5. In Figure 1, what is the sum of the measures of the angles formed by the coplanar rays with a common vertex but with no common interior points?6. In Figure 2, what is the sum of the measures of the angles formed by the radii of a circle with no common interior points?7. In Figure 2, what is the intercepted arc of FAB? How about BAC ? CAD? EAD? EAF ? Complete the table below. Central Angle Measure Intercepted Arca. FABb. BACc. CADd. EADe. EAFDEPED COPY8. What do you think is the sum of the measures of the intercepted arcs of FAB, BAC , CAD, EAD, and EAF ? Why? 9. What can you say about the sum of the measures of the central angles and the sum of the measures of their corresponding intercepted arcs? Were you able to measure the angles accurately and find the sumof their measures? Were you able to determine the relationship betweenthe measures of the central angle and its intercepted arc? For sure youwere able to do it. In the next activity, you will find out how circles areillustrated in real-life situations.Activity 4:Use the situation below to answer the questions that follow. Rowel is designing a mag wheel like the one shown below. He decidedto put 6 spokes which divide the rim into 6 equal parts. 139 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Questions: a. What is the degree measure of each arc along the rim? How about each angle formed by the spokes at the hub? b. If you were to design a wheel, how many spokes will you use to divide the rim? Why? How did you find the preceding activities? Are 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 in real-life situations. But how do the relationships among chords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions? You will find these out in the activities in the next section. Before doing these activities, read and understand first some important notes on this lesson and the examples presented.DEPED COPYCentral Angle and Arcs arc C Recall that a central angle of a circle is an B centralangle formed by two rays whose vertex is the center angleof the circle. Each ray intersects the circle at a point,dividing it into arcs. A In the figure on the right, BAC is a central arc Dangle. Its sides divide A into arcs. One arc is thecurve containing points B and C. The other arc is thecurve containing points B, D, and C.Definition: Sum of Central Angles 12 43 The sum of the measures of the central anglesof a circle with no common interior points is 360degrees. In the figure, m1 m2 m3 m4 360 .(Note: All measures of angles and arcs are in degrees.) 140 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Arcs of a Circle An arc is a part of a circle. The symbol for arc is . A semicircle isan arc with a measure equal to one-half the circumference of a circle. It isnamed by using the two endpoints and another point on the arc.Example: The curve from point N to point Z is an C arc. It is part of O and is named as ON arc NZ or NZ. Other arcs of O are CN, CZ, CZN, CNZ, and NCZ. If mCNZ is one-half the circumference of O, then it is a semicircle.DEPED COPY A minor arc is an arc of the circle that Zmeasures less than a semicircle. It is named usuallyby using the two endpoints of the arc. NExamples: JN, NE, and JE A UA major arc is an arc of a circle that measures Egreater than a semicircle. It is named by using the twoendpoints and another point on the arc. JExamples: JEN, JNE, and EJNDegree Measure of an Arc1. The degree measure of a minor arc is the measure of the central angle which intercepts the arc.Example: GEO is a central angle. It intercepts E at O points G and O. The measure of GO is equal E to the measure of GEO. If mGEO 118, then mGO = 118. M2. The degree measure of a major arc is equal to 360 minusthe measure of the minor arc with the same endpoints.Example: If mGO = 118, then mOMG = 360 – mGO. G That is, mOMG = 360 – 118 = 242.Answer: mOMG = 2423. The degree measure of a semicircle is 180 . 141 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Congruent Circles and Congruent ArcsCongruent circles are circles with congruent radii. AExample: MA is a radius of A. M TH is a radius of T. If MA TH , then A T. H TCongruent arcs are arcs of the same circle or of Tcongruent circles with equal measures.Example: In I, TM KS . M 65°DEPED COPY I If I E, then TM NW 65° S N and KS NW . 65° W K ETheorems on Central Angles, Arcs, and Chords1. 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, ST NO .If E I and SET NEO BIG, then ST NO BG . TO B GS 50° E 50° N 50° I 142 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Proof of the Theorem The proof has two parts. Part 1. Given are two congruent circles and acentral angle from each circle which are congruent. The two-column proofbelow shows that their corresponding intercepted arcs are congruent.Given: E I G SET BIG BProve: ST BG I SEDEPED COPYProof: T Statements Reasons1. E I 1. Given SET BIG 2. The degree measure of a minor arc is2. In E , mSET mST. the measure of the central angle In I , mBIG mBG. which intercepts the arc.3. mSET mBIG 3. From 1, definition of congruent angles 4. From 2 & 3, substitution4. mST mBG 5. From 4, definition of congruent arcs5. ST BG Part 2. Given are two congruent circles and intercepted arcs from eachcircle which are congruent. The two-column proof on the next page showsthat their corresponding angles are congruent. GGiven: E I B ST BG IProve: SET BIG SE T 143 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Proof: Statements Reasons1. E I 1. GivenST BG 2. The degree measure of a minor arc is the measure of the central angle2. In E, mST mSET. which intercepts the arc. In I , mBG mBIG. 3. From 1, definition of congruent arcs3. mST mBG 4. From 2 & 3, substitution4. mSET mBIG 5. From 4, definition of congruent anglesDEPED COPY5. SET BIGCombining parts 1 and 2, the theorem is proven.2. In a circle or in congruent circles, two minor arcs are A congruent if and only if their corresponding chords are T congruent. C In T on the right, BA CH . Since B Hthe two chords are congruent, then BA CH . N If T N and BA CH OE , then EBA CH OE . O 144 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Proof of the Theorem The proof has two parts. Part 1. Given two congruent circles T N and two congruent corresponding chords AB and OE , the two-column proof below shows that the corresponding minor arcs AB and OE arecongruent.Given: T N A E AB OE N TProve: AB OE BDEPED COPYProof: O Statements Reasons 1. T N AB OE 1. Given 2. Radii of the same circle or of 2. TA TB NO NE congruent circles are congruent. 