This is the Grade 7 Mathematics Self-Learning Module (SLM) for Quarter 3, Module 6: Circles. It is an Alternative Delivery Mode (ADM) module from DepEd SOCCSKSARGEN (Region XII), First Edition, 2020. It helps learners name and draw a circle and its parts, from the radius and chord to arcs, secants, tangents, and central and inscribed angles.
On this page 7 sections
What learners will learn
After going through this module, learners are expected to:
- Illustrate a circle and other terms related to it: radius, diameter, chord, center, arc, central angle, and inscribed angle.
The module has one lesson, Lesson 1: Circles. It builds on the earlier lessons on polygons, their convexity, angles, and sides.
Key terms: the circle and its lines
A circle is a closed curve whose points are all the same distance from a fixed point called the center. A circle is named by its center using a capital letter. A circle with center O is called circle O, written ⊙O.
| Term | Meaning in the module | Example from the module |
|---|---|---|
| Radius | A segment with one endpoint at the center and the other on the circle. All radii of a circle are equal. | OJ and OM are radii of circle O |
| Chord | A segment whose endpoints are two different points on the circle | JK and MN are chords of circle O |
| Diameter | A chord that passes through the center. It is twice as long as a radius, and all diameters of a circle are equal. | JL and MP are diameters of circle O |
| Secant | A line that contains a chord | Lines AC and BD are secants of circle O |
| Tangent | A line in the plane of the circle that touches the circle at exactly one point, called the point of tangency | Line XY is a tangent of circle O, with point of tangency E |
A quick way to tell these apart: a radius and a chord are segments, while a secant and a tangent are lines. A secant cuts through the circle at two points. A tangent only touches it at one point.
Key terms: arcs and angles
- Central angle: an angle whose vertex is the center of the circle. Each of its two rays meets the circle at a point and divides the circle into arcs. Example: ∠AOD.
- Arc: a part of a circle. The curve from point V to point E on circle O is arc VE.
- Semicircle: an arc equal to one-half of the circle’s circumference. It is named with its two endpoints and another point on the arc, such as arc VEN.
- Minor arc: an arc smaller than a semicircle. It is usually named with just its two endpoints, such as arc JN, NE, or JE.
- Major arc: an arc larger than a semicircle. It is named with its two endpoints and another point on the arc, such as arc JEN, JNE, or EJN.
- Inscribed angle: an angle whose vertex lies on the circle and whose two sides are chords of the circle. The arc inside the angle, with endpoints on the angle, is the intercepted arc.
The module shows three kinds of inscribed angles:
- ∠HOT, which has the center inside the angle
- ∠TOP, which has one side that is a diameter
- ∠GEM, which has the center outside the angle
Their intercepted arcs are HT, TP, and GM.
The naming rule matters in the tests. Two letters name a minor arc. Three letters are used for a semicircle or a major arc.
Parts of the module
- What I Know: a 10-item multiple-choice pretest on circle terms.
- What’s In: a review of polygons. Learners complete a table of polygon names and their number of sides, then tell whether shapes are polygons and whether they are convex or nonconvex.
- What’s New: a word search with 10 circle words, from radius and chord to semicircle and tangent.
- What is It: the lesson itself, with the definitions and labeled illustrations above.
- What’s More: learners identify labeled parts of a circle, then name radii, diameters, a secant, a central angle, an inscribed angle, and a tangent from a figure.
- What I Have Learned: a 10-item fill-in-the-blank review using a word box.
- What I Can Do: learners draw their own circle and illustrate 10 given parts, such as chord QR, center A, diameter DE, and tangent OB.
- Assessment: a 10-item multiple-choice post-test.
- Additional Activities: learners draw circles with given measurements, such as circle P with a 3 cm diameter and circle A with a 3.5 cm radius AB and tangent line YZ at point B.
- Answer Key: answers for the pretest, activities, and assessment.
How to use this module
- Learners: Start with the pretest, then go through the parts in order. Write all answers on a separate sheet of paper and do not mark the module. Check your work with the answer key honestly, and take the post-test at the end to see what you have learned. A ruler and compass will help with the drawing tasks.
- Parents and facilitators: Guide your child through each part. Encourage them to ask the teacher whenever a task is unclear.
- Teachers: The pretest shows whether a learner needs extra help before starting. The What I Can Do and Additional Activities tasks are good ways to check whether learners can draw and label circle parts on their own.
Frequently asked questions
What grade and quarter is the Circles module for?
It is Grade 7 Mathematics, Quarter 3, Module 6. It is published by DepEd Region XII (SOCCSKSARGEN) as a First Edition, 2020 self-learning module.
