Mathematics Grade 10

Answer Key6. Determine the measure of the central angle that intercepts the same arc. The measure of the central angle is equal to the measure of its intercepted arc. mAD= 111.14 mEFA= 180 mDE= 68.86 mEF= 111.14 mDEA= 248.86 mAF= 68.867. mDCE  2mDAE mDE  2mDAE. Since mDCE  mDE, then mDE  2mDAE. DEPED COPY8. mAD  2mDAB mEFA  2mEAG9. mBGD  1 mAD  mAF 2Activity 3: Find Out by Yourself!Answer Key2. RST is a central angle of S.4. mRST  1 mST  26. Yes. mRST  1 mRVT  mRT 28. Yes. mRST  1 mRT  mNT 210. Yes. mRST  1 mRT  mMN 212. Yes. mRST  1 mRT  mMN 2 160 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.

DEPED COPY Let the students give their realizations of the activities done before proceeding to the next activities. Provide them with an opportunity to relate or connect their responses to the activities given in their lesson, tangents and secants of a circle. Let the students read and understand some important notes on tangents and secants of a circle and study carefully the examples given. What to PROCESS In this section, let the students use the geometric concepts and relationships they have studied and the examples presented in the preceding section to answer the succeeding activities. Present to the students the figure given in Activity 4. In this activity, the students should be able to identify the tangents and secants in the figure including the angles that they form and the arcs that these angles intercept. They should be able to determine also the unknown measure of the angle formed by secants intersecting in the exterior of the circle. Give emphasis to the geometric relationship the students applied in finding the measure of the angle. Provide them opportunities to compare their answers and correct their errors, if there are any. Activity 4: Tangents or Secants? Answer Key 1. KL and LM. Each line intersects the circle at exactly one point. 2. KN and MP. Each line intersects the circle at two points. 3. KNK and N; MPM and P; KLK; LMM 4. There are other angles formed but only these are considered. KOM is formed by two secant lines. KLM is formed by two tangent lines. LMP, LKN, PMR, and NKS. Each is formed by a secant and a tangent. 5. MP  PMR , NP  KOM , KN  NKS , KM  KLM , KPM  KLM 6. mKLM  50 ; mNP = 30 161 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.

In Activity 5, provide the students with opportunities to apply thedifferent geometric relationships in finding the measures of the angles formedby tangents and secants and the arcs that these angles intercept. Let themalso determine the lengths of segments tangent to circle/s and othersegments drawn on a circle. Ask them to support their answers by stating thegeometric relationships applied.Activity 5: From One Place to AnotherAnswer Key DEPED COPY7. mPQO  61 mPQR  1191. mABC  402. mMQL  40 8. a. mPW  1253. mPTR  47 b. mRPW  27.5 c. mPRW  62.5 mRTS  133 d. mWRE  27.54. a. x  10 e. mWER  62.5 f. mWER  62.5 b. mCG  65 c. mAR  55 9. PQ  6  4 55. mMC  71 10. a. x  66. OR  4 85 RS  24 b. ST  19 c. RT  19 KS  4 85  24 d. AT  19What to REFLECT on and UNDERSTAND Let the students think deeply and test further their understanding of thedifferent geometric relationships involving tangents and secants of circles bydoing Activity 6. In this activity, they will apply these geometric relationships insolving problems. 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.

DEPED COPYActivity 6: Think of These Relationships Deeply! Answer Key 1. a. mRON  90; mRON  90. The radius of a circle is perpendicular to a tangent line at the point of tangency. b. NRO  NUD c. mNRO  59 d. mNDU  41; mDUO  131 e. RO  5; DN  12; DU  6 3 NROis not congruent to DUN . The lengths of their sides are not equal. 2. LU is tangent to I. SC is also tangent to I. 3. a. RL  LI . If two segments from the same exterior point are tangent to a circle, then the two segments are congruent. b. LTR  LTI by HyL Theorem. c. mILT  38; mITL  52; mRTL  52 d. TL  26; LI  24 ; AL  16 4. a. SZ  6 b. DZ  3 c. CX  7.5 d. CY  7.5 If two segments from the same exterior point are tangent to a circle, then the two segments are congruent. 5. 5 55 m 6. a. mP  55 mR  55 mS  55 b. The angle that I will make with the lighthouse must be less than 55°. Provide the students with opportunities to prove theorems involving tangents and secants of circles. Let them perform Activity 7. Guide the students in writing the proof. If needed, provide hints. 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.

Activity 7: Is this true?Answer Key C at D.1. Given: AB is tangent to Prove: AB  CD To prove: a. Draw AC b. Assume AB is not perpendicular toCD and AB  ACProof: Statement DEPED COPY ReasonAB is not perpendicular AssumptiontoCD and AB  AC .E is a point on AD such that Ruler PostulateDE  2DADA  AE Betweenness and CongruenceCAE  CAD of SegmentsAC  AC Right angles are congruent.CDA  CEA Reflexive PropertyCD  CE SAS Congruence PostulateCD  CE CPCTCD and E are on C.D and E are the points of intersection The lengths of congruentof tangent line AB and segments are equal. Definition of circle C is not true. A tangent intersects the circle atAB  CD exactly one point. Only one line can be drawn on a circle that is tangent to it at the point of tangency.2. Given: RS is a radius of S. PQ  RS Prove: PQ is tangent to S at R. To prove: Draw QS . 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.

Answer KeyProof: Reason Statement Given RS is a radius of S and PQ  RS . QS >RS The shortest segment from the center of a circle to a line tangent Q is not on S. to it is the perpendicular PQ is tangent to S at R. segment. No other point of a tangent line other than the point of tangency lies on a circle. A tangent intersects the circle at exactly one point.DEPED COPY3. Given: EM and EL are tangent to S at M and L, respectively.Prove: EM  ELTo prove: Draw MS , LS , and ES .Proof: Statement Reason MS  LS Radii of the same circle are congruent.EL  LS and EM  MS . A line tangent to a circle is perpendicular to the radius.ES  ES Reflexive PropertyESM  ESL Hypotenuse-Leg CongruenceEM  EL Theorem CPCTC 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.