3. ATB ONE 3. SSS Postulate4. ATB ONE 4. Corresponding Parts of Congruent5. AB OE Triangles are Congruent (CPCTC) 5. From the previous theorem, “In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.”Part 2. Given two congruent circles T and N and two congruent minorarcs AB and OE , the two-column proof on the next page shows that thecorresponding chords AB and OE are congruent. AGiven: T NAB OE TEProve: AB OE BN O 145 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Proof: Reasons Statements 1. Given 1. T N 2. Definition of congruent arcs AB OE 2. mAB mOE 3. Definition of central angles 3. BTA and ONE are 4. The degree measure of a minor arc is central angles. the measure of the central angle 4. mBTA mBA which intercepts the arc. 5. From 2, 4, substitution mONE mOE 6. Radii of the same circle or of congruent circles are congruent. 5. mBTA mONE 7. SAS Postulate 8. Corresponding Parts of Congruent 6. TA TB NO NE Triangles are Congruent (CPCTC) 7. ATB ONE 8. AB OEDEPED COPYCombining parts 1 and 2, the theorem is proven.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 G U SGN is a chord. If ES GN , then GI IN and IGE EN . E NThe proof of the theorem is given as an exercise in Activity 9. 146 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Arc Addition Postulate The measure of an arc formed by two adjacent arcs L Eis the sum of the measures of the two arcs. VExample: Adjacent arcs are arcs with exactly one O point in common. In E, LO and OV are adjacent arcs. The sum of their measures is equal to the measure of LOV. If mLO = 71 and mOV = 84, then mLOV = 71 + 84 = 155.DEPED COPYSector and Segment of a Circle A sector of a circle is the region bounded by an arc of the circle andthe two radii to the endpoints of the arc. To find the area of a sector of acircle, get the product of the ratio measureof the arc and the area of the 360circle.Example: The radius of C is 10 cm. If mAB = 60, what is the area of sector ACB?Solution: To find the area of sector ACB: A 60° a. Determine first the ratio C 10 cm B m easure of the arc . 360 m easure of the arc 60 360 360 1 6 b. Find the area (A) of the circle using the equation A = r 2 , where r is the length of the radius. A = r 2 = 10 cm2 = 100 cm2 147 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
c. Get the product of the ratio measureof the arc and the 360 area of the circle. Area 1 of sector ACB = 6 100 cm2 = 50 cm2 3 The area of sector ACB is 50 cm2 . 3 DEPED COPY A segment of a circle is the region bounded by an arc and thesegment joining its endpoints. Example: The shaded region in the figure below is a segment of T. It is the region bounded by PQ and PQ . To find the area of the shaded segmentin the figure, subtract the area of triangle PTQfrom the area of sector PTQ. If mPQ = 90 and the radius of the circleis 5 cm, then the area of sector PTQ is one-fourth of the area of the whole circle. That is,Area of sector PTQ = 1 5 cm2 4 = 1 4 25 cm2 = 25 cm2 4 In the same figure, the area of ΔPTQ = 1 5 cm5 cm or 2ΔPTQ = 25 cm2 . 2 The area of the shaded segment, then, is equal to 25 25 cm2 4 2which is approximately 7.135 cm2. 148 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Arc Length A The length of an arc can be determined by using the proportion360 l 2r , where A is the degree measure of the arc, r is the radius of thecircle, and l is the arc length. In the given proportion, 360 is the degreemeasure of the whole circle, while 2r is the circumference.Example: An arc of a circle measures 45°. If the radius of the circle is 6 cm, what is the length of the arc?Solution: In the given problem, A = 45 and r = 6 cm. To find l, the equationDEPED COPY A 360 l can be used. Substitute the given values in the 2r equation. A l 45 l 1 l 360 2r 360 2(6) 8 12 12 l l 4.71 8 The length of the arc is approximately 4.71 cm.Learn more about Chords, http://www.cliffsnotes.com/math/geometry/Arcs, Central Angles, circles/central-angles-and-arcsSector, and Segment of aCircle through the WEB. http://www.mathopenref.com/arc. htmlYou may open thefollowing links. http://www.mathopenref.com/chord.html http://www.mathopenref.com/circlecentral. html http://www.mathopenref.com/arclength.html http://www.mathopenref.com/arcsector.html http://www.mathopenref.com/segment.html http://www.math-worksheet.org/arc-length-and- sector-area 149 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Your goal in this section is to apply the key concepts of chords,arcs, and central angles of a circle. Use the mathematical ideas and theexamples presented in the preceding section to answer the activitiesprovided.Activity 5:DEPED COPYUse A below to identify and name the following. Then, answer the questionsthat follow. KL1. 2 semicircles in the figure2. 4 minor arcs and their corresponding J AM major arcs3. 4 central angles H GQuestions:a. How did you identify and name the semicircles? 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. Were you able to identify and name the arcs and central angles inthe given circle? In the next activity, you will apply the theorems on arcsand central angles that you have learned. 150 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Activity 6:In A below, mLAM 42, mHAG 30, and KAH is a right angle. Find thefollowing measure of an angle or an arc, and explain how you arrived at youranswer.1. mLAK 6. mLK K L2. mJAK 7. mJK3. mLAJ 8. mLMG J AM4. mJAH 9. mJHDEPED COPY5. mKAM 10. mKLM H G In the activity you have just done, were you able to find the degreemeasure of the central angles and arcs? I am sure you did! In the nextactivity, you will apply the relationship among the chords, arcs, andcentral angles of a circle.Activity 7:In the figure, JI and ON are diameters of S. Use the figure and the given Jinformation to answer the following.1. Which central angles are congruent? Why? O2. If mJSN 113, find: S a. mISOb. mNSIc. mJSO N I3. Is OJ IN ? How about JN and OI ? Justify your answer.4. Which minor arcs are congruent? Explain your answer.5. If mJSO 67, find:a. mJO d. mIOb. mJN e. mNJOc. mNI f. mNIO6. Which arcs are semicircles? Why? 151 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Were you able to apply the relationship among the chords, arcs andcentral angles of a circle? In Activity 8, you will use the theorems onchords in finding the lengths of chords.Activity 8:In M below, BD = 3, KM = 6, and KP = 2 7 . Use the figure and the giveninformation to find each measure. Explain how you arrived at your answer.1. AM 5. DS A2. KL 6. MP DEPED COPY C3. MD 7. AK M 6 KL4. CD 8. KP 27 B 3D P S Were you able to find the length of the segments? In the nextactivity, you will complete the proof of a theorem on central angles, arcs,and chords of a circle.Activity 9:Complete the proof of the following theorem. In a circle, a diameter bisects a chord and an arc with the sameendpoints if and only if it is perpendicular to the chord. EGiven: ES is a diameter of U and N G perpendicular to chord GN at I. IProve: 1. NI GI U 2. EN EG 3. NS GS S 152 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Proof of Part 1: Show that ES bisects GN and the minor arc GN. StatementsDEPED COPY Reasons 1. U with diameter ES and chord Two points determine a line. GN; ES GN Given 2. GIU and NIU are right angles. Lines that are perpendicular tm 3. GIU NIU right angles. Radii of a circle are congruent. 4. UG UN Reflexive Property of Congruence. 5. UI UI 6. GIU NIU Corresponding parts of congruent 7. GI NI triangles are congruent. Corresponding parts of congruent 8. ES bisects GN . triangles are congruent In a circle, congruent central angles 9. GUI NUI intercept congruent arcs. Two angles that form a linear pair10. GUI and GUE are the same are supplementary. angles. NUI and NUE are the same Supplements of congruent angles angles. are congruent. In a circle, congruent central angles11. mGUE mNUE intercept congruent arcs.12. mEG mGUE mEN mNUE13. mEN mEG14. mGUS mNUS15. mGS mGUS mNS mNUS16. mNS mGS17. ES bisects GN . 153 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Given: ES is a diameter of U; ES bisects GN E at I and the minor arc GN. N I G U S SProof of Part 2: Show that ES GN . DEPED COPY Statements Reasons1. U with diameter ES , ES bisects Two points determine a line. GN at I and the minor arc Given GN. Radii of a circle are congruent.2. GI NI Reflexive Property of Congruence. Corresponding parts of congruent GE NE triangles are congruent.3. UI UI4. UG UN5. GIU NIU6. UIG UIN7. UIG and UIN are right angles.8. IU GN9. ES GN Was the activity interesting? Were you able to complete the proof?You will do more of this in the succeeding lessons. Now, use the ideasyou have learned in this lesson to find the arc length of a circle. 154 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Activity 10:The radius of O below is 5 units. Find the length of each of the followingarcs given the degree measure. Answer the questions that follow.1. mPV = 45; length of PV = ________ Q2. mPQ = 60; length of PQ = ________ P3. mQR = 90; length of QR = ________ r=5 O V R4. mRTS = 120;length of RTS = ________ TDEPED COPY5. mQRT = 95; length of QRT = ________ SQuestions:a. How did you find the length of each arc?b. What mathematics concepts or principles did you apply to find the length of each arc? Were you able to find the arc length of each circle? Now, find thearea of the shaded region of each circle. Use the knowledge learnedabout segment and sector of a circle in finding each area.Activity 11:Find the area of the shaded region of each circle. Answer the questions thatfollow.1. A 2. 3. 135° QX 90° 45° C 6 cm B S 12 cm R Z 8 cm Y 155 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
4. B 5. J 6. S 100° E T SW 5 cm M 6 cm OR X 4 cm A YQuestions: a. How did you find the area of each shaded region? b. What mathematics concepts or principles did you apply to find the area of the shaded region? Explain how you applied these concepts. How was the activity you have just done? Was it easy for you to find the area of segments and sectors of circles? It was easy for sure! In this section, the discussion was about the relationship among chords, arcs, and central angles of circles, arc length, segment and sector of a circle, and the application of these concepts in solving problems.DEPED COPY Go back to the previous section and compare your initial ideas withthe discussion. How much of your initial ideas are found in thediscussion? Now that you know the important ideas about this topic, let us godeeper by moving on to the next section. 156 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Your goal in this section is to take a closer look at some aspects of thetopic. You are going to think deeper and test further your understanding ofcircles. After doing the following activities, you should be able to answer thisimportant question: “How do the relationships among chords, arcs, andcentral angles of a circle facilitate finding solutions to real-life problems andmaking decisions?”Activity 12:DEPED COPYAnswer the following questions.1. Five points on a circle separate the circle into five congruent arcs.a. What is the degree measure of each arc?b. If the radius of the circle is 3 cm, what is the length of each arc?c. Suppose the points are connected consecutively with line segments. How do you describe the figure formed?2. Do you agree that if two lines intersect at the center of a circle, then the lines intercept two pairs of congruent arcs? Explain your answer.3. In the two concentric circles on the right, C WN R CON intercepts CN and RW. O a. Are the degree measures of CN and RW equal? Why?b. Are the lengths of the two arcs equal? Explain your answer.4. The length of an arc of a circle is 6.28 cm. If the circumference of the circle is 37.68 cm, what is the degree measure of the arc? Explain how you arrived at your answer.5. Mr. Lopez would like to place a fountain in his circular garden on the right. He wants the pipe, where the water will pass through, to be located at the center of the garden. Mr. Lopez does not know where it is. Suppose you were asked by Mr. Lopez to find the center of the garden, how would you do it? 157 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