Does the module have an answer key?
Yes. The answer key covers the pretest, the polygon review, What’s More, What I Have Learned, and the final assessment.
What is the difference between a secant and a tangent?
A secant is a line that contains a chord, so it crosses the circle at two points. A tangent touches the circle at exactly one point, called the point of tangency.
What is the difference between a central angle and an inscribed angle?
A central angle has its vertex at the center of the circle. An inscribed angle has its vertex on the circle itself, and its two sides are chords.
How do you name a minor arc and a major arc?
A minor arc is usually named with its two endpoints, such as arc JN. A major arc or a semicircle is named with its two endpoints and another point on the arc, such as arc JEN.
Full text of the module
The text below is a transcription of the PDF above, as printed.
Mathematics Quarter 3 – Module 6: Circles
Cover
7
Mathematics
Quarter 3 – Module 6: Circles
[illustration: students studying outdoors and indoors, a teacher with a student, a globe, and a lantern]
CO_Q3_Mathematics 7_Module 6
Mathematics – Grade 7 Alternative Delivery Mode Quarter 3 – Module 6: 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.
Published by the Department of Education Secretary: Leonor Magtolis Briones Undersecretary: Diosdado M. San Antonio
| Development Team of the Module | |
|---|---|
| Writer: | Venus Jane E. Rosete |
| Editor: | Jacqueline C. Marcos |
| Reviewer: | Reynaldo C. Tagala |
| Layout Artist: | Maylene F. Grigana |
| Management Team: | Allan G. Farnazo Gilbert B. Barrera Arturo D. Tingson Jr. Peter Van C. Ang-ug Donna S. Panes Elizabeth G. Torres Judith B. Alba |
Printed in the Philippines by: _________________________
Department of Education – SOCCSKSARGEN – Region XII
Office Address: Regional Center, Brgy. Carpenter Hill, City of Koronadal Telefax: (083) 2288825/ (083)2281893 E-mail Address: region12@deped.gov.ph
7
Mathematics
Quarter 3 – Module 6: Circles
Introductory Message
This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.
Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.
Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.
In addition to the material in the main text, Notes to the Teacher are also provided to our facilitators and parents for strategies and reminders on how they can best help you on your home-based learning.
Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.
If you have any questions in using this SLM or any difficulty in answering the tasks in this module, do not hesitate to consult your teacher or facilitator.
Thank you.
What I Need to Know
[illustration: hand touching a circle]
This module was designed and written with you in mind. It is here to help you learn about circles. This module may be used in many different learning situations. The language used recognizes your vocabulary. The lessons are arranged to follow the standard sequence of the course. But the order in which you study them can be rearranged to correspond with the textbook you are using.
After going through this module, you are expected to:
- Illustrate a circle and other terms related to it: radius, diameter chord, center, arc, chord, central angle and inscribed angle.
CO_Q3_Mathematics 7_Module 6
What I Know
[illustration: hand holding a pencil]
Choose the letter of the best answer. Write your answer on a separate sheet of paper.
Which refers to a closed curve and whose points are all equidistant from a fixed point called the center?
a. arc
b. center
c. circle
d. semicircleWhich of the following is NOT true about circle?
a. A circle has exactly one center.
b. A radius is a chord of a circle.
c. A chord is not a diameter.
d. All radii of a circle are equal in measures.
For numbers 3-6, refer to the illustration on the right.
[illustration: circle O with points A (top), B (lower left), and C (lower right); a line segment runs from A to C, and a segment runs from O to B]
Which of the following is the name of the circle?
a. ⊙A
b. ⊙B
c. ⊙C
d. ⊙OWhich of the following is NOT a radius?
a. $\overline{OA}$
b. $\overline{OB}$
c. $\overline{OC}$
d. $\overline{AC}$Which of the following is a semicircle?
a. $\widehat{AB}$
b. $\widehat{BC}$
c. $\widehat{ABC}$
d. $\widehat{ACB}$Which of the following is a major arc?
a. $\widehat{AB}$
b. $\widehat{BC}$
c. $\widehat{ABC}$
d. $\widehat{ACB}$
What I Know (continued)
[illustration: hand holding a pencil]
- What is an angle whose vertex is the center of the circle?
a. central angle
b. inscribed angle
c. obtuse angle
d. right angle
For numbers 8-10, refer to the illustration on the right.
