Math Grade 10

Were you able to identify the tangents and secants in the figure,including the angles that they form? Were you able to identify the arcsthat these angles intercept? Were you able to determine the unknownmeasure of the angle? I am sure you were! In the next activity, you willfurther apply the different ideas learned about tangents and secants infinding the measures of angles, arcs, and segments in some geometricfigures.Activity 5:Use the figure and the given information to answer the questions that follow.Explain how you arrived at your answer.DEPED COPY1. If mADC = 160 and mEF = 80, 2. If mMKL = 220 and mML = 140, what is mABC ? what is mMQL? A DECF B3. If mPR = 45 and mQS = 49, 4. Suppose mCG = 6x + 5, what is mPTR ? mRTS ? mAR = 4x + 15, and mAEC  120. P R Find: a) x b) mCG c) mAR T G ERQS C A 189 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. If mLGC = 149 and mLSC  39, 6. OK is tangent to R at C. What is mMC? Suppose KC  OC , OK = 56, and RC = 24. Find: OR, RS, L and KS. M S R GS K C CDEPED COPY O7. If mQNO = 238, what is 8. PR is a diameter of O and mPQO ? mPQR ? O mRW = 55. Find: a. mPW d. mWRE b. mRPW e. mWER c. mPRW f. mEWRN WE QP P R OR9. Circles P and Q are tangent to each other at point S. P B S AB is tangent to both P and Q at S. Suppose Q AB = 16, AP = 12, and AQ = 10. What is the length of PQ if it bisects AB ? A 190 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.

10. AT is tangent to both circles K and J at A. STis tangent to K at S and RT is tangent to T R J at R. If ST  2x  7 andRT  3x  1, find:a. x c. RT Sb. ST d. AT K ADEPED COPY J How was the activity you have just done? Was it easy for you todetermine the measures of the different angles, arcs, and segments? Itwas easy for sure! In this section, the discussion was about the different geometricrelationships involving tangents and secants of a circle. Now that you know the important ideas about this topic, let us godeeper and move on to the next section. Your goal in this section is to think deeper and test further yourunderstanding of the different geometric relationships involving tangentsand secants of a circle. After doing the following activities, you should beable to find out how the different geometric relationships involvingtangents and secants of a circle facilitate finding solutions to real-lifeproblems and making wise decisions. 191 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:Answer the following.1. In the figure on the right, RO and DN are tangent to U at O and Nrespectively. R Oa. What is the measure of RON ? DNO?Explain how you arrived at your answer.b. Suppose ONR  NDU . Which angle is U congruent to NRO? Why?DEPED COPYc. If mONR  31, what is mNRO? N Dd. If mDUN 49, what is mNDU ? How about mDUO?e. Suppose OU = 6, RN = 13, and DN  ON , what is RO equal to? How about DN? DU?Is NRO  DUN ? Justify your answer.2. In the figure on the right, is LU tangent L 35 to I? Why? 3 I How about SC ? Justify your answer. U 6 A4 S 8C3. LR and LI are tangents to T from an external point L. R a. Is RL congruent to LI ? Why? A Lb. Is ∆LTR congruent to ∆LTI? Justify T your answer.c. Suppose mRLT 38. What is mILT equal to?How about mITL? mRTL? Id. If RT  10 and RL  24 , what is the length of TL ?How about the length of LI ? AL ? 192 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. In the figure on the right, ∆CDS is circumscribed C about M. Suppose the perimeter of ∆CDS is 33 units, SX = 6 units, and DY = 3 units. What Y Mare the lengths of the following segments? Explain Dhow you arrived at your answer. Za. SZ c. CX Xb. DZ d. CY SDEPED COPY5. From the main entrance of a park, there are two pathways where visitors can walk along going to the circular garden. The pathways are both tangent to the garden whose center is 40 m away from the main entrance. If the area of the garden is about 706.5 m2, how long is each pathway?Garden Main Entrance6. The map below shows that the waters within ARC, a 250° arc, isdangerous for shipping vessels. In the diagram, two lighthouses arelocated at points A and C and points P, R, and S are the locations of theship at a certain time, respectively. Aa. What are the possible measures of P,R, and S ?b. If you were the captain of a ship, how P shore would you make sure that your ship is in safe water? RC S 193 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 was the activity you have just performed? Did you gain betterunderstanding of the lesson? Were you able to use the mathematicsconcepts and principles learned in solving problems? Were you able torealize the importance of the lesson in the real world? I am sure youwere! In the next activity you will be proving geometric relationshipsinvolving tangents and secants.Activity 7:Show a proof of the following theorems involving tangents and secants.DEPED COPY1. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.Given: AB is tangent to C at D. AProve: AB  CD D CB2. 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.Given: RS is a radius of S. P S PQ  RS RProve: PQ is tangent to S at R. Q 194 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 S L point are tangent to a circle, then the two M segments are congruent. Given: EM and EL are tangent to S at M and L, respectively.Prove: EM  EL E4. If two tangents, a secant and a tangent, or 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.DEPED COPYa. Given: RS and TS are tangent to V R at R and T, respectively, and intersect at the exterior point S. V QProve: mRST  1 mTQR  mTR S T 2b. Given: KL is tangent to O at K. K L NL is a secant that passes P through O at M and N. KL and NL intersect at the O exterior point L. M NProve: mKLN  1 mNPK  mMK  2c. Given: AC is a secant that passes A through T at A and B. B EC is a secant that passes TC through T at E and D. D AC and EC intersect at the E exterior point C.Prove: mACE  1 mAE  mBD 2 195 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. 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.Given: AC and EC are secants P intersecting in the interior of V at T. PS and QR are the intercepted V arcs of PTS and QTR. ST QDEPED COPYProve:mPTS1mPS  mQR R 26. If a secant and a tangent intersect at the point of tangency, then themeasure of each angle formed is one-half the measure of its interceptedarc. KGiven: MP and LN are secant and tangent, respectively, and P O intersect at C at the point of tangency, M. LProve: mNMP  1 mMP  and M 2 1 N 2 mLMP  m MKP  Were you able to prove the different geometric relationshipsinvolving tangents and secants? Were you convinced that thesegeometric relationships are true? I know you were! Find out by yourselfhow these geometric relationships are illustrated or applied in the realworld. In this section, the discussion was about your understanding of thedifferent geometric relationships involving tangents and secants and howthey are illustrated in real life. What new realizations do you have about the different geometricrelationships involving tangents and secants? How would you connectthis to real life? How would you use this in making wise decisions? 