DEPED COPY6. 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? e. Suppose the radius of the circle graph is 25 cm. What is the area of each sector in the circle graph? How about the length of the arc of each sector? In this section, the discussion was about your understanding of chords, arcs, central angles, area of a segment and a sector, and arc length of a circle including their real-life applications. What new realizations do you have about the lesson? How would you connect this to real life? Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. 158 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Your goal in this section is to apply your learning to real-lifesituations. You will be given a practical task which will demonstrate yourunderstanding of circles. Activity 13:Answer the following. Use the rubric provided to rate your work.1. Name 5 objects or cite 5 situations in real life where chords, arcs, and central angles of a circle are illustrated. Formulate problems out of these objects or situations, then solve.2. Make a circle graph showing the different school fees that students like you have to pay voluntarily. Ask your school cashier how much you would pay for the following school fees: Parents-Teachers Association, miscellaneous, school paper, Supreme Student Government, and other fees. Explain how you applied your knowledge of central angles and arcs of a circle in preparing the graph.3. Using the circle graph that you made in number 2, formulate at least two problems involving arcs, central angles, and sectors of a circle, then solve.DEPED COPYRubric for a Circle GraphScore Descriptors 4 The circle graph is accurately made, presentable, and appropriate. 3 The circle graph is accurately made and appropriate but not presentable. 2 1 The circle graph is not accurately made but appropriate. The circle graph is not accurately made and not appropriate. 159 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Rubric on Problems Formulated and SolvedScore Descriptors Poses a more complex problem with 2 or more correct possible6 solutions and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. Poses a more complex problem and finishes all significant parts of5 the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes all significant parts of the4 solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes.DEPED COPY Poses a complex problem and finishes most significant parts of the3 solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details. Poses a problem and finishes some significant parts of the solution2 and communicates ideas unmistakably but shows gaps on theoretical comprehension.1 Poses a problem but demonstrates minor comprehension, not being able to develop an approach.Source: D.O. #73, s. 2012 In this section, your task was to name 5 objects or cite 5 situations inreal life where chords, arcs, and central angles of a circle are illustrated.Then, formulate and solve problems out of these objects or situations. Youwere also asked to make a circle graph. How did you find the performance task? How did the task help yourealize the importance of the lesson in real life?SUMMARY/SYNTHESIS/GENERALIZATION: This lesson was about the relationships among chords, arcs, and centralangles of a circle, area of a segment and a sector, and arc length of a circle. Inthis lesson, you were asked to determine the relationship between the measuresof the central angle and its intercepted arc. You were also given the opportunityto apply the different geometric relationships among chords, arcs, and centralangles in solving problems, complete the proof of a theorem related to theseconcepts, find the area of a segment and the sector of a circle, and determine thelength of an arc. Moreover, you were asked to name objects and cite real-life situationswhere chords, arcs, and central angles of a circle are illustrated and applied. Yourunderstanding of this lesson and other previously learned mathematics conceptsand principles will facilitate your learning of the next lesson, Arcs and InscribedAngles of Circles. 160 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Start Lesson 1B of this module by checking your prior mathematical knowledge and skills that will help you in understanding the relationships among arcs and inscribed angles of a circle. As you go through this lesson, think of this important question: How are the relationships among arcs and inscribed angles of a circle used in finding solutions to real-life problems and in 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. Activity 1:Name the angles and their intercepted arcs in the figure below. Answer thequestions that follow. MC D GDEPED COPY SAngles Arc That the Angle Intercepts 161 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
DEPED COPYQuestions:1. How did you identify and name the angles in the figure? How about the arcs that these angles intercept?2. How many angles did you identify in the figure? How about the arcs that these angles intercept?3. When do you say that an angle intercepts an arc? Were you able to identify and name the angles and their intercepted arcs? I am sure you were! This time, find out the relationships that exist among arcs and inscribed angles of a circle by doing the next activity. Activity 2:Perform the following activity by group. Answer every question that follows.Procedure:1. Use a compass to draw a circle. Mark and label the center of the circle as point E.2. Draw a diameter of the circle. Label the endpoints as D and W.3. From the center of the circle, draw radius EL. Using a protractor, what is the measure of LEW ? How about the degree measure of LW? Why?4. Draw LDW by connecting L and D with a line segment. Using a protractor, what is the measure of LDW ?5. LDW is an inscribed angle. How do you describe an inscribed angle?6. LW is the intercepted arc of LDW. Compare the measure of LDW with the degree measure of LW. What statements can you make? 162 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
7. Draw other inscribed angles in the circle. Determine the measures of these angles and the degree measures of their respective intercepted arcs. How does the measure of each inscribed angle compare with the degree measure of its intercepted arc? What conclusion can you make about the relationship between the measure of an inscribed angle and the measure of its intercepted arc? Were you able to determine the relationship between the measureof an inscribed angle and the measure of its intercepted arc? If yes, thenyou are now ready to determine the relationship that exists when aninscribed angle intercepts a semicircle by performing the next activity.DEPED COPYActivity 3:Perform the following activity by group. Answer every question that follows.Procedure:1. Draw a circle whose radius is 3 cm. Mark the center and label it C.2. Extend the radius to form a diameter of 6 cm. Mark and label the endpoints of the diameter with M and T.3. On the semicircle, mark and label three points O, U, and N.4. Draw three different angles whose vertices are O, U, and N, respectively, and whose sides contain M and T.5. Find the measure of each of the following angles using a protractor.a. MOT b. MUT c. MNTWhat can you say about the measures of the angles?What statements can you make about an inscribed angle intercepting asemicircle?How would you compare the measures of inscribed angles interceptingthe same arc? 163 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