[illustration: circle O with points D (lower left on the circle), E (on the right side of the circle), and F (on the left); line through F and E, line through X and Y passing through E, a ray through D and E, and point O at the center]
What is the point of tangency in circle O?
a. D
b. E
c. X
d. YWhich of the following is an inscribed angle?
a. ∠FED
b. ∠DEY
c. ∠FEX
d. ∠FEYWhich of the following is NOT a secant line?
a. $\overleftrightarrow{EF}$
b. $\overleftrightarrow{DE}$
c. $\overleftrightarrow{XY}$
d. none of the above
Lesson 1: Circles
[illustration: hand holding a puzzle piece]
In the previous lessons, you have learned about polygons, its convexity, angles and sides. In this module, you will learn about circles. Do you know that many practical problems in real-life situations can be solved using the concepts of circles?
Let us begin by studying circles and the other terms related to it such as radius, diameter chord, center, arc, chord, central angle and inscribed angle. Good luck!
What’s In
Before we will discuss circles, let us review the previous lessons about polygons by doing the activity below. Write your answer on a separate sheet of paper.
A. Complete the table below.
| Polygon | No. of Sides / Angles | Polygon | No. of Sides / Angles |
|---|---|---|---|
| triangle | 3 | octagon | |
| quadrilateral | 9 | ||
| 5 | decagon | ||
| hexagon | 11 | ||
| 7 | dodecagon |
B. Tell whether the following is a polygon or not. If it is a polygon, classify whether it is convex or nonconvex. Write your answer on a separate sheet of paper.
_____________1. [illustration: pentagon-like shape with one point facing right]
_____________2. [illustration: shape with wavy, curved sides]
_____________3. [illustration: cross-like shape with notched sides]
_____________4. [illustration: trapezoid]
_____________5. [illustration: shape with a rounded, semicircular end]
_____________6. [illustration: arrow pointing left]
What’s New
This time, we will be looking for 10 words in the grid that are related to circles. Encircle the words which may be written horizontally, vertically or diagonally.
| M | D | I | A | M | E | T | E | R | M | M | S |
|---|---|---|---|---|---|---|---|---|---|---|---|
| I | E | N | H | E | R | A | P | A | I | A | E |
| N | U | S | T | C | H | O | R | D | S | J | M |
| O | R | C | I | A | X | T | E | I | T | O | I |
| R | D | R | E | M | N | Y | I | U | Z | R | C |
| A | S | I | R | A | N | G | L | S | H | A | I |
| R | A | B | C | X | A | W | E | T | W | R | R |
| C | N | E | M | A | K | L | S | N | I | C | C |
| U | S | D | I | V | F | B | P | E | T | E | L |
| C | E | N | T | R | A | L | A | N | G | L | E |
Nice try! Check if you found all the 10 words below.
radius diameter chord inscribed central angle minor arc major arc secant tangent semicircle
What is It
Let us familiarize ourselves with circles, its definitions and illustrations and the terms related to it.
A circle is a closed curve, all points of which are equidistant from a fixed point called the center.
Circles are named by their centers using capital letters. The circle on the right is called circle O. In symbols, $\odot$ O.
[illustration: circle with center O]
Lines related to circle:
- A radius is a segment whose one endpoint is in the center of the circle and the other endpoint is any point on the circle.
Examples:
[illustration: circle O with center O and points J and M on the circle; labels J, O, M]
$\overline{OJ}$ is a radius of circle O. $\overline{OM}$ is also a radius of circle O. All radii of a circle are equal in measures.
- A chord is a segment whose endpoints are any of two different points in the circle.
Examples:
[illustration: circle O with points J, K, N, M on the circle; labels J, K, O, N, M]
$\overline{JK}$ is a chord of circle O. $\overline{MN}$ is also a chord of circle O.
What is It
- A diameter is a chord which passes through the center of the circle. It passes through the center of a circle. It is twice the length of a radius.
Examples:
[illustration: circle O with diameters JL and MP; labels J, P, O, M, L]
$\overline{JL}$ is a diameter of circle O. $\overline{MP}$ is also a diameter of circle O. All diameters of a circle have equal measures.
- A secant is a line that contains a chord.
Examples:
[illustration: circle O with secant lines AC and BD; labels A, B, C, D, O]
$\overleftrightarrow{AC}$ is a secant of circle O. $\overleftrightarrow{BD}$ is also a secant of circle O.
- A tangent is a line in the plane of a circle that intersects the circle at exactly one point. This point is called the point of tangency.
Example:
[illustration: circle O with tangent lines XY and ST; labels X, E, Y, O, S, I, T]
$\overleftrightarrow{XY}$ is a tangent of circle O whose point of tangency is at point E. $\overleftrightarrow{ST}$ is also a tangent $\odot$ O whose point of tangency is at point I.