196 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 COPY Now that you have a deeper understanding of the topic, you are ready to 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 the different geometric relationships involving tangents and secants. Activity 8: Answer the following. Use the rubric provided to rate your work. 1. The chain and gears of bicycles or motorcycles or belt around two pulleys are some real-life illustrations of tangents and circles. Using these real-life objects or similar ones, formulate problems involving tangents, then solve. 2. The picture below shows a bridge in the form of an arc. It also shows how secant is illustrated in real life. Using the bridge in the picture and other real- life objects, formulate problems involving secants, then solve them. 197 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 6 Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth 5 comprehension of the pertinent concepts and/or processes and 4 provides explanations wherever appropriate. 3 2 Poses a more complex problem and finishes all significant parts 1 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 formulate then solve problemsinvolving the different geometric relationships involving tangents andsecants. How did you find the performance task? How did the task help yourealize the importance of the topic in real life?SUMMARY/SYNTHESIS/GENERALIZATION: This lesson was about different geometric relationships involvingtangents and secants and their applications in real life. The lesson providedyou with opportunities to find the measures of angles formed by secants andtangents and the arcs that these angles intercept. You also applied theserelationships involving tangents and secants in finding the lengths of segmentsin some geometric figures. You were also given the opportunities to formulateand solve real-life problems involving tangents and secants of a circle. Yourunderstanding of this lesson and other previously learned mathematicsconcepts and principles will facilitate your learning in the succeeding lessons. 198 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 2B of this module by assessing your knowledge of thedifferent mathematical concepts previously studied and mathematicalskills learned. These knowledge and skills will help you understand therelationships among tangent and secant segments. As you go throughthis lesson, think of this important question: “How do the relationshipsamong tangent and secant segments facilitate finding solutions to real-life problems and making decisions?” To find the answer, perform eachactivity. If you find any difficulty in answering the exercises, seek theassistance of your teacher or peers or refer to the modules you havestudied earlier. You may check your work with your teacher.DEPED COPYActivity 1:Solve the following equations. Answer the questions that follow.1. 3x  27 6. x2  25 7. x2  642. 4x  20 8. x2  123. 6x  3124. 63  7x 9. x2  45 10. x2  805. 815  10x a. How did you find the value of x in each equation? b. What mathematics concepts or principles did you apply in solving the equations? Were you able to find the value of x in each equation? Were you able to recall how the equations are solved? The skill applied in the previous activity will be used as you go on with the module. 199 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 2:Use the figure below to answer the following questions.1. Which of the lines or line segments is a tangent? secant? chord? N Name these lines or line segments. S2. AT intersects LN at E. What are the A T different segments formed? Name these E segments.3. What other segments can be seen in theDEPED COPYJ figure? Name these segments. L4. SJ and LJ intersect at point J. How would you describe point J in relation to the given circle? Was it easy for you to identify the tangent and secant lines and chords and to name all the segments? I am sure it was! This time, find out the relationships among tangent, and secant segments, and external secant segments of circles by doing the next activity. Activity 3:Perform the following activity.Procedure:1. In the given circle below, draw two intersecting chords BT and MN. 200 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. Mark and label the point of intersection of the two chords as A.3. With a ruler, measure the lengths of the segments formed by the intersecting chords.What is the length of each of the following segments?a. BA c. MAb. TA d. NA4. Compare the product of BA and TA with the product of MA and NA.5. Repeat #1 to #4 using other pairs of chords of different lengths.DEPED COPYWhat conclusion can you make? Were you able to determine the relationship that exists amongsegments formed by intersecting chords of a circle? For sure you wereable to do it. In the next activity, you will see how tangent and secantsegments are used in real-life situations. Activity 4:Use the situation below to answer the questions that follow. You are in a hot air balloon and your eye level is 60 meters over theocean. Suppose your line of sight is tangent to the radius of the earth like theillustration shown below. 1. How far away is the farthest point you can see over the ocean if the radius of the earth is approximately 6378 kilometers? 201 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. What mathematics concepts would you apply to find the distance from where you are to any point on the horizon? How did you find the preceding activities? Are you ready to learn about tangent and secant segments? I am sure you are! From the activities done, you were able to find out how tangent and secant segments of circles are illustrated in real life. But how do the relationships among tangent and secant segments of circles 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 tangent and secant segments of circles and the examples presented.DEPED COPYTheorem on Two Intersecting ChordsIf two chords of a circle intersect, then the product of the measures ofthe segments of one chord is equal to the product of the measures of thesegments of the other chord.In the circle shown on the right, SN SLintersects DL at A. From the theorem, ASA  NA  DA  LA . D NExternal Secant Segment An external secant segment is the part of a secant segment that isoutside a circle. GIn the figure, GM and SM are secants. S A EMAM and EM are external secantsegments. 202 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.