DEPED COPYWere you able to determine the measure of an inscribed angle that intercepts a semicircle? For sure you were able to do it. In the next activity, you will find out how inscribed angles are illustrated in real-life situations. Activity 4:Use the situation below to answer the questions that follow. Janel works for a realtor. One of her jobs is to take photographs ofhouses that are for sale. She took a photograph of a house two months agousing a camera lens with 80° field of view like the one shown below. She hasreturned 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 Janel 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? 164 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
How did you find the preceding activities? Are you ready to learnabout the relations among arcs and inscribed angles of a circle? I amsure you are! From the activity done, you were able to find out howinscribed angles are used in real-life situations. But how does theconcept of inscribed angles of a circle facilitate finding solutions to real-life problems and making decisions? You will find these out through theactivities in the next section. Before doing these activities, read andunderstand first some important notes on this lesson and the examplespresented.Inscribed Angles and Intercepted ArcsDEPED COPYAn inscribed angle is an angle whose vertex is on a circle and whosesides contain chords of the circle. The arc that lies in the interior of aninscribed angle and has endpoints on the angle is called the intercepted arcof the angle. L TPExamples: P GC A OM 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. 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 Angles1. If an angle is inscribed in a circle, then the measure of the angle equalsone-half the measure of its intercepted arc (or the measure of theintercepted arc is twice the measure of the inscribed angle).Note: The theorem has three cases and the proof of each case is given asan exercise in Activity 8 and Activity 9. AExample: In the figure on the right, ACT is an inscribed angle and AT is its intercepted arc. C If mAT = 120, then mACT = 60. T 165 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
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 P O Figure 2 IL T SEDEPED COPY IL MPExample 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 aright angle. N SExample: O In the figure, NTE intercepts NSE. NSE is a semicircle, then NTE is If right angle. T a E4. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.Example: Quadrilateral DREA is inscribed in M. mRDA mREA 180. mDRA mDAE 180 . R DM E A 166 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Learn more about Arcs and http://www.cliffsnotes.com/math/geometry/circles/arcInscribed Angles of a Circle s-and-inscribed-anglesthrough the WEB. You mayopen the following links. http://www.ck12.org/book/CK-12-Geometry-Honors- Concepts/section/8.4/ http://www.math-worksheet.org/inscribed-angles http://www.mathopenref.com/circleinscribed.html http://www.onlinemathlearning.com/circle- theorems.htmlDEPED COPY Your goal in this section is to apply the key concepts of arcs andinscribed angles of a circle. Use the mathematical ideas and theexamples presented in the preceding section to answer the activitiesprovided.Activity 5:In the figure below, CE and LA are diameters of N. Use the figure toanswer the following.1. Name all inscribed angles in the figure.2. Which inscribed angles intercept the following arcs? La. CL c. LE C2 3b. AE d. AC 1 4N 56 78 9 E A 167 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
3. If mLE = 124, what is the measure of each of the following angles?a. 1 d. 4 g. 7b. 2 e. 5 h. 8c. 3 f. 6 i. 94. If m126, what is the measure of each of the following arcs?a. CL c. AEb. AC d. LEDEPED COPY Were you able to identify the inscribed angles and their interceptedarcs including their degree measures? In the next activity, you will applythe theorems on arcs and inscribed angles that you have learned.Activity 6:In F, AB , BC , CD , BD and AC are chords. Use the figure and the giveninformation to answer the following questions.1. Which inscribed angles are congruent? D Explain your answer.2. If mCBD 54, what is the measure of CD?3. If mAB 96, what is the measure of ACB ? A F E4. If mABD 5x 3 and mDCA 4x 10 , find:a. the value of x c. mDCA B Cb. mABD d. mAD5. If mBDC 6x 4 and mBC 10x 2, find:a. the value of x c. mBCb. mBDC d. mBAC In the activity you have just done, were you able to apply thetheorems on arcs and inscribed angles? I am sure you were! In the nextactivity, you will still apply the theorems you have studied in this lesson. 168 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Activity 7:Use the given figures to answer the following.1. ∆GOA is inscribed in L. If mOGA 75 andmAG 160, find:a. mOA c. mGOA O A Lb. mOG d. mGAO 75° G 160°DEPED COPY2. Isosceles ∆CAR is inscribed inE. If mCR 130, find: a. mCAR A b. mACRc. mARC CEd. mACe. mAR 130° R3. DR is a diameter of O. If mMR 70, find: M 70° Ra. mRDM d. mDM Ob. mDRM e. mRD Dc. mDMR4. Quadrilateral FAIT is inscribed in H. F A If mAFT 75 and mFTI 98, find: 75° a. mTIA b. mFAI H T 98° I 169 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
