Mathematics Grade 10

6. Which of the following could be the graph of y  x4  5x2  4 ? yy x C. x A. y x y x DEPED COPY B. D.7. If you will draw the graph of y  x2(x  1) , how will the graph behave at the x-axis? A. The graph crosses both (0, 0) and (1, 0). B. The graph crosses (0, 0) and is tangent to the x-axis at (1, 0). C. The graph crosses (1, 0) and is tangent to the x-axis at (0, 0). D. The graph is tangent to the x-axis at both (0, 0) and (1, 0).8. You are asked to graph f (x)  x6  x5  5x4  x3  3x2  x using its properties. Which of these will be your graph? y y 2y y 11 2 1 -6 -5 -4 -3 -6 -2 -5 -1 -4 O -3 1 x -2 -1 O 1x 1-5 -4 -3 -2 -1 O 1x -1 -1-6 -5 -4 -3 -2 -1 O 1x A. -1 B. C. -2 D. -2 -19. Given that-2 f (x)  7x 3n  x2, what value -3 be -3 n to should assigned to make f a function of degree-2 7? -4 -4 A. 7 -3 B.  3 3 7 3 -4 -3 C. 7-5 D.-5 3 7 -4 -6 -6 -5 -5 110 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. If you were to choose from 2, 3, and 4, which pair of values for a and n would you consider so that y = axn could define the graph below? y A. a = 2 , n = 3 2 B. a = 3 , n = 2 C. a = 2 , n = 4 1 D. a = 3 , n = 3-8 -7 -6 -5 -4 -3 -2 -1 O x11. A car -1 determines that its profit, P, in thousands of manufacturer pesos, can be m-2 odeled by the function P(x) = 0.001 25x4 + x – 3,DEPED COPY where x represents the number of cars sold. What is the profit at x =150? -3 A. Php 75.28 C. Php 3,000,000.00 B. Php 632,9-54 9.50 D. Php 10,125,297.0012. Your friend Aaro-5n Marielle asks your help in drawing a rough sketch of the graph of y  (x2  1)(2x 4  3) by means of the Leading Coefficient Test.-6 How will you explain the behavior of the graph? A. The graph is falling to the left and rising to the right. B. The graph is rising to both left and right. C. The graph is rising to the left and falling to the right. D. The graph is falling to both left and right.13. Lein Andrei is tasked to choose from the numbers –2, –1, 3, and 6 to form a polynomial function in the form y = axn. What values should he assign to a and n so that the function could define the graph below? y x A. a = 3 , n = -2 B. a = 3 , n = 6 C. a = 6 , n = 3 D. a = -1 , n = 614. Consider this Revenue-Advertising Expense situation. A drugstore that sells a certain brand of vitamin capsule estimates that the profit P (in pesos) is given by P  50x3  2400x2  2000 , 0  x  32 111 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.

where x is the amount spent on advertising (in thousands of pesos). An advertising agency provides four (4) different advertising packages with costs listed below. Which of these packages will yield the highest revenue for the company? A. Package A: Php 8,000.00 B. Package B: Php 16,000.00 C. Package C: Php 32,000.00 D. Package D: Php 48,000.00Part 2Read and analyze the situation below. Then, answer the questions orperform the required task. An open box with dimensions 2 inches by 3 inches by 4 inchesneeds to be increased in size to hold five times as much material as thecurrent box. (Assume each dimension is increased by the sameamount.)Task: (a) Write a function that represents the volume V of the new box. (b) Find the dimensions of the new box. (c) Using hard paperboard, make the two boxes - one with the original dimensions and another with the new dimensions. (d) On one face of the bigger box, write your mathematical solution in getting the new dimensions.Additional guidelines: 1. The boxes should look presentable and are durable enough to hold any dry material such as sand, rice grains, etc. 2. Consider the rubric below.DEPED COPYRubric for Rating the Output:Point Descriptor 3 Polynomial function is correctly presented as model, 2 appropriate mathematical concepts are used in the solution, 1 and the correct final answer is obtained. Polynomial function is correctly presented as model, appropriate mathematical concepts are partially used in the solution, and the correct final answer is obtained. Polynomial function is not correctly presented as model, other alternative mathematical concepts are used in the solution, and the final answer is incorrect. 112 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.