4. RS and TS are tangent to V at R and T, respectively, a. Given: and intersect at the exterior S. DEPED COPYProve:mRST1mTQR mTR 2 To prove: Draw RV , TV , and SV .Proof: Reason Statement (Proven) Acute angles of a right SVR  SVT triangle are complementary. mRVS  mRSV  90 and Angle Addition Postulate mTVS  mTSV  90 mRVS  mTVS  mRVT Substitution mRVT  90  x  90  x  The measure of a central angle is equal to the  180  2x measure of its intercepted arc. mTR  180  2x The degree measure of a circle is 360. mTQR  mTR  360 Substitution and Addition mTQR  180  2x Property of Equality mRSV  mTSV  mRST Angle Addition Postulate mRSV  mTSV  x  x By Substitution and Addition  2x mRST  2x Transitive Property mTQR  mTR  180  2x  180  2x By Substitution and  22x Subtraction 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.

Answer KeymTQR  mTR  2mRST  By Substitution Multiplication PropertymRST  1 mTQR  mTR 2b. Given:KL is tangent to O at K. NL is a secant that passes through O at M and N.DEPED COPYKL and NL intersect at the exterior point L. Prove: mKLN  1 mNPK  mMK 2 To prove: Draw KM , MO , and KO . Let mMKL  x so that mMKO  90  x and mKMO  90  x .Proof: Statement Reason The measure of anmNMK  1 mNPK  inscribed angle is one-half 2 the measure of its intercepted arc.mNMK  mMKL  mNLK The measure of themKOM  mKM exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. The measure of a central angle is equal to the measure of its intercepted arc. 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.

mMKL  mMKO  90 The sum of the measures ofmKMO  mMKO  mKOM  180 complementary angles is 90.mKOM  2x The sum of the measures of the interiormKM  2x angle of a triangle is 180 Addition Property2mMKL   mKM mMKL 1 mKM  Transitive Property 2 Multiplication Property By Subtraction DEPED COPY or 1 mNPK   1 mKM   mNMK  mMKL2 2 mMKL  mNLK  mMKL mNLKmNLK  1 mNPK  mKM By Substitution 2 c. Given: AC is a secant that passes through T at A and B. EC is a secant that passes through T at E and D. AC and EC intersect at the exterior point C. Prove: mACE  1 mAE  mBD 2 To prove: Draw AD and BE . 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.

Answer KeyProof: Statement Reason The measure of themADE  mDAC  mACE exterior angle of a triangle is equal to themADE  1 mAE and sum of the measures ofDEPED COPY 2 its remote interior angles.mDAB  1 mBD The measure of an 2 inscribed angle is one- half the measure of its1 mAE  1 mBD  mADE  mDAB intercepted arc.2 2 By SubtractionmADE  mDAC  mACE Addition PropertymACE  1 mAE  1 mBD or 2 2 Transitive PropertymACE  1 mAE  mBD 25. Given: PR and QS are secants intersecting in the interior of V at T. PS and QR are the intercepted arcs of PTS and QTR .Prove: mPTS  1 mPS  mQR  2To prove: Draw RS . 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.

Proof: Statement Reason The measure of anmPRS  1 mPS and inscribed angle is one-half 2 the measure of its intercepted arc.mQSR  1 mQR  2 The measure of the exterior angle of a trianglemQTR  mPRS  mQSR is equal to the sum of the measures of its remote interior angles. Substitution The measures of vertical angles are equal. Transitive PropertymQTR DEPED COPY1mPS 1 mQR  or 2 2mQTR  1 mPS  mQR 2mQTR  mPTSmPTS  1 mPS  mQR 26. Given: MP and LN are secant and Prove: tangent, respectively, and intersect at C at the point of tangency, M. mNMP  1 mMP  and 2 mLMP  1 mMKP  2 To prove: Draw OP and OM . Let mNMP  x so that mOMP  90  x and mOPM  90  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.

Answer KeyProof: Statement Reason The measure of a centralmMOP  mMP angle is equal to the measure of its interceptedmNMP  mOMP  90 arc. The sum of the measuresDEPED COPYmOMP  mOPM  mMOP  180of complementary angles is 90.mMOP  2x The sum of the measures of a triangle is 180.mMP  2x Addition PropertymMP  2mNMP  Transitive PropertymNMP  1 mMP  Substitution 2 Multiplication PropertymMP  mMKP  360 The degree measure of amMKP  360  2x circle is 360. By Substitution andmMKP  2180  x Subtraction By FactoringmLMP  90  90  x ormLMP  180  x Angle Addition PostulatemMKP  2mLMP  SubstitutionmLMP  1 mMKP  Multiplication Property 2 Before the students move to the next section of this lesson, give ashort test (formative test) to find out how well they understood the lesson. Askthem also to write a journal about their understanding of tangents and secantsof a circle. Refer to the Assessment Map. 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.

DEPED COPYWhat to TRANSFER Give the students opportunities to demonstrate their understanding ofthe different geometric relationships involving tangents and secants of circlesby doing a practical task. Let them perform Activity 8. You can ask thestudents to work individually or in a group. In this activity, the students willformulate and solve problems involving tangents and secants of circles asillustrated in some real-life objects.Activity 8: My Real World Answer Key Evaluate students’ product. You may use the rubric provided.Summary/Synthesis/Generalization: This lesson was about the geometric relationships involving tangentsand secants of a circle, the angles they form and the arcs that these anglesintercept. The lesson provided the students with opportunities to derivegeometric relationships involving radius of a circle drawn to the point oftangency, investigate relationships among arcs and angles formed by secantsand tangents, and apply these in solving problems. Moreover, they were giventhe chance to prove the different theorems on tangents and secants anddemonstrate their understanding of these concepts by doing a practical task.Their understanding of this lesson and other previously learned mathematicsconcepts and principles will facilitate their learning of the wide applications ofcircles in real life. 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.