5. Rectangle TEAM is inscribed in B. If mTE 64 and mTEM 58 , find:a. mTM Eb. mMA Tc. mAE B Ad. mMEA Me. mTAMDEPED COPY How was the activity you have just done? Was it easy for you toapply the theorems on arcs and inscribed angles? It was easy for sure! Now, let us complete the proof of a theorem on inscribed angle andits intercepted arc.Activity 8:Complete the proof of the theorem on inscribed angle and its intercepted arc. The proofs of cases 2 and 3 of this theorem are given in Activity 9. If an angle is inscribed in a circle, then the measure of the angleequals one-half the measure of its intercepted arc (or the measure of theintercepted arc is twice the measure of the inscribed angle).Case 1: Q xGiven: PQR inscribed in S and S PQ is a diameter. PRProve: mPQR 1 mPR 2Draw RS and let mPQR x . 170 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Statements Reasons1. PQR is inscribed in S and PQ is a diameter.2. QS RS3. QRS is an isosceles .4. PQR QRS5. mPQR mQRS6. mQRS x7. mPSR 2x8. mPSR mPRDEPED COPY9. mPR 2x10. mPR 2mPQR11. mQRS 1 mPR 2 Were you able to complete the proof of the first case of the theorem? Iknow you did! In this section, the discussion was about the relationship among arcsand inscribed angles of a circle. Go back to the previous section and compare your initial ideas with thediscussion. How much of your initial ideas are found in the discussion?Which ideas are different and need modification? Now that you know the important ideas about this topic, let us godeeper by moving on to the next section. 171 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Your goal in this section is to take a closer look at some aspects of thetopic. You are going to think deeper and test further your understanding ofthe relationships among inscribed angles and their intercepted arcs. Afterdoing the following activities, you should be able to answer this importantquestion: How are the relationships among inscribed angles and theirintercepted arcs applied in real-life situations?Activity 9:DEPED COPYWrite a proof of each of the following theorems.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 arc is twice the measure of the inscribed angle).Case 2: LGiven: KLM inscribed in O.Prove: mKLM 1 mKM O 2 KCase 3: MGiven: SMC inscribed in A. MProve: mSMC 1 mSC AC 2 S 172 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
2. If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent. AGiven: In T, PR and AC are the C intercepted arcs of PQR and ABC , respectively. PPR AC TProve: PQR ABC B RQDEPED COPY3. If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle.Given: In C, GML intercepts E G semicircle GEL. M CProve: GML is a right angle. L4. If a quadrilateral is inscribed in a circle, then its opposite angles are Isupplementary.Given: Quadrilateral WIND is inscribed in Y . W YNProve: 1. W and N are supplementary.2. I and D are supplementary. D 173 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Were you able to prove the theorems on inscribed angles andintercepted arcs? In the next activity, you will use these theorems toprove congruence of triangles.Activity 10:Write a two-column proof for each of the following. C1. MT and AC are chords of D. If MC AT , D prove that CHM THA . MHDEPED COPY T A2. Quadrilateral DRIV is inscribed in E. RV is a diagonalthat passes through the center of the circle. If DV IV ,prove that RVD RVI. R3. In A, SE NE and SC NT . EIProve that CSE TNE . DV C S T E A N Were you able to use the theorems on inscribed angles to provecongruence of triangles? In the next activity, you will further understandinscribed angles and how they are used in real life. 174 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Activity 11:Answer the following questions.1. There are circular gardens having paths in the shape of an inscribed regular star like the one shown on the right. a. Determine the measure of an arc intercepted by an inscribed angle formed by the star in the garden. b. What is the measure of an inscribed angle in a garden with a five-pointed star? Explain.2. What kind of parallelogram can be inscribed in a circle? Explain.3. The chairs of a movie house are arranged consecutively like an arc of a circle. Joanna, Clarissa, and Juliana entered the movie house but seated away from each other as shown below. E Movie Screen GDEPED COPYJoanna 380 Juliana F Clarissa Let E and G be the ends of the screen and F be one of the seats. Theangle formed by E, F, and G or EFG is called the viewing angle of theperson seated at F. Suppose the viewing angle of Clarissa in the abovefigure measures 38°. What are the measures of the viewing angles ofJoanna and Juliana? Explain your answer.4. A carpenter’s square is an L-shaped tool used to draw right angles. Mang Ador would like to make a copy of a circular plate using the available wood that he has. Suppose he traces the plate on a piece of wood. How could he use a carpenter’s square to find the center of the circle? 175 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
5. Ramon made a circular cutting board by M R sticking eight 1- by 2- by 10-inch boards P SQ together, as shown on the right. Then, he drew and cut a circle with an 8-inch diameter from the boards. a. In the figure, if PQ is a diameter of the circular cutting board, what kind of triangle is PQR ? b. How is RS related to PS and QS ? Justify your answer. c. Find PS, QS, and RS. d. What is the length of the seam of the cutting board that is labeled RT ? How about MN ?DEPED COPY NT In this section, the discussion was about your understanding ofinscribed angles and how they are used in real life. What new realization do you have about inscribed angles? How wouldyou connect this to real life? Now that you have a deeper understanding of the topic, you are readyto do the tasks in the next section. Your goal in this section is to apply your learning to real-life situations. You will be given a practical task which will demonstrate your understanding of inscribed angles. Activity 12:Make a design of a stage where a special event will be held. Include in thedesign some circular objects that illustrate the use of inscribed angles and arcs ofa circle. Explain how you applied your knowledge of inscribed angles andintercepted arcs of a circle in preparing the design. Then, formulate and solveproblems out of this design that you made. Use the rubric provided to rate yourwork. 176 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Rubric for a Stage’s DesignScore Descriptors 4 The stage’s design is accurately made, presentable, and appropriate. 3 The stage’s design is accurately made and appropriate but not presentable. 2 1 The stage’s design is not accurately made but appropriate. The stage’s design is not accurately made and not appropriate.Rubric on Problems Formulated and SolvedScore Descriptors 6 5 Poses a more complex problem with 2 or more correct possible 4 solutions and communicates ideas unmistakably, shows in-depth 3 comprehension of the pertinent concepts and/or processes and 2 provides explanations wherever appropriate. 1 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details. Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension. Poses a problem but demonstrates minor comprehension, not being able to develop an approach.DEPED COPYSource: D.O. #73, s. 2012 In this section, your task was to design a stage, formulate, and solveproblems where inscribed angles of circles are illustrated. How did you find the performance task? How did the task help you realizethe importance of the topic in real life?SUMMARY/SYNTHESIS/GENERALIZATION: This lesson was about arcs and inscribed angles of a circle. In this lesson,you were given the opportunity to determine the geometric relationships that existamong arcs and inscribed angles of a circle, apply these in solving problems, andprove related theorems. Moreover, you were given the chance to formulate and solvereal-life problems involving these geometric concepts. Your understanding of thislesson and other previously learned mathematics concepts and principles will facilitateyour learning of the next lesson, Tangent and Secant Segments. 177 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
DEPED COPYStart Lesson 2A of this module by assessing your knowledge of the different mathematical concepts previously studied and other mathematical skills learned. These knowledge and skills will help you understand the different geometric relationships involving tangents and secants of a circle. As you go through this lesson, think of this important question: How do the different geometric relationships involving tangents and secants of a circle facilitate finding solutions to real-life problems and making wise 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. Activity 1:Perform the following activity. Answer every question that follows.Procedure:1. Use a compass to draw S.2. Draw line m such that it intersects S at exactly one point. Label the point of intersection as T.3. Connect S and T with a line segment. What is TS in the figure drawn?4. Mark four other points on line m such that two of these points are on the left side of T and the other two points are on the right side. Label these points as M, N, P, and Q, respectively.5. Using a protractor, find the measures of MTS, NTS, PTS, and QTS . How do the measures of the four angles compare?6. Repeat steps 2 to 5. This time, draw line n such that it intersects the circle at another point. What statement can you make about the measures of angles in item #5 and those in item #6?7. Draw MS, NS, PS , and QS . 178 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
DEPED COPY8. Using a ruler, find the lengths of TS , MS , NS , PS , and QS . How do the lengths of the five segments compare? What do you think is the shortest segment from the center of a circle to the line that intersects it at exactly one point? Explain your answer. In the activity you have just done, were you able to compare the measures of different angles drawn? Were you able to determine the shortest segment from the center of a circle to the line that intersects it at exactly one point? I know you were! The activity you have done has something to do with your new lesson. Do you know why? Find this out in the succeeding activities! Activity 2: In the figure below, C is the center of the circle. Use the figure to answer the questions that follow. 1. Which lines intersect circle C at two points? How about the lines that intersect the circle at exactly one point? 2. What are the angles having A as the vertex? C as the vertex? D as the vertex? G as the vertex? Make a list of these angles, then describe each. 3. What arc/s does each angle intercept? 179 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
4. Which angles intercept the same arc?5. Using a protractor, find the measures of the angles identified in item #2?6. How would you determine the measures of the arcs intercepted by the angles? Give the degree measure of each arc.7. Compare the measures of DCE and DAE . How about the mDE and m DAE ? Explain your answer.8. How is the mAD related to the m DAB? How about mEFA and m EAG ?9. What relationship exists among mAD, mAF, and m BGD ?DEPED COPY Were you able to measure the different angles and arcs shown inthe figure? Were you able to find out the different relationships amongthese angles and arcs? Learn more about these relationships in thesucceeding activities.Activity 3:Prepare the following materials, then perform the activity that follows. Answerevery question asked.Materials: Circular cardboard with radius 6 cm that is equally divided into 72 arcs so that each arc measures 5° 2 pieces of string, each measures about 40 cm self-adhesive tape cardboard or any flat surfaceProcedure:1. Attach the endpoints of the strings to R the cardboard or any flat surface using self-adhesive tape to form an S angle of any convenient measure. T Label the angle as RST. 180 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
22.. Locate the center of the circular R cardboard. Slide it underneath the S strings until its center coincides with T their point of intersection, S. If the edge of the circular cardboard represents a circle, what is RST in relation to the circle? What are the measures of RST and RT? Explain how you arrived at your answer.DEPED COPY3. Slide the circular cardboard so that RS R intersects the circle at S and ST S intersects the circle at two points, S and T. T4. Find the measure of ST using the circular cardboard. How would you compare the measure of RST with that of ST?5. Slide the circular cardboard so that S is S R in the exterior of the circle and RS and V T ST intersect the circle at R and T, respectively. Mark and label another point V on the circle.6. Find the measures of RVT and RT. Is there any relationship among the measures of RST , RVT, and RT? Describe the relationship, if there is any.7. Slide the circular cardboard so that S is R in the exterior of the circle, ST intersects the circle at T, and RS intersects the N circle at two points, R and N. S T 181 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