Theorems on Secant Segments, Tangent Segments,and External Secant Segments1. If two secant segments are drawn to a circle from an exterior point, thenthe product of the lengths of one secant segment and its external secantsegment is equal to the product of the lengths of the other secant segmentand its external secant segment. A IRAR and NR are secant segments drawn Eto the circle from an exterior point R. Fromthe theorem, AR IR  NR  ER.DEPED COPY N2. If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment. COYO is a secant segment drawn to the Ncircle from exterior point O. CO is atangent segment that is also drawn tothe circle from the same exterior pointO. From the theorem, CO2 YO  NO. YLearn more about Tangent http://www.regentsprep.org/Regents/math/geomand Secant Segments of a etry/GP15/CircleAngles.htmCircle through the WEB.You may open the http://www.cliffsnotes.com/math/geometry/circlefollowing links. s/segments-of-chords-secants-tangents http://www.mathopenref.com/secantsintersecting. html http://www.ck12.org/book/CK-12-Geometry- Honors-Concepts/section/8.8/ http://www.math-worksheet.org/tangents 203 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 tangent and secant segments of a circle. Use the mathematical ideas and the examples presented in the preceding section to answer the given activities. Activity 5: DEPED COPYName the external secant segments in each of the following figures.1. E 4. G I M R IL Y L2. M 5. O L CT F DS W RE3. J 6. I O H GF S JE KD E ABC Were you able to identify the external secant segments in the given circles? In the next activity, you will apply the theorems you have learned in this lesson. 204 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:Find the length of the unknown segment (x) in each of the following figures.Answer the questions that follow.1. L 6. R A F 6 3 4 A 12 x 5x S N 87 O 5M 7. J 4T x U 56 E 5N2.DEPED COPY G G Dx 12 E 4 U 63. 5I 8. S S I x 16 Hx 8 10 F 5 O 9A 16 R M T4. A 9. 4 E4 x 5S 11 x J5 N 12 6 C T S5. xA 6 10. G M 25 L 6I 5 Ox V 8C 5 E 10 205All 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. How did you find the length of the unknown segment? What geometric relationships or theorems did you apply to come up with your answer? b. Compare your answers with those of your classmates. Did you arrive at the same answer? Explain. In the activity you have just done, were you able to apply the theoremsyou have learned? I am sure you were! In the next activity, you will use thetheorems you have studied in this lesson.Activity 7:DEPED COPYAnswer the following.1. Draw and label a circle that fits the following descriptions.a. has center Lb. has secant segments MO and QOc. has external secant segments NO and POd. has tangent segment RO S2. In the figure on the right, SU and X 12 WU are secant segments and XU is a tangent segment. If WU  14, 5 W ST  12 , and TU  4 , find: T a. VU 4V b. XU 5 14 U 5 How was the activity you have just done? Was it easy for you to applythe theorems on secant segments and tangent segments? It was easy forsure! In this section, the discussion was about tangent and secant segmentsand their applications in solving real-life problems. 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. 206 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 oftangents and secant segments. After doing the following activities, youshould be able to answer this important question: How do tangents andsecant segments of circles facilitate finding solutions to real-life problemsand making decisions?DEPED COPYActivity 8:Show a proof of each of the following theorems.1. If two chords of a circle intersect, then the product of the measures of thesegments of one chord is equal to the product of the measures of thesegments of the other chord. AGiven: AB and DE are chords of Cintersecting at M. D C E MProve: AM  BM  DM  EM B2. If two secant segments are drawn to a circle from an exterior point, then the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment.Given: DP and DS are secant P Q D R segments of T drawn T from exterior point D. SProve: DP  DQ  DS  DR 207 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 a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment.Given: KL and KM are tangent L and secant segments, O respectively, of O drawn from exterior point K. KM intersects O at N. K N MProve: KL2  KM  KNDEPED COPY Were you able to prove the theorems on intersecting chords, secantsegments, and tangent segments? I am sure you did! Let us find out more about these theorems and their applications.Perform the next activity.Activity 9:Answer the following questions.1. Jurene and Janel were asked to find the length of AB in the figure below. The following are their solutions. A Jurene: 7x 9 10 x B 7 Janel: 7x  7  99 10 E 10 D 9 C Who do you think would arrive at the correct answer? Explain youranswer. 208 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. The figure below shows a sketch of a circular children’s park and the different pathways from the main road. If the distance from the main road to Gate 2 is 70 m and the length of the pathway from Gate 2 to the Exit is 50 m, about how far from the main road is Gate 1? Gate 1 Gate 2 Main RoadExitDEPED COPY Gate 33. Anton used strings to hang two small light balls on the ceiling as shown in the figure on the right. The broken line represents the distance from the point of tangency of the two light balls to the ceiling. a. Suppose the diameter of each light ball is 10 cm and the length of the string used to hang it is 40 cm. How far is the point of tangency of the two light balls from the ceiling? b. Suppose Anton hangs 40 pairs of light balls on the ceiling of a hall in preparation for an event. How long is the string that he needs to hang these light balls if each has a diameter of 12 cm and the point of tangency of each pair of balls is 30 cm from the ceiling? How did you find the activity? Were you able to find out some real- life applications of the different geometric relationships involving tangents and secant segments? Do you think you could cite some more real-life applications of these? I am sure you could. Try doing the next activity. 209 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 this section, the discussion was about your understanding oftangent and secant segments and how they are used in real life. What new realizations do you have about tangent and secantsegments? How would you connect this to real life? Now that you have a deeper understanding of the topic, you areready to do the tasks in the next section. DEPED COPY 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 tangent and secant segments.Activity 10:Make a design of an arch bridge that would connect two places which areseparated by a river, 20 m wide. Indicate on the design the differentmeasurements of the parts of the bridge. Out of the design and themeasurements of its parts, formulate problems involving tangent and secantsegments, and then solve. Use the rubric provided to rate your work.Rubric for the Bridge’s DesignScore Descriptors 4 The bridge’s design is accurately made, presentable, and 3 appropriate. 2 1 The bridge’s design is accurately made and appropriate but not presentable. The bridge’s design is not accurately made but appropriate. The bridge’s design is made but not appropriate. 210 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 6 Poses a more complex problem with 2 or more correct possible 5 solutions and communicates ideas unmistakably, shows in-depth 4 comprehension of the pertinent concepts and/or processes, and 3 provides explanations wherever appropriate. 2 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 formulate problems where tangentand secant segments of circles are illustrated. How did you find the performance task? How did the task help yourealize the importance of the topic in real life?SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about the geometric relationships involving tangentand secant segments. In this lesson, you were able to find the lengths ofsegments formed by tangents and secants. You were also given the opportunityto design something practical where tangent and secant segments areillustrated or applied. Then, you were asked to formulate and solve problemsout of this design. Your understanding of this lesson and other previouslylearned mathematics concepts and principles will facilitate your learning of thesucceeding lessons in mathematics. 211 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.