Lesson 2B: Tangent and Secant SegmentsWhat to KNOW Find out how much students have learned about the differentmathematics concepts previously studied and their skills in performingmathematical operations. Checking these will facilitate teaching and students’understanding of the geometric relationships involving tangent and secantsegments. Tell them that as they go through this lesson, they have to think ofthis important question: How do geometric relationships involving tangent andsecant segments facilitate solving real-life problems and making decisions? Provide the students with opportunities to enhance further their skills infinding solutions to mathematical sentences previously studied. Let themperform Activity1. In this activity, the students will solve linear and quadraticequations in one variable. These mathematical skills are prerequisites tolearning the geometric relationships involving tangent and secant segments. Ask the students to explain how they arrived at the solutions and howthey applied the mathematics concepts or principles in solving eachmathematical sentence.Activity 1: What is my value?DEPED COPYAnswer Key 6. x  5 7. x  8 1. x  9 8. x  2 3 2. x  5 9. x  3 5 3. x  6 10. x  4 5 4. x  9 5. x  12Questions: a. Applying the Division Property of Equality and Extracting Square Roots b. Division Property of Equality and Extracting Square Roots Present to the students the figure in Activity 2. Then, let them identifythe tangent and secant lines and the chords, name all the segments they cansee, and describe a point in relation to the circle. This activity has somethingto do with the lesson. Let the students relate this to the succeeding activities. 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.

Activity 2: My Segments Answer Key 1. JL - tangent; JS - secant; AS ; AT ; LN - chords 2. NE ; ET ; AE ; EL 3. AS ; AJ ; JL 4. A point outside the circle Ask the students to perform Activity 3 to determine the relationship thatexists among segments formed by intersecting chords of a circle. In thisactivity, the students might not be able to arrive at the accuratemeasurements of the chords due to the limitations of the measuringinstrument to be used. If possible, use math freeware like GeoGebra inperforming the activity.Activity 3: What is true about my chords? Answer Key DEPED COPY1-2.3. a. BA = 2.8 units c. MA = 1.95 units b. TA = 2.8 units d. NA = 4.02 units4. The product of BA and TA is equal to the product of MA and NA .5. 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. (Emphasize this idea.) Present to the students a situation that would capture their interest anddevelop their understanding of the lesson. Let them perform Activity 4. In thisactivity, the students will determine the mathematics concepts or principles tosolve the given problem. 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.

DEPED COPYActivity 4: Fly Me to Your World Answer Key 1. d = 27.67 km 2. External secant segment, tangent, Pythagorean theorem Ask the students to summarize the activities done before proceeding to the next activities. Provide them with an opportunity to relate or connect their responses in the activities given to their new lesson, Tangent and Secant Segments. Let the students read and understand some important notes on tangent and secant segments and study carefully the examples given. What to PROCESS Let the students use the different geometric relationships involving tangent and secant segments and the examples presented in the preceding section to answer the succeeding activities. In Activity 5, the students will name the external secant segments in the given figures. This activity would familiarize them with the geometric concept and facilitate problem solving. Activity 5: Am I away from you? Answer Key 1. IM and IL 2. TS and DS 3. OS 4. IR 5. LF and WE 6. IH , FG , IJ , EF , AK , DC 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.

Have the students apply the different theorems involving chords andtangent and secant segments to find the unknown lengths of segments on acircle and solve related problems. Ask the students to perform Activity 6 andActivity 7.Activity 6: Find My Length!Answer Key 6. x  10.5units 7. x  4.8 units 1. x  8 units 8. x  15units 2. x  8 units 9. x  2 10 6.32 units 3. x  9 units 10. x  4 units 4. x  5 units 5. x  6.64units DEPED COPY Questions: a. The theorems on two intersecting chords, secant segments, tangent segments, and external secant segments were applied. b. Evaluate students’ responses.Activity 7: Try to Fit! Answer Key 1. Possible answer: 2. a. VU = 4.57 units b. XU = 8 units 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.

What to REFLECT on and UNDERSTAND Test further students’ understanding of the different geometricrelationships involving tangent and secant segments including chords bydoing Activity 8 and Activity 9. Let the students prove the different theoremson intersecting chords, secant segments, tangent segments, and externalsecant segments and solve problems involving these concepts.Activity 8: Prove Me Right! Answer KeyDEPED COPY1. Given: AB and DE are chords of C intersecting at M.Prove: AM  BM  DM  EMTo prove: Draw AE and BD .Proof: Statement ReasonmBAE  1 mBE and The measure of an inscribed 2 angle is one-half the measure of its interceptedmBDE  1 mBE arc. 2 Inscribed angles intercepting the same arc are congruent.BAE  BDE AA Similarity Theorem Lengths of sides of similarAME ~ DMB triangles are proportional.EM  BM Multiplication PropertyAM DMAM  BM  DM  EM 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.

Answer Key2. Given: DP and DS are secant Prove: segments of T drawn from exterior point D. DP  DQ  DS  DR To prove: Draw PR and QS .Proof: DEPED COPY Statement Reason Inscribed anglesQPR  RSQ and PQS  SRP intercepting the same arc are congruent.DQS  DRP Supplements of congruent angles are congruentDQS ~ DRP AA Similarity TheoremDP DSDR  DQ Lengths of sides of similarDP  DQ  DS  DR triangles are proportional. Multiplication Property3. Given: KL and KM are tangent Prove: and secant segments, respectively of O drawn from exterior point K. KM intersects O at N. KM  KN  KL 2 To prove: Draw LM and LN . 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.

Answer KeyProof: Statement Reason The measure of an inscribedmNLK  1 mLN and angle is one-half the measure 2 of its intercepted arc.mLMN  1 mLN Transitive Property 2 Angles with equal measures are congruent.mNLK  mLMN The measure of the exterior angle of a triangle is equal toNLK  LMN the sum of the measures of its remote interior angles.mLNK  mNLM  mLMNDEPED COPYSubstitution Angle Addition PostulatemLNK  mNLM  mNLK Transitive PropertymKLM  mNLM  mNLK Angles with equal measuresmLNK  mKLM are congruent. AA Similarity TheoremLNK  KLM Lengths of sides of similar triangles are proportional.MKL ~ LNM Multiplication PropertyKM  KLKL KNKM  KN  KL 2Activity 9: Understand Me More … Answer Key 1. Janel. She used the theorem “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.” 2. Gate 1 is 91.65 m from the main road. 3. a. The point of tangency of the two light balls from the ceiling is about 44.72 cm. b. Anton needs about 1967.53 cm of string. 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.