Criteria for Rating the Output (Box):  Each box has the needed dimensions.  The boxes are durable and presentable.Point/s to be Given: 3 points if the boxes have met the two criteria 2 points if the boxes have met only one criterion 1 point if the boxes have not met any of the criteriaAnswer Key for Summative TestDEPED COPYPart I: Part II. 1. B (Use the rubric to rate students’ work/output) 2. D 3. B Solution for finding the dimensions of the desired box: 4. C Let x be the number to be added to each of length, width 5. B and height to increase the size of the box. Then the 6. A dimensions of the new box are x+2 by x+3 by x+4. 7. C 8. C Since the volume of the original box is (2 inches) 9. A (3 inches) (4 inches) = 24 cubic inches, then the volume of the new box is 120 cubic inches.10. B11. B Writing these in an equation, we have12. D (x  2)(x  3)(x  4)  V(x)13. D x3  9x2  26x  24  12014. C x3  9x2  26x  96  0 , (x  2)(x2 11x  48)  0 Therefore, from the last equation, the only real solution is x = 2. Thus, the dimensions of the new box are 4 inches by 5 inches by 6 inches. Note to the Teacher: To validate that the volume of the bigger box is five times the volume of the other box, guide the students to compare the content of both boxes using sand, rice grains, or mongo seeds. 113 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 COPYGlossary of TermsConstant Function – a polynomial function whose degree is 0Cubic Function – a polynomial function whose degree is 3Evaluating a Polynomial – the process of finding the value of thepolynomial at a given value in its domainIntercepts of a Graph – the points on the graph that have zero aseither the x-coordinate or the y-coordinateIrreducible Factor - a factor that can no longer be factored usingcoefficients that are real numbersLeading Coefficient Test - a test that uses the leading term of thepolynomial function to determine the right-hand and the left-handbehaviors of the graphLinear Function - a polynomial function whose degree is 1Multiplicity of a Root - tells how many times a particular number is aroot for a given polynomialNonnegative Integer - zero or any positive integerPolynomial Function - a function denoted byP(x)  an xn  an1xn1  an2xn2  ...  a1x  a0 , where n is a nonnegativeinteger, a0, a1, ..., an are real numbers called coefficients, but an  0,,an xn is the leading term, an is the leading coefficient, and a0 is theconstant termPolynomial in Standard Form - any polynomial whose terms arearranged in decreasing powers of xQuadratic Function - a polynomial function whose degree is 2Quartic Function - a polynomial function whose degree is 4Quintic Function - a polynomial function whose degree is 5Turning Point - point where the function changes from decreasing toincreasing or from increasing to decreasing values 114 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 References Alferez, M. S., Duro, MC.A., & Tupaz, KK. L. (2008). MSA Advanced Algebra. Quezon City, Philippines: MSA Publishing House Berry, J., Graham, T., Sharp, J., & Berry, E. (2003). Schaum’s A-Z Mathematics. London, United Kingdom: Hodder &Stoughton Educational. Cabral, E. A., De Lara-Tuprio, E. P., De Las Penas, ML. N., Francisco, F. F., Garces, IJ. L., Marcelo, R. M., & Sarmiento, J. F. (2010). Precalculus. Quezon City, Philippines: Ateneo de Manila University Press Jose-Dilao, S., Orines, F. B., & Bernabe, J. G. (2003). Advanced Algebra, Trigonometry and Statistics. Quezon City, Philippines: JTW Corporation Lamayo, F. C., & Deauna, M. C. (1990). Fourth Year Integrated Mathematics. Quezon City, Philippines: Phoenix Publishing House, Inc. Larson, R., & Hostetler, R. P. (2012). Algebra and Trigonometry. Pasig City, Philippines: Cengage Learning Asia Pte Ltd Marasigan, J. A., Coronel, A. C., & Coronel, I. C. (2004). Advanced Algebra with Trigonometry and Statistics. Makati City, Philippines: The Bookmark, Inc. Quimpo, N. F. (2005). A Course in Freshman Algebra. Quezon City, Philippines Uy, F. B., & Ocampo, J. L. (2000). Board Primer in Mathematics. Mandaluyong City, Philippines: Capitol Publishing House. Villaluna, T. T., & Van Zandt, GE. L. (2009). Hands-on, Minds-on Activities in Mathematics IV. Quezon City, Philippines: St. Jude Thaddeus Publications. 115 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 4: CirclesA. Learning Outcomes Content Standard: The learner demonstrates understanding of key concepts of circles.Performance Standard: The learner is able to formulate and find solutions to challengingsituations involving circles and other related terms in different disciplinesthrough appropriate and accurate representations. DEPED COPY Unpacking the Standards for UnderstandingSubject: Mathematics 10 Learning CompetenciesQuarter: Second Quarter 1. Derive inductively the relations among chords, arcs, central angles, and inscribedTOPIC: Circles anglesLESSONS: 2. Illustrate segments and sectors of circles1. A. Chords, Arcs, and 3. Prove theorems related to chords, arcs, Central Angles central angles, and inscribed angles B. Arcs and Inscribed 4. Solve problems involving chords, arcs, Angles central angles, and inscribed angles of2. A. Tangents and circles 5. Illustrate tangents and secants of circles Secants 6. Prove theorems on tangents and secants of a Circle 7. Solve problems involving tangents and B. Tangent and Secant secants of circles Segments Essential EssentialWriter: Understanding: Question:Concepcion S. Ternida Students will How do geometric understand that the relationships concept of circles has involving circles wide applications in real help solve real-life life and is a useful tool problems that are in problem-solving and circular in nature? in decision making. 116 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.