Other basic terms related to circles are illustrated and defined as follows:
What is It
A central angle is an angle formed by two rays whose vertex is the center of the circle. Each ray intersects the circle at appoint, dividing it into arcs.
[illustration: circle with center O, rays to A and D, labeled “central angle”]
$\angle$ AOD is a central angle.
An arc is a part or a portion of a circle. The symbol for arc is $\frown$. A semicircle is an arc with a measure equal to one-half the circumference of a circle. It is named using the two endpoints and another point in the arc.
Example:
The curve from point V to point E is an arc. It is part of the circle O and is named as arc VE or $\overset{\frown}{VE}$. Other arcs of circle O are $\overset{\frown}{EN}$, $\overset{\frown}{EV}$, $\overset{\frown}{VEN}$, $\overset{\frown}{VNE}$ and $\overset{\frown}{EVN}$.
[illustration: circle O with labeled points V, E, N]
If $m\overset{\frown}{VEN}$ is one-half the circumference of $\odot$ O, then it is a semicircle.
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.
Examples: $\overset{\frown}{JN}$, $\overset{\frown}{NE}$, and $\overset{\frown}{JE}$
[illustration: circle with center A and labeled points N, E, J]
A major arc is an arc of the circle that measures greater than a semicircle. It is named by using two endpoints and another point on the arc.
Examples: $\overset{\frown}{JEN}$, $\overset{\frown}{JNE}$, and $\overset{\frown}{EJN}$
What is It
An inscribed angle is an angle whose vertex lies on a circle and its two sides are chords a circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc.
Examples:
[illustration: circle with center A, angle HOT with labeled points H, O, T]
HOT is an $\angle$ inscribed angle and its intercepted arc is HT. The center of the circle is in the interior of the angle.
[illustration: circle with labeled points T, O, A, and a line segment with a dot]
of
[transcriber’s note: the word “of” appears beside the TOP figure in the printed page]
TOP is an inscribed $\angle$ angle and its intercepted arc is TP. One side the angle is the diameter of the circle.
[illustration: circle with center A and labeled points E, G, M]
GEM is an inscribed angle and its intercepted arc is GM. The center of the circle is in the exterior of the angle.
What’s More
Let us check your understanding about circles by answering the following set of problems.
A. Identify the parts of the circle. Write your answer on the space provided before each number.
[illustration: circle O with labeled points S, M, O, T, R, P, N, Q; a secant line through S and T, a tangent line through T, and chords and diameters drawn]
_____________________ 1. O
_____________________ 2. $\overline{MT}$
_____________________ 3. $\overline{PR}$
_____________________ 4. $\overline{OM}$
_____________________ 5. $\overleftrightarrow{QT}$
_____________________ 6. $\angle$TOS
_____________________ 7. $\overleftrightarrow{ST}$
_____________________ 8. $\overset{\frown}{TSM}$
_____________________ 9. $\angle$TSN
_____________________ 10. $\overset{\frown}{PNR}$
B. Name each of the following. Refer to the circle below.
[illustration: circle O with labeled points X, M, Y, A, H, T; tangent line XY through M, diameters MT and AH]
_____________________ 1. two radii
_____________________ 2. two diameters
_____________________ 3. a secant
_____________________ 4. a central angle
_____________________ 5. an inscribed angle
_____________________ 6. a tangent
What I Have Learned
Here is another activity that will let you apply what you have learned about circle. Choose the word/expressions from the box to fill in each blank to make a statement true. Write your answer on a separate sheet of paper.
| radius | diameter | chord |
|---|---|---|
| Inscribed angle | central angle | minor arc |
| major arc | secant | tangent |
| semicircle | center | point of tangency |
A circle is a closed curve, all points of which are equidistant from a fixed point called the _______________________.
A ___________________ is a segment one whose endpoint is in the center of the circle and the other endpoint is any point on the circle.
A ___________________ is a segment whose endpoints are any of two different points on the circle.
A ____________________ is a chord which passes through the center of the circle. It is twice the length of a radius.
A tangent is a line in the plane of a circle that intersects the circle at exactly one point. This point is called the _______________________.
A _________________ is an arc with a measure equal to one-half the circumference of a circle.
A _____________________ is an angle whose vertex is the center of the circle.
A ___________________ is an arc of the circle that measures less than a semicircle.
A ___________________ is an arc of the circle that measures greater than a semicircle.
An __________________ is an angle whose vertex lies on a circle and its two sides are chords of a circle.
Nice work! Now you’re up for the next challenge of this lesson.
What I Can Do
Apply what you have learned about circles and the terms related to it.