DEPED COPYFind out how well the students understood the lesson by giving a shorttest (formative test) before proceeding to the next section. Ask them also towrite a journal about their understanding of tangent and secant segments.Refer to the Assessment Map.What to TRANSFER Give the students opportunities to demonstrate their understanding oftangent and secant segments including chords of a circle by doing a practicaltask. Let them perform Activity 10. You can ask the students to workindividually or in a group. In Activity 10, the students will make a design of an arch bridge thatwould connect two places which are separated by a river, 20 m wide. Tellthem to indicate on the design the different measurements of the parts of thebridge. The students are expected to formulate and solve problems involvingtangent and secant segments out of the design and the measurements of itsparts.Activity 10: My True World! Answer Key Evaluate students’ product. You may use the rubric provided.Summary/Synthesis/Generalization: This lesson was about the different geometric relationships involvingtangents, secants, and chords of a circle. The lesson provided the studentswith opportunities to derive geometric relationship involving intersectingchords, identify tangent and secant segments, and prove and apply differenttheorems on chords, tangent, and secant segments. These theorems wereused to solve various geometric problems. Understanding the ideaspresented in this lesson will facilitate their learning of the succeeding lessons. 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.

SUMMATIVE TESTPart IChoose the letter that you think best answers each of the following questions.1. In the figure on the right, which is an inscribed angle? A. RST B. PQR C. QVT D. QST2. In F below, AG is a diameter. What is mAD if mDFG  65? A. 65° B. 115° C. 130° D. 230°DEPED COPY3. Which of the following lines is tangent to F as shown in the figure below? A. DE B. AGC. BDD. AE 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.

4. Quadrilateral ABCD is inscribed in a circle. Which of the following is true about the angle measures of the quadrilateral?I. mA  mC  180II. mB  mD  180III. mA  mC  90A. I and II B. I and III C. II and III D. I, II, and III5. An arc of a circle measures 72°. If the radius of the circle is 6 cm, abouthow long is the arc?A. 1.884 cm B. 2.4 cm C. 3.768 cm D. 7.54 cmDEPED COPY6. What is the total measure of the central angles of a circle with no commoninterior points?A. 480 B. 360 C. 180 D. 1207. What kind of angle is the inscribed angle that intercepts a semicircle?A. straight B. obtuse C. right D. acute8. What is the length of AS in the figure on the right?A. 6.92 units C. 14.4 unitsB. 117 units D. 130 units 10 99. Line AB is tangent to C at D. If mDF = 166 and mDE = 78, what is mABF ? A. 44 B. 61 C. 88 D. 122 182 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.

10. How many line/s can be drawn through a given point on a circle that istangent to the circle?A. four B. three C. two D. one11. In U on the right, what is mPRE if mPUE  56 ? EP UA. 28 C. 56B. 34 D. 124DEPED COPY R12. In the figure below, TA and HA are secants. If TA = 18 cm, LA = 8 cm, and AE = 10 cm, L T A E Hwhat is the length of AH in the given figure?A. 18 cm C. 22.5 cmB. 20 cm D. 24.5 cm13. In O on the right, mHT = 45 and the length of theradius is 8 cm. What is the area of the shaded region Tin terms of  ? 45°A. 6  cm 2 C. 10  cm 2B. 8  cm 2 D. 12  cm 2 HO 8 cm 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.

14. In the circle on the right, what is the measure of SRT if AST is a semicircle and mSRA  74?A. 16 SB. 74C. 106D. 154 AR TDEPED COPY15. Quadrilateral LUCK is inscribed inS. If mLUC  96 and mUCK  77, find mULK . UC 96° 77° A. 77 B. 84 LS C. 96 D. 103 K16. In S on the right, what is RT if QS = 18 units and VW = 4 units? A. 4 2 units B. 8 2 units C. 14 units D. 16 2 units17. A circular garden has a radius of 2 m. Find the area of the smallersegment of the garden determined by a 90 arc.A.   2 m2 D. 4  2 m2 B. 2 m2 C.  m2 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.

18. Karen has a necklace with a circular pendant hanging 30° from a chain around her neck. The chain is tangent to the pendant. If the chain is extended as shown in the diagram on the right, it forms an angle of 30° below the pendant. What is the measure of the arc at the bottom of the pendant? A. 60° B. 75° C. 120° D. 150°DEPED COPY19. Mang Jose cut a circular board with a diameter 80 cm. Then, he dividedthe board into 20 congruent sectors. What is the area of each sector?A. 80 cm2 B. 320 cm2 C. 800 cm2 D. 1600 cm220. Mary designed a pendant. It is a regular octagon set in a circle. Supposethe opposite vertices are connected by line segments and meet at thecenter of the circle. What is the measure of each angle formed at thecenter? C. 67.5 D. 135A. 22.5 B. 45Part IISolve each of the following problems. Show your complete solutions.1. Mr. Jaena designed an arch for the top part of a subdivision’s main gate. The arch will be made out of bent iron. In the design, the 16 segments between the two concentric semicircles are each 0.7 meter long. Suppose the diameter of the outer semicircle is 8 meters. What is the length, in whole meters, of the shortest iron needed to make the arch?2. A rope fits tightly around two pulleys. What is the distance between the centers of the pulleys if the radii of the bigger and smaller pulleys are 10 cm and 6 cm, respectively, and the portion of the rope tangent to the two pulleys is 50 cm long? 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.

Rubric for Problem Solving 4 3 2 1Used an Used an Used an Attempted toappropriate appropriate appropriate solve the problemstrategy to come strategy to come strategy but but used anup with a correct up with a came up with an inappropriatesolution and solution, but a entirely wrong strategy that ledarrived at a part of the solution that led to a wrongcorrect answer solution led to an to an incorrect solution incorrect answer answerPart III A: GRASPS AssessmentDEPED COPYPerform the following.Goal: To prepare the different student formations to be done during a field demonstrationRole: Student assigned to prepare the different formations to be followed in the field demonstrationAudience: The school principal, your teacher, and your fellow studentsSituation: Your school has been selected by the municipal/city government to perform a field demonstration as part of a big local event where many visitors and spectators are expected to arrive and witness the said occasion. The principal of your school designated one of your teachers to organize and lead the group of students who will perform the field demonstration. Being one of the students selected to perform during the activity, your teacher asked you to plan the different student formations for the field demonstration. In particular, your teacher instructed you to include arrangements that show geometric figures such as circles, arcs, tangents, and secants. Your teacher also asked you to make a sketch of the various formations and include the order in which these will be performed by the group. 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.