Transfer Goal: Students will be able to apply the key concepts of circles in finding solutions and in making decisions for certain real-life problems.B. Planning for AssessmentProduct/Performance The following are products and performances that students are expected to come up with in this module.1. Objects or situations in real life where chords, arcs, and central angles of circles are illustrated2. A circle graph applying the knowledge of central angles, arcs, and sectors of a circle3. Sketch plans or designs of a stage with circular objects that illustrate the use of inscribed angles and arcs of a circle4. Sketch plans or designs of an arch bridge that illustrate the applications of secants and tangents5. Deriving geometric relationships involving circles6. Proof of theorems and other geometric relationships involving circles7. Formulated and solved real-life problemsDEPED COPYAssessment Map TYPE KNOWLEDGE PROCESS/ UNDERSTANDING PERFORMANCE SKILLSPre- Pre-Test: Pre-Test:Assessment/ Part I Pre-Test: Part IDiagnostic Part I Part II Identifying Finding the Solving problems inscribed angle length of an arc involving the key of a circle given concepts of Identifying the its radius circles external secant segment Finding the measure of a Describing the central angle opposite angles given its of a quadrilateral intercepted arc inscribed in a circle Finding the lengths of Identifying the segments sum of the formed by measures of the intersecting central angles of chords a circle 117 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 Describing the Finding the inscribed angle measure of the intercepting a angle formed by semicircle two secants Determining the Finding the number of line length of a chord that can be that is drawn tangent to perpendicular to the circle a radiusDEPED COPY Finding the length of a secant segment Finding the area of a sector of a circle Finding the measure of a central angle given its supplement Finding the measure of an angle of a quadrilateral inscribed in a circle Finding the measure of an inscribed angle given the measure of a central angle intercepting the same arc Pre-Test: Pre-Test: Pre-Test: Pre-Test: Part III Part III Part III Part III Situational Situational Situational Situational Analysis Analysis Analysis Analysis Planning the Illustrating every Explaining how to Making designs design of a part or portion of prepare the of gardens garden the garden designs of the including their garden measurements and accessories 118 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 PERFORMANCEFormative SKILLS Determining the Quiz: mathematics Formulating Lesson 1A concepts or problems that principles describe the involved in the situations design of the garden Solving the problems Quiz: formulated Lesson 1A Quiz: Lesson 1ADEPED COPY Identifying and Solving the Justifying why describing terms degree measure angles or arcs are related to circles of the central congruent angles and arcs Explaining why Finding the an arc is a length of the semicircle unknown segments in a Explaining how to circle find the degree measure of an Determining the arc reasons to support the Explaining how to given find the center of statements in a a circular garden two-column proof of a Solving real-life theorem problems involving the Solving the chords, arcs, and length of an arc central angles of of a circle given circles its degree measure Finding the area of the shaded region of circles Quiz: Quiz: Quiz: Lesson 1B Lesson 1B Lesson 1B Identifying the Finding the Explaining why inscribed angles measure of an the inscribed and their inscribed angle angles are intercepted arcs and its congruent intercepted arc Proving theorems on inscribed 119 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 angles and Determining the intercepted arcs measure of an using two-column inscribed angle proofs that intercepts a semicircle Proving congruence of Determining the triangles using reasons to the theorems on support the inscribed angles given statements in a Solving real-life two-column problems proof of a involving arcs and theorem inscribed anglesDEPED COPY Quiz: Quiz: Explaining the Lesson 2A Lesson 2A kind of parallelogram that can be inscribed in a circle Quiz: Lesson 2A Identifying Determining the Proving theorems tangents and measures of the on tangents and secants different angles, secants using including the arcs, and two-column angles they form segments proofs Explaining how to find the measure of an angle given a circle with tangents Quiz: Quiz: Solving real-life Lesson 2B Lesson 2B problems involving tangents and secants of a circle Quiz: Lesson 2B Identifying the Finding the Proving theorems external secant length of the on intersecting segment in a unknown chords, secant circle segment in a segments, and circle tangent segments Explaining why the solution for finding the length 120 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 