GLOSSARY OF TERMSArc – a part of a circleArc Length – the length of an arc which can be determined by using theproportion A l , where A is the degree measure of this arc, r is the 360 2rradius of the circle, and l is the arc lengthCentral Angle – an angle formed by two rays whose vertex is the center ofthe circleDEPED COPYCommon External Tangents – tangents which do not intersect the segmentjoining the centers of the two circlesCommon Internal Tangents – tangents that intersect the segment joiningthe centers of the two circlesCommon Tangent – a line that is tangent to two circles on the same planeCongruent Arcs – arcs of the same circle or of congruent circles with equalmeasuresCongruent Circles – circles with congruent radiiDegree Measure of a Major Arc – the measure of a major arc that is equalto 360 minus the measure of the minor arc with the same endpointsDegree Measure of a Minor Arc – the measure of the central angle whichintercepts the arcExternal Secant Segment – the part of a secant segment that is outside acircleInscribed Angle – an angle whose vertex is on a circle and whose sidescontain chords of the circleIntercepted Arc – an arc that lies in the interior of an inscribed angle and hasendpoints on the angleMajor Arc – an arc of a circle whose measure is greater than that of asemicircle 212 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 COPYMinor Arc – an arc of a circle whose measure is less than that of a semicircle Point of Tangency – the point of intersection of the tangent line and the circle Secant – a line that intersects a circle at exactly two points. A secant contains a chord of a circle Sector of a Circle – the region bounded by an arc of the circle and the two radii to the endpoints of the arc Segment of a Circle – the region bounded by an arc and the segment joining its endpoints Semicircle – an arc measuring one-half the circumference of a circle Tangent to a Circle – a line coplanar with the circle and intersects it at one and only one point LIST OF THEOREMS AND POSTULATES ON CIRCLES Postulates: 1. Arc Addition Postulate. The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. 2. At a given point on a circle, one and only one line can be drawn that is tangent to the circle. Theorems: 1. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent. 2. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. 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. 4. 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). 213 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 COPY5. If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent. 6. If an inscribed angle of a circle intercepts a semicircle, then the angle is a right angle. 7. If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. 8. If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. 9. 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.10. If two segments from the same exterior point are tangent to a circle, then the two segments are congruent.11. 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.12. 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.13. 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.14. 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.15. 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.16. If two chords of a circle intersect, then the product of the measures of the segments of one chord is equal to the product of the measures of the segments of the other chord. 214 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 COPY17. If two secant segments are drawn to a circle from an exterior point, then the product of the lengths of one secant segment and its external secant segment is equal to the product of the lengths of the other secant segment and its external secant segment. 18. If a tangent segment and a secant segment are drawn to a circle from an exterior point, then the square of the length of the tangent segment is equal to the product of the lengths of the secant segment and its external secant segment. DepEd Instructional Materials That Can Be Used as Additional Resources 1. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third Year Mathematics. Module 18: Circles and Their Properties. 2. Distance Learning Module (DLM) 3, Modules 1 and 2: Circles. REFERENCES AND WEBSITE LINKS USED IN THIS MODULE: References: Bass, Laurie E., Randall, I. Charles, Basia Hall, Art Johnson, and Kennedy, D. Texas Geometry. Pearson Prentice Hall, Boston, Massachusetts 02116, 2008. Bass, Laurie E., Rinesmith Hall B., Johnson A., and Wood, D. F. Prentice Hall Geometry Tools for a Changing World. Prentice-Hall, Inc., NJ, USA, 1998. Boyd, Cummins, Malloy, Carter, and Flores. Glencoe McGraw-Hill Geometry. The McGraw-Hill Companies, Inc., USA, 2008. Callanta, Melvin M. Infinity, Worktext in Mathematics III. EUREKA Scholastic Publishing, Inc., Makati City, 2012. Chapin, Illingworth, Landau, Masingila, and McCracken. Prentice Hall Middle Grades Math, Tools for Success, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1997. Cifarelli, Victor, et al. cK-12 Geometry, Flexbook Next Generation Textbooks, Creative Commons Attribution-Share Alike, USA, 2009. 215 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 COPYClemens, Stanley R., Phares G. O’Daffer, Thomas J. Cooney, and John A. Dossey. Addison-Wesley Geometry. Addison-Wesley Publishing Company, Inc., USA, 1990.Clements, D. H., Jones, K. W., Moseley, L. B., and Schulman, L. Math in My World, McGraw-Hill Division, Farmington, New York, 1999.Department of Education. K to 12 Curriculum Guide Mathematics, Department of Education, Philippines, 2012.Gantert, Ann Xavier. AMSCO’s Geometry. AMSCO School Publications, Inc., NY, USA, 2008.Renfro, Freddie L. Addison-Wesley Geometry Teacher’s Edition. Addison- Wesley Publishing Company, Inc., USA, 1992.Rich, Barnett and Christopher Thomas. Schaum’s Outlines Geometry Fourth Edition. The McGraw-Hill Companies, Inc., USA, 2009.Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and Marvin L. Bittinger. Addison-Wesley Informal Geometry. Addison- Wesley Publishing Company, Inc., USA, 1992.Wilson, Patricia S., et al. Mathematics, Applications and Connections, Course I, Glencoe Division of Macmillan/McGraw-Hill Publishing Company, Westerville, Ohio, 1993. 