Products: Sketches of the different formations to be followed in the field demonstrations including the order and manner on how each will be performedStandards: The sketches of the different formations must be accurate and presentable, and the sequencing must also be systematic.Rubric for Sketches of the Different Formations 4 3 2 1The sketches of The sketches of The sketches of The sketches ofthe different the different the different the differentformations are formations are formations are not formations areaccurately made, accurately made accurately made made but notpresentable, and and the but the accurate and thethe sequencing is sequencing is sequencing is sequencing is notsystematic. systematic but not systematic. systematic.DEPED COPY presentable.Part III BUse the prepared sketches of the different formations in Part III A informulating problems involving circles, then solve. 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.

Rubric on Problems Formulated and Solved Score Descriptors 6 Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in- 5 depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. 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- 4 depth comprehension of the pertinent concepts and/or processes. DEPED COPY Poses a complex problem and finishes most significant parts of 3 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 2 solution and communicates ideas unmistakably but shows gaps on theoretical comprehension. Poses a problem but demonstrates minor comprehension, not 1 being able to develop an approach.Source: D.O. #73 s. 2012Answer Key Part I 11. A Part II (Use the rubric to rate students’ works/outputs) 12. C 1. 35 m 1. B 2. 50.16 cm 2. B 13. B 3. D 14. C Part III A (Use the rubric to rate students’ works/outputs) 4. A 15. D Part III B (Use the rubric to rate students’ works/outputs) 5. D 16. D 6. B 17. A 7. C 18. D 8. D 19. A 9. A10. D 20. B 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.

GLOSSARY OF TERMSArc – a part of a circleArc Length – the length of an arc which can be determined by using the lproportion A = 2 r , where A is the degree measure of an arc, r is the 360radius 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 endpoints.Degree 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 angle 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.

DEPED COPYMajor Arc – an arc of a circle whose measure is greater than that of asemicircleMinor Arc – an arc of a circle whose measure is less than that of a semicirclePoint of Tangency – the point of intersection of the tangent line and thecircleSecant – a line that intersects a circle at exactly two points. A secant containsa chord of a circleSector of a Circle – the region bounded by an arc of the circle and the tworadii to the endpoints of the arcSegment of a Circle – the region bounded by an arc and a segment joiningits endpointsSemicircle – an arc measuring one-half the circumference of a circleTangent to a Circle – a line coplanar with the circle and intersects it at oneand only one point 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.

DEPED COPYList 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). 5. 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. 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.

DEPED COPY10. 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.17. 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. 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.

DEPED COPYDEPED INSTRUCTIONAL MATERIALS THAT CAN BE USED AS ADDITIONAL RESOURCES FOR THE LESSON ON CIRCLES: 1. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third Year Mathematics. Module 18: Circles and Their Properties. 2. Distance Learning Module (DLM) 3, Module 1 and 2: Circles. References And Website Links Used in This Module: References: Bass, L. E., Charles, R.I., Hall, B., Johnson, A., & Kennedy, D. (2008). Texas Geometry. Boston, Massachusetts: Pearson Prentice Hall. Bass, L. E., Hall B.R., Johnson A., & Wood, D.F. (1998). Prentice Hall Geometry Tools for a Changing World. NJ, USA: Prentice-Hall, Inc. Boyd, C., Malloy, C., & Flores. (2008). McGraw-Hill Geometry. USA: The McGraw-Hill Companies, Inc. Callanta, M. M. (2002). Infinity, Worktext in Mathematics III. Makati City: EUREKA Scholastic Publishing, Inc. Chapin, I., Landau, M. & McCracken. (1997). Prentice Hall Middle Grades Math, Tools for Success. Upper Saddle River, New Jersey: Prentice- Hall, Inc. Cifarelli, V. (2009) cK-12 Geometry, Flexbook Next Generation Textbooks. USA: Creative Commons Attribution-Share Alike. Clemens, S. R., O’Daffer, P. G., Cooney, T.J., & Dossey, J. A. (1990). Geometry. USA: Addison-Wesley Publishing Company, Inc. Clements, D. H., Jones, K.W., Moseley, L. G., & Schulman, L. (1999). Math in My World. Farmington, New York: McGraw-Hill Division. Department of Education. (2012) K to 12 Curriculum Guide Mathematics. Department of Education, Philippines. Gantert, A. X. (2008) AMSCO’s Geometry. NY, USA: AMSCO School Publications, Inc. Renfro, F. L. (1992) Addison-Wesley Geometry Teacher’s Edition. USA: Addison-Wesley Publishing Company, Inc. 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.

DEPED COPYRich, B. and Thomas, C. (2009). Schaum’s Outlines Geometry (4th ed.) USA: The McGraw-Hill Companies, Inc.Smith, S. A., Nelson, C.W., Koss, R. K., Keedy, M. L., & Bittinger, M. L. (1992) Addison-Wesley Informal Geometry. USA: Addison-Wesley Publishing Company, Inc.Wilson, P. S. (1993) Mathematics, Applications and Connections, Course I., Westerville, Ohio: Glencoe Division of Macmillan/McGraw-Hill Publishing Company.Website Links as References and Source of for Learning Activities:CK-12 Foundation. cK-12 Inscribed Angles. (2014). Retrieved fromhttp://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.7/CK-12 Foundation. cK-12 Secant Lines to Circles. (2014). Retrieved fromhttp://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.8/CK-12 Foundation. cK-12 Tangent Lines to Circles. (2014). Retrieved fromhttp://www.ck12.org/book/CK-12-Geometry-Honors-Concepts/ section/8.4/Houghton Mifflin Harcourt. Cliffs Notes. Arcs and Inscribed Angles. (2013).Retrieved from http://www.cliffsnotes.com/math/geometry/circles/arcs-and-inscribed-anglesHoughton Mifflin Harcourt. Cliffs Notes. Segments of Chords, Secants, andTangents. (2013). Retrieved fromhttp://www.cliffsnotes.com/math/geometry/circles/segments-of-chords-secants-tangentsMath Open Reference. Arc. (2009). Retrieved fromhttp://www.mathopenref.com/arc.htmlMath Open Reference. Arc Length. (2009). Retrieved fromhttp://www.mathopenref.com/arclength.htmlMath Open Reference. Central Angle. (2009). Retrieved fromhttp://www.mathopenref.com/circlecentral.htmlMath Open Reference. Central Angle Theorem. (2009). Retrieved fromhttp://www.mathopenref.com/arccentralangletheorem.html 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.