Direction: Draw your own circle inside the box and illustrate the following.
[illustration: empty rectangular box for drawing a circle]
- chord QR
- center A
- diameter DE
- radius AC
- central angle PAC
- minor arc BD
- radius AP
- major arc BDS
- semicircle DBE
- tangent OB
Great work! You did a good job in applying what you have learned!
Assessment
Multiple Choice. Choose the letter with the correct answer. Write your answer on a separate sheet of paper.
A circle is a closed curve, all points of which are equidistant from a fixed point. What do you call this point?
A. arc
B. center
C. circle
D. semicircleWhich of the following is NOT a true statement about circle?
A. A tangent passes through the center of a circle.
B. A secant contains a chord.
C. A tangent of a circle intersects a radius.
D. A diameter divides the circle into two equal parts.
For numbers 3-6, refer to the illustration on the right.
[illustration: circle with center P; points X and C on the upper arc, B on the left, Y at the bottom; a line through P from X to Y and a line from B through P to C]
- Which of the following is the name of the circle?
A. ⊙ X
B. ⊙ Y
C. ⊙ B
D. ⊙ P
[transcriber’s note: in the printed questions 4 to 6, segment, arc, and line names carry a bar, arc, or double-arrow mark over the letters; they are written here as plain letters]
Which of the following is NOT a radius?
A. PX
B. PY
C. BC
D. PCWhich of the following is a semicircle?
A. XC
B. BY
C. BXC
D. XYBWhich of the following is a minor arc?
A. XC
B. XCY
C. BXC
D. XYBWhat is an angle whose vertex lies on a circle and its two sides are chords of a circle?
A. central angle
B. inscribed angle
C. obtuse angle
D. right angle
For numbers 8-10, refer to the illustration on the right.
[illustration: circle with center M; points E at the top of the circle, D on the left and F on the right of the lower arc, X at the bottom; a horizontal line through D, X, and F touches the circle at X; two lines extend from E through the circle]
What is the point of tangency in circle M?
A. D
B. E
C. F
D. XWhich of the following is an inscribed angle?
A. ∠ DEF
B. ∠ EDF
C. ∠ DFE
D. ∠ FDEWhich of the following is a tangent line?
A. DE
B. DF
C. EF
D. all of the above
Additional Activities
Direction: On a separate sheet of paper, draw a circle with the corresponding measurement.
- A circle P with a diameter of 3 cm
- A circle M with a radius 5 cm
- A circle X with a chord 1 cm
- A circle O with 2 cm chord which is a part of secant XZ
- A circle A with 3.5 cm radius AB and tangent line YZ whose point of tangency at B
Answer Key
What I Know
- C
- B
- D
- D
- C
- D
- A
- B
- A
- C
What’s In
A.
- Triangle – 3 sides
- Quarilateral – 4 sides
- Pentagon – 5 sides
- Hexagon – 6 sides
- Heptagon – 7 sides
- Octagon – 8 sides
- Nonagon – 9 sides
- Decagon – 10 sides
- Undecagon – 11 sides
- Dodecagon – 12 sides
B.
- polygon, convex
- not polygon
- polygon, nonconvex
- polygon, convex
- polygon, nonconvex
What’s New
[illustration: word search puzzle grid with circled words]
What’s More
A.
- center
- diameter
- chord
- radius
- tangent
- central angle
- secant
- semicircle
- inscribed angle
- minor arc
B.
- OA, OH, OM, OT
- MT, AH
- HT
- MOA, MOH, AOT, HOT
- MTH
- XY
What I Have Learned
- center
- radius
- chord
- diameter
- point of tangency
- semicircle
- central angle
- minor arc
- major arc
- inscribed angle
What I Can Do
[illustration: circle with center A; labeled points include B, O, Q, R, S, C, D, P, and E]
Assessment
- B
- A
- D
- C
- C
- A
- B
- D
- A
- B
References
Department of Education-Bureau of Learning Resources (DepEd-BLR) (2016) Grade 10 Mathematics Learner’s Module. Lexicon Press Inc., Philippines
De Leon, Cecille M., et.al, Integrated Mathematics Textbook for First Year. JTW Corporation 2000.
For inquiries or feedback, please write or call:
Department of Education – Bureau of Learning Resources (DepEd-BLR)
Ground Floor, Bonifacio Bldg., DepEd Complex Meralco Avenue, Pasig City, Philippines 1600
Telefax: (632) 8634-1072; 8634-1054; 8631-4985
Email Address: blr.lrqad@deped.gov.ph * blr.lrpd@deped.gov.ph
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