216 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 COPYWebsite Links as References and Sources of Learning Activities: CK-12 Foundation. cK-12 Inscribed Angles. (2014). Retrieved June 29, 2014, from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.7/ CK-12 Foundation. cK-12 Secant Lines to Circles. (2014). Retrieved June 29, 2014, from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.8/ CK-12 Foundation. cK-12 Tangent Lines to Circles. (2014). Retrieved June 29, 2014, from http://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.4/ Houghton Mifflin Harcourt. CliffsNotes. Arcs and Inscribed Angles. (2013). Retrieved June 29, 2014, from http://www.cliffsnotes.com/math/geometry/ circles/arcs-and-inscribed-angles Houghton Mifflin Harcourt. CliffsNotes. Segments of Chords, Secants, and Tangents. (2013). Retrieved June 29, 2014, from http://www.cliffsnotes.com/math/geometry/circles/segments-of-chords- secants-tangents Math Open Reference. Arc. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/arc.html Math Open Reference. Arc Length. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/arclength.html Math Open Reference. Central Angle. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/circlecentral.html Math Open Reference. Central Angle Theorem. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/arccentralangletheorem.html Math Open Reference. Chord. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/chord.html Math Open Reference. Inscribed Angle. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/circleinscribed.html Math Open Reference. Intersecting Secants Theorem. (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/secantsintersecting.html 217 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 COPYMath Open Reference. Sector. (2009). Retrieved June 29, 2014, fromhttp://www.mathopenref.com/arcsector.htmlMath Open Reference. Segment. (2009). Retrieved June 29, 2014, fromhttp://www.mathopenref.com/segment.htmlmath-worksheet.org. Free Math Worksheets. Arc Length and Sector Area.(2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/arc-length-and-sector-areamath-worksheet.org. Free Math Worksheets. Inscribed Angles. (2014).Retrieved June 29, 2014, from http://www.math-worksheet.org/inscribed-anglesmath-worksheet.org. Free Math Worksheets. Secant-Tangent Angles. (2014).Retrieved June 29, 2014, from http://www.math-worksheet.org/secant-tangent-anglesmath-worksheet.org. Free Math Worksheets. Tangents. (2014). RetrievedJune 29, 2014, from http://www.math-worksheet.org/tangentsOnlineMathLearning.com. Circle Theorems. (2013). Retrieved June 29, 2014,from http://www.onlinemathlearning.com/circle-theorems.htmlRoberts, Donna. Oswego City School District Regents exam Prep Center.Geometry Lesson Page. Formulas for Angles in Circles Formed by Radii,Chords, Tangents, Secants. (2012). Retrieved June 29, 2014, fromhttp://www.regentsprep.org/Regents/math/geometry/ GP15/CircleAngles.htmWebsite Links for Videos:Coach, Learn. NCEA Maths Level 1 Geometric reasoning: Angles WithinCircles. (2012). Retrieved June 29, 2014, fromhttp://www.youtube.com/watch?v=jUAHw-JIobcKhan Academy. Equation for a circle using the Pythagorean Theorem.Retrieved June 29, 2014, fromhttps://www.khanacademy.org/math/geometry/cc-geometry-circlesSchmidt, Larry. Angles and Arcs Formed by Tangents, Secants, and Chords.(2013).Retrieved June 29, 2014, from http://www.youtube.com/watch?v=I-RyXI7h1bMSophia.org. Geometry. Circles. (2014). Retrieved June 29, 2014, fromhttp://www.sophia.org/topics/circles 218 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 COPYWebsite Links for Images: Cherry Valley Nursery and Landscape Supply. Seasonal Colors Flowers and Plants. (2014). Retrieved June 29, 2014 from http://www.cherryvalleynursery.com/ eBay Inc. Commodore Holden CSA Mullins pursuit mag wheel 17 inch genuine - 4blok #34. (2014). Retrieved June 29, 2014, from http://www.ebay.com.au/itm/Commodore-Holden-CSA-Mullins-pursuit-mag- wheel-17-inch-genuine-4blok-34-/221275049465 Fort Worth Weekly. Facebook Fact: Cowboys Are World’s Team. (2012) . Retrieved June 29, 2014 from http://www.fwweekly.com/2012/08/21/ facebook-fact-cowboys-now-worlds-team/ GlobalMotion Media Inc. Circular Quay, Sydney Harbour to Historic Hunter's Hill Photos. (2013). Retrieved June 29, 2014 from http://www.everytrail.com/ guide/circular-quay-sydney-harbour-to-historic-hunters-hill/photos HiSupplier.com Online Inc. Shandong Sun Paper Industry Joint Stock Co.,Ltd. Retrieved June 29, 2014, from http://pappapers.en.hisupplier.com/product- 66751-Art-Boards.html Kable. Slip-Sliding Away. (2014). Retrieved June 29, 2014, from http://www.offshore-technology.com/features/feature1674/feature1674-5.html Materia Geek. Nikon D500 presentada officialmente. (2009). Retrieved June 29, 2014 from http://materiageek.com/2009/04/nikon-d5000-presentada- oficialmente/ Piatt, Andy. Dreamstime.com. Rainbow Stripe Hot Air Balloon. Retrieved June 29, 2014, from http://thumbs.dreamstime.com/z/rainbow-stripe-hot-air- balloon-788611.jpg Regents of the University of Colorado. Nautical Navigation. (2014). Retrieved June 29, 2014, from http://www.teachengineering.org/view_activity.php?url= collection/cub_/activities/cub_navigation/cub_navigation_lesson07_activity1.xml 219 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 COPYSambhav Transmission. Industrial Pulleys. Retrieved June 29, 2014 fromhttp://www.indiamart.com/sambhav-transmission/industrial-pulleys.htmlshadefxcanopies.com. Flower Picture Gallery, Garden Pergola Canopies.Retrieved June 29, 2014, from http://www.flowerpicturegallery.com/v/halifax-public-gardens/Circular+mini+garden+with+white+red+flowers+and+dark+grass+in+the+middle+at+Halifax+Public+Gardens.jpg.htmlTidwell, Jen. Home Sweet House. (2012). Retrieved June 29, 2014 fromhttp://youveneverheardofjentidwell.com/2012/03/02/home-sweet-house/Weston Digital Services. FWR Motorcycles LTD. CHAINS ANDSPROCKETS. (2014). Retrieved June 29, 2014 fromhttp://fwrm.co.uk/index.php?main_page=index&cPath=585&zenid=10omr4hehmnbkktbl94th0mlp6 220 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 COPYI. INTRODUCTION Look around! What geometric figures do you see in your classroom, school buildings, houses, bridges, roads, and other structures? Have you ever asked yourself how geometric figures helped in planning the construction of these structures? In your community or province, was there any instance when a stranger or a tourist asked you about the location of a place or a landmark? Were you able to give the right direction and how far it is? If not, could you give the right information the next time somebody asks you the same question? Find out the answers to these questions and determine the vast applications of plane coordinate geometry through this module. 221 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.