DEPED COPYMath Open Reference. Chord. (2009). Retrieved from http://www.mathopenref.com/chord.html Math Open Reference. Inscribed Angle. (2009). Retrieved from http://www.mathopenref.com/circleinscribed.html Math Open Reference. Intersecting Secants Theorem. (2009). Retrieved from http://www.mathopenref.com/secantsintersecting.html Math Open Reference. Sector. (2009). Retrieved from http://www.mathopenref.com/arcsector.html Math Open Reference. Segment. (2009). Retrieved from http://www.mathopenref.com/segment.html math-worksheet.org. Free Math Worksheets. Arc Length and Sector Area. (2014). Retrieved from http://www.math-worksheet.org/arc-length-and-sector- area math-worksheet.org. Free Math Worksheets. Inscribed Angles. (2014). Retrieved from http://www.math-worksheet.org/inscribed-angles math-worksheet.org. Free Math Worksheets. Secant-Tangent Angles. (2014). Retrieved from http://www.math-worksheet.org/secant-tangent-angles math-worksheet.org. Free Math Worksheets. Tangents. (2014). Retrieved from tangents OnlineMathLearning.com. Circle Theorems. (2013). Retrieved from http://www.onlinemathlearning.com/circle-theorems.html Roberts, 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 from http://www.regentsprep.org/Regents/math/geometry/ GP15/CircleAngles.htm 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.

DEPED COPYWebsite Links for Videos:Coach, Learn. NCEA Maths Level 1 Geometric reasoning: Angles WithinCircles. (2012). Retrieved from http://www.youtube.com/watch?v=jUAHw-JIobcKhan Academy. Equation for a circle using the Pythagorean Theorem.Retrieved from https://www.khanacademy.org/math/geometry/cc-geometry-circlesSchmidt, Larry. Angles and Arcs Formed by Tangents, Secants, and Chords.(2013). Retrieved from http://www.youtube.com/watch?v=I-RyXI7h1bMSophia.org. Geometry. Circles. (2014). Retrieved fromhttp://www.sophia.org/topics/circlesWebsite Links for Images:Cherry Valley Nursery and Landscape Supply. Seasonal Colors Flowers andPlants. (2014). Retrieved from http://www.cherryvalleynursery.com/eBay Inc. Commodore Holden CSA Mullins pursuit mag wheel 17 inchgenuine - 4blok #34. (2014). Retrieved fromhttp://www.ebay.com.au/itm/Commodore-Holden-CSA-Mullins-pursuit-mag-wheel-17-inch-genuine-4blok-34-/221275049465Fort Worth Weekly. Facebook Fact: Cowboys Are World’s Team. (2012) .Retrieved from http://www.fwweekly.com/2012/08/21/facebook-fact-cowboys-now-worlds-team/GlobalMotion Media Inc. Circular Quay, Sydney Harbour to Historic Hunter'sHill Photos. (2013). Retrieved from http://www.everytrail.com/ guide/circular-quay-sydney-harbour-to-historic-hunters-hill/photosHiSupplier.com Online Inc. Shandong Sun Paper Industry Joint Stock Co.,Ltd.Retrieved from http://pappapers.en.hisupplier.com/product-66751-Art-Boards.htmlKable. Slip-Sliding Away. (2014). Retrieved from http://www.offshore-technology.com/features/feature1674/feature1674-5.htmlMateria Geek. Nikon D500 presentada officialmente. (2009). Retrieved fromhttp://materiageek.com/2009/04/nikon-d5000-presentada-oficialmente/ 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 COPYPiatt, Andy. Dreamstime.com. Rainbow Stripe Hot Air Balloon. Retrieved from http://thumbs.dreamstime.com/z/rainbow-stripe-hot-air-balloon-788611.jpg Regents of the University of Colorado. Nautical Navigation. (2014). Retrieved from http://www.teachengineering.org/view_activity.php?url= collection/cub_/activities/cub_navigation/cub_navigation_lesson07_activity1.x ml Sambhav Transmission. Industrial Pulleys. Retrieved from http://www.indiamart.com/sambhav-transmission/industrial-pulleys.html shadefxcanopies.com. Flower Picture Gallery, Garden Pergola Canopies. Retrieved 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.html Tidwell, Jen. Home Sweet House. (2012). Retrieved from http://youveneverheardofjentidwell.com/2012/03/02/home-sweet-house/ Weston Digital Services. FWR Motorcycles LTD. CHAINS AND SPROCKETS. (2014). Retrieved from http://fwrm.co.uk/index.php?main_page=index&cPath=585&zenid=10omr4he hmnbkktbl94th0mlp6 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.

Module 5: Plane Coordinate GeometryA. Learning OutcomesContent Standard: The learner demonstrates understanding of key concepts ofcoordinate geometry.Performance Standard: The learner is able to formulate and solve problems involvinggeometric figures on the rectangular coordinate plane with perseverance andaccuracy.DEPED COPY Unpacking the Standards for UnderstandingSubject: Learning CompetenciesMathematics 10  Derive the distance formulaQuarter: Second  Apply the distance formula to prove some geometricQuarter propertiesTopic: Plane  Illustrate the center-radius form of the equation of aCoordinate Geometry circle  Determine the center and radius of a circle given itsLessons: equation and vice versa1. The Distance  Graph a circle and other geometric figures on the coordinate plane Formula2. The Equation of a  Solve problems involving geometric figures on theCircle coordinate planeWriter: Essential EssentialMelvin M. Callanta Understanding: Question: Students will understand that How do the key the concepts involving plane concepts of plane coordinate geometry are coordinate geometry useful tools in solving real-life facilitate finding problems like finding solutions to real-life locations, distances, problems involving mapping, etc. geometric figures? 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.

