
2. EL, EJ
8. LEJ, JEN
3. JNE, JLE
Answer Key
25
1. A radius is half the measure of the diameter.
2. A diameter is twice the measure of the radius and it is the longest
chord.
3. A chord is a segment joining any two points on the circle.
4. A semicircle is an arc measuring one-half the circumference of a
circle.
5. A minor arc is an arc of a circle that measures less than the
semicircle.
6. A major arc is an arc of a circle that measures greater than the
semicircle.
7. A central angle is an angle whose vertex is the center of the circle
and with two radii as its sides.
8. An inscribed angle is an angle whose vertex is on a circle and
whose sides contain chords of the circle.
What I
Know
1. B
2. C
3. C
4. B
5. C
6. B
7. C
8. A
9. C
10. B
11. D
12. D
13. A
14. D
15. D
26
Activity 3. Get My Length
1. 8 units
2. 2 units
6. 8 units
3. 5 units
Note: Evaluate students’ explanations.
Activity 2. Find Me
1. JSO and NSI ; JSN and OSI . They are vertical angles.
2. a. 113
b. 67
c. 67
3. Yes. Yes. Opposite sides of rectangles are congruent.
4. JO and NI ; JN and OI . The central angles that intercept the arcs
are congruent.
5. a. 67
d. 113
b. 113
e. 180
c. 67
f. 180
6. NJO ; NIO; JOI ; JNI . The arcs measure 180°. Each arc or
semicircle contains the endpoints of the diameter.
27
What I Have learned
7. 180
2. Central
6. intercepted
1. Radius
3. Diameter
8. 360
10. chord
5. Semicircle
9. major
4. Inscribed
Activity 4. Inscribe and Intercept
1.
LCA,
LCE ,
ACE ,
ALC ,
CAE ,
CAL ,
LAE , and
AEC
2.
a. CAL
b. ACE
c. LCE and LAE
3.
d. ALC and AEC
f.
c. m3  62
e.
b. m2  62
d.
a. m1  28
.
4.
a.
mCL  52
b.
i. m9  62
m6  56
h. m8  62
m5  124
g. m7  28
m4  56
mAE  52
c.
d.
mAC  128
mLE  128
28
the lowest budget which is Php4,500.00.
3. A
Php12,000.00 Savings & Utilities, because they have t
2. C
a. Education, because it has the highest budget which is
1. D
Additional Activities
Assessment
4. B
5. B
6. C
7. D
8. B
b. Education. It should be given the greater allocation
because it is a very good investment.
c. Education – 120 Food – 90 Utilities – 45 Savings – 45
Other expenses - 60
d. Get the percentage for each item by dividing the
it by 360.
10. D
allotted budget by the monthly income, then multiply
9. D
11. B
12. A
13. B
14. C
15. A
What Can I Do
80°
40°
1. New location where Janel could photograph the entire house with the
telephoto lens
2. Relationship between the central angle or inscribed angle and the arc that the
angle intercepts.
3. Go farther from the house until the entire house is seen on the eye piece or on
the LCD screen viewer of the camera.
References
Mathematics Learner’s Module for Grade 10
Mathematics Teachers Guide for Grade 10
29
EDITOR’S NOTE
This Self-learning Module (SLM) was developed by DepEd SOCCSKSARGEN
with the primary objective of preparing for and addressing the new normal.
Contents of this module were based on DepEd’s Most Essential Learning
Competencies (MELC). This is a supplementary material to be used by all
learners of Region XII in all public schools beginning SY 2020-2021. The
process of LR development was observed in the production of this module.
This is version 1.0. We highly encourage feedback, comments, and
recommendations.
For inquiries or feedback, please write or call:
Department of Education – SOCCSKSARGEN
Learning Resource Management System (LRMS)
Regional Center, Brgy. Carpenter Hill, City of Koronadal
Telefax No.: (083) 2288825/ (083) 2281893
Email Address: region12@deped.gov.ph
30