II. LESSONS AND COVERAGE: In this module, you will examine the questions asked in the precedingpage when you take the following lessons:Lesson 1 – The Distance Formula, The Midpoint, and The Coordinate ProofLesson 2 – The Equation of a CircleIn these lessons, you will learn to:  derive the distance formula;  apply the distance formula in proving some geometric properties;  graph geometric figures on the coordinate plane; and  solve problems involving the distance formula.  illustrate the center-radius form of the equation of a circle;  determine the center and radius of a circle given its equation and vice versa;  graph a circle on the coordinate plane; and  solve problems involving circles on the coordinate plane.Lesson 1DEPED COPYLesson 2 Here is a simple map of the lessons that will be covered in this module: Plane Coordinate Geometry The Distance Formula Problems Involving The Midpoint Formula Geometric Figures on the Coordinate Coordinate Proof Plane The Equation and Graph of a Circle 222 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.

III. PRE-ASSESSMENTPart IFind out how much you already know about this module. Choose the letterthat you think best answers each of the following questions. Take note ofthe items that you were not able to answer correctly and find the rightanswer as you go through this module.1. Which of the following represents the distance d between the two points x1, y1 and x 2, y 2  ?        A. d  x 2  x1 2  y 2  y1 2 C. d  x 2  x1 2  y 2  y1 2        B. d  x 2  x1 2  y 2  y1 2 D. d  x 2  x1 2  y 2  y1 2DEPED COPY2. Point L is the midpoint of KM . Which of the following is true about the distances among K, L, and M?A. KL  KM C. KL  LMB. LM  KM D. 2 KM  KL  LM3. A map is drawn on a grid where 1 unit is equivalent to 1 km. On thesame map, the coordinates of the point corresponding to San Vicenteis (4, 9). Suppose San Vicente is 13 km away from San Luis. Which ofthe following could be the coordinates of the point corresponding toSan Luis?A. (-13, 0) B. (16, 4) C. (4, 16) D. (0, 13)4. What is the distance between the points M(-3,1) and N(7,-3)?A. 6 B. C. 14 D.5. Which of the following represents the midpoint M of the segment whose endpoints are x1, y1 and x 2, y 2  ?A. M   x 1  x 2 , y1  y2  C. M   x 1  y 1 , x2  y2   2 2   2 2     B. M   x 1  x 2 , y1  y2  D. M   x 1  y 1 , x2  y2   2 2   2 2     6. What are the coordinates of the midpoint of a segment whoseendpoints are (-1, -3) and (11, 7)?A. (2, 5) B. (6, 5) C. (-5, -2) D. (5, 2) 223 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. Which of the following equations describe a circle on the coordinateplane with a radius of 4 units?A. x  42  y  42  22 C. x  22  y  22  42B. x  22  y  22  42 D. x  42   y  42  1628. P and Q are points on the coordinate plane as shown in the figure below. yDEPED COPY xIf the coordinates of P and Q are  2,5 and 8,5, respectively, whichof the following would give the distance between the two points?A.  2  5 B. 8  5 C. 8  2 D.  2  89. A new transmission tower will be put up midway between two existingtowers. On a map drawn on a coordinate plane, the coordinates of thefirst existing tower are (–5, –3) and the coordinates of the secondexisting tower are (9,13). What are the coordinates of the point wherethe new tower will be placed?A. (2, 5) B. (7, 8) C. (4, 10) D. (14, 16)10. What proof uses figures on a coordinate plane to prove geometricproperties?A. indirect proof C. coordinate proofB. direct proof D. two-column proof11. The coordinates of the vertices of a square are H(3, 8), I(15, 8), J(15, –4), and K(3, –4). What is the length of a diagonal of the square?A. 4 B. 8 C. 12 D. 12 212. The coordinates of the vertices of a triangle are T(–1, –3), O(7, 5), andP(7, –2). What is the length of the segment joining the midpoint of OTand P?A. 5 B. 4 C. 3 D. 7 224 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.

13. What figure is formed when the points A(3, 7), B(11, 10), C(11, 5), and D(3, 2) are connected consecutively?A. parallelogram C. squareB. trapezoid D. rectangle14. In the parallelogram below, what are the coordinates of Q? P(a, c) QDEPED COPYS(0, 0) R(b, 0)A. (a, b+c) B. (a+b,c) C. (a-b,c) D. (a,b-c)15. Diana, Jolina, and Patricia live in three different places. The location of their houses are shown on a coordinate plane below. y JolinaDiana Patricia xAbout how far is Jolina’s house from Diana’s house?A. 10 units B. 10.58 units C. 11.4 units D. 12 units16. What is the center of the circle x 2  y 2  4x  10y  13  0 ?A. (2, 5) B. (–2, 5) C. (2, –5) D. (–2, –5) 225 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.