DEPED COPY Transfer Goal: Students will be able to apply with perseverance and accuracy the key concepts of plane coordinate geometry in formulating and solving problems involving geometric figures on the rectangular coordinate plane. B. Planning for Assessment Product/Performance The following are products and performances that students are expected to come up with in this module. 1. Ground Plan drawn on a grid with coordinates 2. Equations and problems involving mathematics concepts already learned such as coordinate plane, slope and equation of a line, parallel and perpendicular lines, polygons, distance, angles, etc 3. Finding the distance between a pair of points on the coordinate plane 4. Determining the missing coordinates of the endpoints of a segment 5. Finding the coordinates of the midpoint of the segment whose endpoints are given 6. Describing the figure formed by a set of points on a coordinate plane 7. Determining the missing coordinates corresponding to the vertices of some polygons 8. Solutions to problems involving the distance and the midpoint formulas 9. Coordinate Proofs of some geometric properties 10. Sketch of a municipal, city, or provincial map on a coordinate plane with the coordinates of some important landmarks 11. Formulating and solving real-life problems involving the distance and the midpoint formula 12. Finding the radius of a circle drawn on a coordinate plane 13. Determining the center and the radius of a circle given the equation 14. Graphing a circle given the equation 15. Writing the equation of a circle given the center and the radius 16. Writing the equation of a circle from standard form to general form and vice-versa 17. Determining the equation that describes a circle 18. Solutions to problems involving the equation of a circle 19. Formulating and solving real-life problems involving the equation of a circle 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.

Assessment Map TYPE KNOWLEDGE PROCESS/ UNDERSTANDING PERFORMANCE SKILLSPre- Pre-Test: Pre-Test:Assessment/ Part I Pre-Test: Part I and Part IIDiagnostic Part I Identifying the distance Determining the Solving problems formula distance involving the between a pair Distance Formula Illustrating the of points including the distance Midpoint Formula, between two Determining the and the Equation points on the coordinate of a of a Circle coordinate point given its plane distance from another pointDEPED COPY Illustrating the Determining the midpoint coordinates of formula the midpoint and the Illustrating the endpoints of a midpoint of a segment segment Describing the Defining figure formed by coordinate a set of points proof Determining the Identifying an coordinates of equation of a the vertex of a circle geometric figure Finding the length of the radius of a circle given the endpoints of a diameter Finding the center of a circle given the equation Finding the equation of a circle given the endpoints of a radius 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.

TYPE KNOWLEDGE PROCESS/ UNDERSTANDING PERFORMANCE SKILLSFormative Pre-Test: Pre-Test: Pre-Test: Pre-Test: Part III Part III Part III Part III Situational Situational Situational Analysis Analysis Analysis Situational Determining Illustrating the Analysis the locations of mathematics objects or Explaining how to Making a concepts or groups prepare the ground ground plan for principles plan for the Boy the Boy Scouts involved in a Writing the Scouts Jamboree Jamboree prepared equations that ground plan describe the Solving real-life Formulating situations or problems equations, problems inequalities, andDEPED COPY problems Quiz: Solving Quiz: Lesson 1 Lesson 1 equations Quiz: Lesson 1 Identifying the Finding the Explaining how to coordinates of distance find the distance points to be between each between two substituted in pair of points on points the distance the coordinate formula and in plane Explaining how to the midpoint find the midpoint of formula Finding the a segment coordinates of Identifying the the midpoint of Describing figures figures formed a segment given formed by some by some sets the endpoints sets of points of points Plotting some Explaining how to Identifying sets of points on find the missing parts of some the coordinate coordinates of geometric plane some geometric figures and figures their properties Naming the missing Solving real-life coordinates of problems involving the vertices of the distance some geometric formula and the figures midpoint formula Using coordinate proof to justify claims 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.

TYPE KNOWLEDGE PROCESS/ UNDERSTANDING PERFORMANCE SKILLSSummative Writing a coordinate proof to prove geometric properties Quiz: Quiz: Quiz: Lesson 2 Lesson 2 Lesson 2 Identifying the Determining the Explaining how to equations of center and the determine the circles in radius of a circle center of a circle center-radius form or standard form and in general form DEPED COPY Graphing a Explaining how to circle given the graph circles given equation written the equations in center-radius written in center- form. radius form and general form Writing the equation of a Explaining how to circle given the write the equation center and the of a circle given radius the center and the radius Writing the equation of a Explaining how to circle from write the equation standard form to of a circle from general form standard form to and vice-versa general form and vice-versa Solving problems involving the equation of a circle Post-Test: Post-Test: Post-Test: Post-Test: Part I Part I Part I and Part II Part III A and B Identifying the Determining the Solving problems Preparing distance distance involving the emergency formula between a pair Distance Formula, measures to be of points including the undertaken in Illustrating the Midpoint Formula, times of natural distance Determining the and the Equation calamities and between two coordinate of a of a Circle disasters points on the point given its particularly coordinate distance from typhoons and plane another point floods 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.

TYPE KNOWLEDGE PROCESS/ UNDERSTANDING PERFORMANCE SKILLS Illustrating the Determining the Preparing a grid midpoint coordinates of map of a formula the midpoint municipality and the Illustrating the endpoints of a Formulating and midpoint of a segment solving problems segment Describing the involving the key figure formed by concepts of Defining a set of points plane coordinate coordinate geometry proof Determining the Identifying an coordinates of equation of a the vertex of a circle geometric figureDEPED COPY Finding the length of the radius of a circle given the endpoints of a diameter Finding the center of a circle given the equation Finding the equation of a circle given the endpoints of a radiusSelf- Journal Writing:Assessment Expressing understanding of the distance formula, midpoint formula, coordinate proof, and the equation of a circle. 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.

Assessment Matrix (Summative Test)) Levels of DEPED COPY What will I assess? How will I How Will I Score? Assessment assess? Knowledge The learner Paper and 1 point for every demonstrates Pencil Test correct response 15% understanding of key Part I items 1, 3, 1 point for every concepts of plane 4, 7, 8, and 13 correct responseProcess/Skills coordinate geometry. Part I items 5, 6, 25%  Derive the distance 9, 10, 11, 12, 14, 1 point for every 16, 18, and 19 correct responseUnderstanding formula. Part I items 2, 30%  Apply the distance 15, 17, and 20 Rubric on Problem Solving (maximum of 4 Product/ formula to prove some Part II items 1 points for each Performance geometric properties. and 2 problem)  Illustrate the center- 30% radius form of the Part III A Rubric for the equation of a circle. Prepared Emergency  Determine the center Measures and radius of a circle Rubric for Grip Map of given its equation and the Municipality vice versa. (Total Score: maximum  Graph a circle and of 6 points ) other geometric figures on the coordinate plane.  Solve problems involving geometric figures on the coordinate plane. The learner is able to formulate and solve problems involving geometric figures on the rectangular coordinate plane with perseverance and accuracy. Part III B Rubric on Problems Formulated and Solved (Total Score: maximum of 6 points ) 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.