8. Find the measures of RT and NT. Is there any relationship among the measures of RST , RT, andNT? Describe the relationship, if there is any.9. Slide the circular cardboard so that S is N R in the exterior of the circle, RS SM T intersects the circle at points N and R, and ST intersects the circle at points M and T.DEPED COPY10. Find the measures of RT and MN. Is there any relationship among the measures of RST , RT, and MN? Describe the relationship, if there is any.11. Slide the circular cardboard so that S N R is in the interior of the circle, NT MS T intersects the circle at points N and T, and MR intersects the circle at points M and R.12. Find the measures of RT and MN. Is there any relationship among the measures of RST , RT, and MN? Describe the relationship, if there is any. Was the activity interesting? Were you able to come up with somerelationships involving angles formed by lines and their intercepted arcs?Are you ready to learn about tangents and secants and their real-lifeapplications? I am sure you are! “How do the different geometricrelationships involving tangents and secants of a circle facilitate findingsolutions to real-life problems and making wise decisions?” You will findthese out in the activities in the next section. Before doing theseactivities, read and understand first some important notes on tangentsand secants and the different geometric relationships involving them.Understand very well the examples presented so that you will be guidedin doing the succeeding activities. 182 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Tangent Line A tangent to a circle is a line coplanar with the circle and intersects itin one and only one point. The point of intersection of the line and the circle iscalled the point of tangency. Example: In the figure on the right, PQ CQ intersects C at A. PQ is a A tangent line and A is the point of tangency.DEPED COPYPostulate on Tangent Line P At a given point on a circle, one and only one line can be drawn that istangent to the circle. A To illustrate, consider V on the right. If UU is a point on the circle, then one and only oneline can be drawn through U that is tangent to thecircle. VTheorems on Tangent Line B1. If a line is tangent to a circle, then it is perpendicular R to the radius drawn to the point of tangency. If AB is tangent to Q at R, then it is A perpendicular to radius QR. Q2. If a line is perpendicular to a radius of a circle at its endpoint that is on the circle, then the line is tangent to the circle. If CS is perpendicular to radius LT at C T L, then it is tangent to T. L S 183 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
3. If two segments from the same exterior point are tangent to a circle, thenthe two segments are congruent.If DW and GW are tangent to E, Dthen DW GW . E WGCommon Tangent A common tangent is a line that is tangent to two circles in the sameplane.DEPED COPYCommon internal tangents Common external tangents dointersect the segment joining not intersect the segment joiningthe centers of the two circles the centers of the two circles. cd tDE MN s n Lines c and d are Lines s and t are common internal tangents. common external tangents. tangents. tangents.Tangent and Secant Segments and rays that are contained in the tangent or intersect thecircle in one and only one point are also said to be tangent to the circle.In the figure on the right, MN M Nand QR are tangent to S. S R Q 184 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
A secant is a line that intersects a circle at exactly two points. A secantcontains a chord of a circle.In circle A, MN is a secant line. A N MDEPED COPYTheorems on Angles Formed by Tangents and Secants1. If two secants intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. In the figure below, NX and MY are two secants intersecting outside the circle at point P. XY and MN are the two intercepted arcs of XPY.mXPY 1 mXY mMN 2For example, if mXY = 140 PN Mand mMN = 30, then X YmXPY 1 140 30 2 1 110 2mXPY 552. If a secant and a tangent intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. 185 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
In the figure below, CM is a secant and LM is a tangent intersectingoutside the circle at point M. LEC and LG are the two intercepted arcs ofLMC .mLMC 1 mLEC mLG LM 2 GFor example, if mLEC = 186 E Cand mLG = 70, thenDEPED COPYmLMC 1 186 70 2 1 116 2mLMC 583. If two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. In the figure below, QK and QH are two tangents intersecting outside the circle at point Q. HJK and HK are the two intercepted arcs of KQH .mKQH 1 mHJK mHK 2For example, if mHJK = 250 J Hand mHK = 110, then K QmKQH 1 250 110 2 1 140 2mKQH 704. If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 186 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
In the figure below, WS and RX are two secants intersecting inside thecircle. WR and XS are the two intercepted arcs of 1 while WX and RSare the two intercepted arcs of 2 .m1 1 mWR mXS m2 1 mWX mRS 2 2For example, W For example, Xif mWR = 100 and R if mWX = 80 and 1 mRS = 60, thenmXS = 120, thenDEPED COPY 1 2 m2 1 80 60m1 2 100 120 S 2 1 220 2 1 140m1 110 2 m2 705. If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one-half the measure of its intercepted arc. In the figure below, QS is a secant and RW is a tangent intersecting at S, the point of tangency. QS is the intercepted arc of QSR while QTS is the intercepted arc of QSW .mQSR 1 mQS mQSW 1 mQTS Q 2 2For example, For example, R Sif mQS = 170, then if mQTS = 190, then W 1mQSR 2 170 mQSW 1 190 T 2 mQSW 95mQSR 85 187 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
Learn more about Tangents http://www.regentsprep.org/Regents/math/geometry/and Secants of a circle GP15/CircleAngles.htmthrough the WEB. You may http://www.math-worksheet.org/secant-tangent-open the following links. angles http://www.mathopenref.com/tangentline.html http://www.ck12.org/book/CK-12-Geometry-Honors- Concepts/section/8.7/ http://www.ck12.org/book/CK-12-Geometry- Honors-Concepts/section/8.8/DEPED COPY Your goal in this section is to apply the key concepts of tangentsand secants of a circle. Use the mathematical ideas and the examplespresented in the preceding section to answer the activities provided.Activity 4:In the figure below, KL, KN, MP, and ML intersect Q at some points. Usethe figure to answer the following questions. S 1. Which lines are tangent to the K circle? Why? 2. Which lines are secants? Why?3. At what points does each secant N Q L intersect the circle? OP How about the tangents? M R4. Which angles are formed by two secant lines? two tangents? a tangent and a secant?5. Name all the intercepted arcs in the figure. Which angles intercept each of these arcs?6. Suppose mKOM 50 and mKQM 130, what is mKLM equal to? How about mNP? 188 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
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