17. Point F is 5 units from point D whose coordinates are (6, 2). If the x-coordinate of F is 10 and lies in the first quadrant, what is its y-coordinate?A. -3 B. -1 C. 5 D. 718. The endpoints of a diameter of a circle are L(–3, –2) and G(9, –6).What is the length of the radius of the circle?A. 10 B. 2 10 C. 4 10 D. 8 1019. A radius of a circle has endpoints (4, –1) and (8, 2). What is theequation that defines the circle if its center is at the fourth quadrant?A. x  82   y  22  25 C. x  82   y  22  100B. x  42   y  12  100DEPED COPYD. x  42   y  12  2520. On a grid map of a province, the coordinates that correspond to the location of a cellular phone tower is (–2, 8) and it can transmit signals up to a 12 km radius. What is the equation that represents the transmission boundaries of the tower? A. x 2  y 2  4x  16y  76  0 C. x 2  y 2  4x 16y  76  0 B. x 2  y 2  4x 16y  76  0 D. x 2  y 2  4x 16y  76  0Part IISolve each of the following problems. Show your complete solutions.1. A tracking device in a car indicates that it is located at a point whose coordinates are (17, 14). In the tracking device, each unit on the grid is equivalent to 5 km. How far is the car from its starting point whose coordinates are (1, 2)?2. A radio signal can transmit messages up to a distance of 3 km. If the radio signal’s origin is located at a point whose coordinates are (4, 9), what is the equation of the circle that defines the boundary up to which the messages can be transmitted?Rubric for Problem SolvingScore Descriptors 4 3 Used an appropriate strategy to come up with a correct solution 2 and arrived at a correct answer. 1 Used an appropriate strategy to come up with a solution. But a part of the solution led to an incorrect answer. Used an appropriate strategy but came up with an entirely wrong solution that led to an incorrect answer. Attempted to solve the problem but used an inappropriate strategy that led to a wrong solution. 226 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.

Part IIIRead and understand the situation below, then answer the question orperform what are asked. The Scout Master of your school was informed that the Provincial BoyScouts Jamboree will be held in your municipality. He was assigned toprepare the area that will accommodate the delegates from 30 municipalities.It is expected that around 200 boy scouts will join the jamboree from eachmunicipality. To prepare for the event, he made an ocular inspection of the areawhere the jamboree will be held. The area is rectangular in shape and is largeenough for the delegates to set up their tents and other camping structures.Aside from these, there is also a provision for the jamboree headquarter,medics quarter in case of emergency and other health needs, walkways, androads, security posts, and a large ground where the different boy scoutsevents will be held. Aside from conducting an ocular inspection, he was also tasked toprepare a large ground plan to be displayed in front of the camp site. Copiesof the ground plan will also be given to heads of the different delegations.DEPED COPY1. Suppose you are the Scout Master, how will you prepare the ground plan of the Boy Scouts jamboree?2. Prepare the ground plan. Use a piece of paper with a grid and coordinate axes. Indicate the scale used.3. On the grid paper, indicate the proposed locations of the different delegations, the jamboree headquarter, medics quarter, walkways and roads, security posts, and the boy scouts event ground.4. Determine all the mathematics concepts or principles already learned that are illustrated in the prepared ground plan.5. Formulate equations and problems involving these mathematics concepts or principles, then solve.Rubric for Ground PlanScore Descriptors 4 The ground plan is accurately made, appropriate, and 3 presentable. 2 1 The ground plan is accurately made and appropriate but not presentable. The ground plan is not accurately made but appropriate. The ground plan is not accurately made and not appropriate. 227 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 Equations Formulated and SolvedScore Descriptors 4 All equations are properly formulated and solved correctly. 3 All equations are properly formulated but some are not solved correctly. 2 All equations are properly formulated but at least 3 are not solved correctly. 1 All equations are not properly formulated and solved.Rubric on Problems Formulated and SolvedScoreDEPED COPY Descriptors 6 Poses a more complex problem with 2 or more correct possible 5 solutions and communicates ideas unmistakably, shows in- depth comprehension of the pertinent concepts and/or 4 processes, and provides explanations wherever appropriate 3 Poses a more complex problem and finishes all significant 2 parts of the solution and communicates ideas unmistakably, 1 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 approachSource: D.O. #73, s. 2012IV. LEARNING GOALS AND TARGETS: After going through this module, you should be able to demonstrateunderstanding of key concepts of plane coordinate geometry particularly thedistance formula, equation of a circle, and the graphs of circles and othergeometric figures. Also, you should be able to formulate and solve problemsinvolving geometric figures on the rectangular coordinate plane. 228 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 1 of this module by assessing your knowledge of thedifferent mathematical concepts previously studied and your skills inperforming mathematical operations. These knowledge and skills willhelp you understand the distance formula. As you go through this lesson,think of this important question: How do the distance formula, themidpoint formula, and the coordinate proof facilitate finding solutions toreal-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 youhave studied earlier. You may check your work with your teacher.DEPED COPYActivity 1:Use the number line below to find the length of each of the followingsegments and then answer the questions that follow.ABC DE FG Q1. AB 4. DE2. BC 5. EF3. CD 6. FGQuestions:1. How did you find the length of each segment?2. Did you use the coordinates of the points in finding the length of eachsegment? If yes, how?3. Which segments are congruent? Why?4. How would you relate the lengths of the following segments?d.1) AB , BC , and AC d.2) AC , CE , and AE 229 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. Is the length of AD the same as the length of DA ? How about BF and FB ? Explain your answer. Were you able to determine the length of each segment? Were youable to come up with relationships among the segments based on theirlengths? What do you think is the significance of this activity in relation toyour new lesson? Find this out as you go through this module.Activity 2:The length of one side of each right triangle below is unknown. Determine thelength of this side. Explain how you obtained your answer. DEPED COPY1. ? 4. 3 ? 4 462. 5. 15 9 8 12 ??3. 13 6. 24 5 ? ? 18 In the activity, you have just done, were you able to determine thelength of the unknown side of each right triangle? I know you were ableto do it! The mathematics principles you applied in finding each unknownside is related to your new lesson, the distance formula. Do you knowwhy? Find this out in the succeeding activities! 