DEPED COPYC. Planning for Teaching-Learning This module covers key concepts of plane coordinate geometry. It is divided into two lessons, namely: The Distance Formula and the Equation of a Circle. In Lesson 1 of this module, the students will derive the distance formula and apply it in proving geometric relationships and in solving problems, particularly finding the distance between objects or points. They will also learn about the midpoint formula and its applications. Moreover, the students will graph and describe geometric figures on the coordinate plane. The second lesson is about the equation of a circle. In this lesson, the students will illustrate the center-radius form of the equation of a circle, determine the center and the radius given its equation and vice-versa, and show its graph on the coordinate plane (or by using the computer freeware, GeoGebra). More importantly, the students will solve problems involving the equation of a circle. In learning the equation of a circle, the students will use their prior knowledge and skills through the different activities provided. This is to connect and relate those mathematics concepts and skills that students previously studied to their new lesson. They will also perform varied learning tasks to process the knowledge and skills learned and to further deepen and transfer their understanding of the different lessons in real-life situations. Introduce the main lesson to the students by showing them the pictures below, then ask them the questions that follow: 205 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 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 its distance? If not, could you give the right information the next time somebody asks you the same question? Entice the students to find the answers to these questions and to determine the vast applications of plane coordinate geometry through this module. Objectives: After the learners have gone through the lessons contained in this module, they are expected to: 1. derive the distance formula; 2. find the distance between points; 3. determine the coordinates of the midpoint of a segment; 4. name the missing coordinates of the vertices of some geometric figures; 5. write a coordinate proof to prove some geometric relationships; 6. give/write the center-radius form of the equation of a circle; 7. determine the center and radius of a circle given its equation and vice versa; 8. write the equation of a circle from standard form to general form and vice versa; 9. graph a circle and other geometric figures on the coordinate plane; and 10. solve problems involving geometric figures on the coordinate plane. 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.

PRE-ASSESSMENT:Assess students’ prior knowledge, skills, and understanding of mathematicsconcepts related to the Distance Formula, the Midpoint Formula, theCoordinate Proof, and the Equation of a Circle. These will facilitate teachingand students’ understanding of the lessons in this module.Answer KeyPart I Part II (Use the rubric to rate students’ works/outputs) 1. 100 km 2. x  42  y  92  9 Part III (Use the rubric to rate students’ works/outputs)DEPED COPY1. C11. D2. C 12. A3. B 13. A4. B 14. B5. B 15. C 6. D 16. C 7. B 17. C 8. D 18. B 9. A 19. D10. C 20. BLEARNING GOALS AND TARGETS: Students are expected to demonstrate understanding of key concepts ofplane coordinate geometry, formulate real-life problems involving these concepts,and solve these with perseverance and accuracy.Lesson 1: The Distance Formula, the Midpoint Formula, and the Coordinate ProofWhat to KNOW Check students’ knowledge of the different mathematics conceptspreviously studied and their skills in performing mathematical operations. Thesewill facilitate teaching and students’ understanding of the distance formula andthe midpoint formula and in writing coordinate proofs. Tell them that as they gothrough this lesson, they have to think of this important question: How do thedistance formula, the midpoint formula, and the coordinate proof facilitate findingsolutions to real-life problems and making decisions? Let the students start the lesson by doing Activity 1. Ask them to use thegiven number line in determining the lengths of segments. Let them explain how 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.

they used the coordinates of points in finding each length. Emphasize in thisactivity the relationships among the segments based on their lengths, thedistance between the endpoints of segments whose coordinates on the numberline are known, and the significance of these to the lesson.Activity 1: How long is this part?Answer Key1. 4 units2. 4 units3. 6 unitsDEPED COPY4. 2 units5. 3 units6. 1 unita. Counting the number of units from one point to the other point usingthe number line or finding the absolute value of the difference of thecoordinates of the pointsb. Yes. By counting the number of units from one point to the otherpoint using the number line or finding the absolute value of thedifference of the coordinates of the pointsc. AB  BC , AC  CE , CD  DG , AB  EG . The two segments have the same lengths. d.2) AC + CE = AEd. d.1) AB + BC = AC;e. Yes. The absolute values of the difference of their coordinates areequal.AD = 10  4 = 14DA = 4  10 = 14BF =  6  9 = 15FB = 9   6 = 15 Students’ understanding of the relationships among the sides of a righttriangle is a prerequisite to the derivation of the Distance Formula. In Activity 2,provide the students opportunity to recall Pythagorean theorem by asking themto find the length of the unknown side of a right triangle. Tell them to explain howthey arrived at each length of a side.Activity 2: Why am I right? Answer Key 1. 5 units 2. 12 units 3. 12 units 4. 2 13 units  7.21 units 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.

5. 4 5 units  8.94 units 6. 2 63 units  15.87 units The length of the unknown side of each right triangle is obtained by applying the Pythagorean theorem. Let students relate their understanding of the Pythagorean theorem tofinding the distance between objects or points on the coordinate plane. Thiswould help them understand the derivation of the distance formula. Ask the students to perform Activity 3. In this activity, they will bepresented with a situation involving distances of objects or points on a coordinateplane. If possible, let the students find out how the coordinates of points can beused in finding distances between objects.DEPED COPYActivity 3: Let’s Exercise!Answer Key1. 10 km. By applying the Pythagorean theorem. That is, 62  82  c 2 ; c = 10 km.2. 3 km. distance from City Hall 0,4 to Plaza 3,4= 0  3 = 39 km. distance from City Hall 0,4 to Emilio’s house 9,4 = 0  9 = 93. 9 km. distance from Jose’s house 0,0 to Gasoline Station 9,0 = 0  9 = 94. 0,0 – Jose’s house 3,12 – Diego’s house 9,4 – Emilio’s house 3,4 – Plaza 9,0 – Gasoline Station5. 0,4 – City Hall6. By finding the absolute value of the difference of the coordinates of the points corresponding to Emilio’s house and the City Hall and Jose’s house and the Gasoline Station, respectivelyDistance from Emilio’s house 9,4 to City Hall 0,4 = 0  9Answer: 9 km =9Distance from Jose’s house 0,0 to Gasoline Station 9,0 = 9  0Answer: 9 km = 9 kmThe distances of the houses of Jose, Emilio, and Diego from each othercan be determined by applying the Pythagorean Theorem. Jose’s house 0,0 to Emilio’s house 9,4 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.