230 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 3:Use the situation below to answer the questions that follow. Jose lives 5 km away from the plaza. Every Saturday, he meets Emilioand Diego for a morning exercise. In going to the plaza, Emilio has to travel6 km to the west while Diego has to travel 8 km to the south. The location oftheir houses and the plaza are illustrated on the coordinate plane as shownbelow. y Diego’s houseDEPED COPY PlazaCity Hall Emilio’s houseJose’s xhouse Gasoline Station1. How far is Emilio’s house from Diego’s house? Explain your answer.2. Suppose the City Hall is 4 km north of Jose’s house. How far is it from the plaza? from Emilio’s house? Explain your answer.3. How far is the gasoline station from Jose’s house if it is km south of Emilio’s house? Explain your answer.4. What are the coordinates of the points corresponding to the houses of Jose, Emilio, and Diego? How about the coordinates of the point corresponding to the plaza? 231 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 COPY5. If the City Hall is km north of Jose’s house, what are the coordinates of the point corresponding to it? How about the coordinates of the point corresponding to the gasoline station if it is km south of Emilio’s house? 6. How are you going to use the coordinates of the points in determining the distance between Emilio’s house and the City Hall? Jose’s house and the gasoline station? The distances of the houses of Jose, Emilio, and Diego from each other? Explain your answer. Did you learn something new about finding the distance between two objects? How is it different from or similar with the methods you have learned before? Learn about the distance formula and its derivation by doing the next activity. Activity 4:Perform the following activity. Answer every question that follows. 1. Plot the points A(2,1) and B(8,9) on the coordinate plane below. y x 2. Draw a horizontal line passing through A and a vertical line containing B. 3. Mark and label the point of intersection of the two lines as C. What are the coordinates of C? Explain how you obtained your answer. What is the distance between A and C? How about the distance between B and C? 232 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 COPY 4. Connect A and B by a line segment. What kind of triangle is formed by A, B, and C? Explain your answer. How will you find the distance between A and B? What is AB equal to? 5. Replace the coordinates of A by (x1, y1) and B by (x2, y2). What would be the resulting coordinates of C? What expression represents the distance between A and C? How about the expression that represents the distance between B and C? What equation will you use to find the distance between A and B? Explain your answer. How did you find the preceding activities? Are you ready to learn about the distance formula and its real-life applications? I am sure you are! From the activities done, you were able to find the distance between two points or places using the methods previously learned. You were able to derive also the distance formula. But how does the distance formula facilitate solving real-life problems and making wise 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 the distance formula including the midpoint formula and the coordinate proof. Understand very well the examples presented so that you will be guided in doing the succeeding activities. 233 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.

Distance between Two Points The distance between two points is always nonnegative. It is positivewhen the two points are different, and zero if the points are the same. If P andQ are two points, then the distance from P to Q is the same as the distancefrom Q to P. That is, PQ = QP. Consider two points that are aligned horizontally or vertically on thecoordinate plane. The horizontal distance between these points is theabsolute value of the difference of their x-coordinates. Likewise, the verticaldistance between these points is the absolute value of the difference of theiry-coordinates.DEPED COPYExample 1: Find the distance between P(3, 2) and Q(10, 2).Solution: y Q P x Since P and Q are aligned horizontally, then PQ  10  3 or PQ  7.Example 2: Determine the distance between A(4, 3) and B(4, –5). ySolution: x Points A and B are on the same vertical line. So the distance between them is AB  3   5 . This can be simplified to AB  3  5 or AB  8 . 234 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.

The Distance Formula The distance between two points, whether or not they are alignedhorizontally or vertically, can be determined using the distance formula. Consider the points P and Q whose coordinates are (x1, y1) and(x2, y2), respectively. The distance d between these points can be determined   using the distance formula d  x 2  x1 2  y 2  y1 2 or   PQ  x 2  x1 2  y 2  y1 2 . y Q(x2, y2) PQDEPED COPY P(x1, y1) xExample 1: Find the distance between P(1, 3) and Q(7, 11).Solution: To find the distance between P and Q, the following procedures can be followed. 1. Let  x1,y1 = (1, 3) and  x2,y2  = (7, 11). 2. Substitute the corresponding values of x1, y1, x 2, and y 2 in the distance formula x2  x1 2  2    PQ  y2  y1 . 235 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. Solve the resulting equation. PQ  7  12  11 32  62  82  36  64  100 PQ  10 The distance between P and Q is 10 units.Example 2: Determine the distance between A(1, 6) and B(5, –2).DEPED COPYSolution: Let x1  1, y1  6 , x2  5 , and y2  2. Then substitute these values in the formula AB  x2  x1 2  y2  y1 2 . AB  5 12   2  62 Simplify. AB  5 12   2  62  42   82  16  64  80  16  5 AB  4 5 or AB  8.94 The distance between A and B is 4 5 units or approximately 8.94 units. The distance formula has many applications in real life. In particular, itcan be used to find the distance between two objects or places.Example 3: A map showing the locations of different municipalities and cities is drawn on a coordinate plane. Each unit on the coordinate plane is equivalent to 6 kilometers. Suppose the coordinates of Mabini City is (2, 2) and Sta. Lucia town is (6, 8). What is the shortest distance between these two places? 236 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 COPYSolution: Let x1  2 , y1  2 , x 2  6 , and y 2  8. Then substitute these    values into the distance formula d  x 2  x1 2  y 2  y1 2 . d  6  22  8  22 Simplify the expression. d  6  22  8  22  42  62  16  36  52 d  2 13 units or d  7.21 units Since 1 unit on the coordinate plane is equivalent to 6 units, multiply the obtained value of d by 6 to get the distance between Sta. Lucia town and Mabini City. 7.216  43.26 The distance between Sta. Lucia town and Mabini City is approximately 43.26 km. 237 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.

The Midpoint FormulaIf L x1, y1 and N x 2,y 2  are the endpoints of a segment and M is themidpoint, then the coordinates of M =  x1  x2 , y1  y2  . This is also referred  2 2   to as the Midpoint Formula. y N x2,y2 DEPED COPY M  x1  x 2 , y1  y 2   2 2    L x1,y1 xExample: The coordinates of the endpoints of LG are  3,2 andSolution: (8, 9), respectively. What are the coordinates of its midpoint M? Let x1  3 , y1  2 , x2  8 , and y2  9 . Substitute these values into the formula M   x 1  x 2 , y 1  y 2   2 2 .   M   3  8 , 2  9  or M   5 , 7  2   2   2   2  The coordinates of the midpoint of LG are  5 , 7  .  2 2    238 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.