PERFORMANCESummative SKILLS of a segment is correct or Drawing a circle incorrect with appropriate labels and Solving real-life description problems involving tangent and secant segments Pre-Test: Pre-Test: Pre-Test: Post-Test: Part I Part I Part I Part III A and B Part II Preparing Solving problems sketches of the involving the key different concepts of formations to be circles followed in theDEPED COPY Identifying an Finding the field inscribed angle measure of an demonstrations arc intercepted including their Identifying a by a central sequencing and tangent angle presentation on how each will be Describing the Finding the performed angles of a length of an arc quadrilateral Formulating and inscribed in a Finding the solving problems circle lengths of involving the key segments concepts of Identifying the formed by circles sum of the intersecting measures of the chords central angles of a circle Finding the measure of the Describing the angle formed by inscribed angle a tangent and a intercepting a secant semicircle Finding the Determining the measure of an number of lines inscribed angle that can be given the drawn tangent to measure of a the circle central angle intercepting the same arc Finding the length of a secant segment Finding the area of a sector of a circle 121 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 PERFORMANCESelf- SKILLSAssessment Finding the measure of a central angle given its supplement Finding the measure of an angle of a quadrilateral inscribed in a circle DEPED COPY Finding the length of a chord that is perpendicular to a radius Journal Writing: Expressing understanding of the key concepts of circles Expressing understanding of the different geometric relationships involving circlesAssessment Matrix (Summative Test) Levels of What will I assess? How will I How Will IAssessment assess? Score? The learner demonstrates Paper and Pencil Knowledge understanding of key Test 1 point for 15% concepts of circles. every correct Part I items 1, 3, 4, responseProcess/Skills 1. Derive inductively the 6, 7, and 10 25% relations among 1 point for chords, arcs, central Part I items 2, 5, 8, every correctUnderstanding angles, and inscribed 9, 11, 12, 13, 14, response 30% angles. 15, and 16 1 point for 2. Illustrate segments Part I items 17, 18, every correct and sectors of circles. 19, and 20 response 3. Prove theorems related to chords, arcs, central angles and inscribed angles 4. Solve problems involving chords, arcs, central angles, and inscribed angles of circles 122 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. Illustrate tangents and Part II items 1 and Rubric on secants of circles 2 Problem Solving 6. Prove theorems on (maximum of tangents and secants 4 points for each 7. Solve problems problem) involving tangents and secants of circles The learner is able to Part III A Rubric for formulate and find Part III B Sketches of solutions to challenging the Different situations involving Formations circles and other related terms in different (Total Score: disciplines through maximum of appropriate and accurate 6 points ) representations.DEPED COPYProduct/ Rubric onPerformance Problems Formulated 30% and Solved (Total Score: maximum of 6 points )C. Planning for Teaching-Learning This module covers key concepts of circles. It is divided into fourlessons namely: Chords, Arcs, and Central Angles, Arcs and InscribedAngles, Tangents and Secants of a Circle, and Tangent and SecantSegments. Lesson 1A is about the relations among chords, arcs and centralangles of a circle, area of a segment and a sector, and arc length of acircle. In this lesson, the students will determine the relationship betweenthe measures of the central angle and its intercepted arc, apply thedifferent geometric relationships among chords, arcs, and central anglesin solving problems, complete the proof of a theorem related to theseconcepts, find the area of a segment and the sector of a circle, anddetermine the length of an arc. (Note that all measures of angles and arcsare in degrees.) Moreover, the students will be given the opportunity to demonstratetheir understanding of the lesson by naming objects and citing real-lifesituations where chords, arcs, and central angles of a circle are illustratedand applied. 123 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 The concepts about arcs and inscribed angles of a circle are contained in Lesson 1B. In this lesson, the students will determine the geometric relationships that exist among arcs and inscribed angles of a circle, apply these in solving problems, and prove related theorems. Moreover, they will formulate and solve real-life problems involving these geometric concepts. The geometric relationships involving tangents and secants and their applications in real life will be taken up in Lesson 2A. In this lesson, the students will find the measures of angles formed by secants and tangents and the arcs that these angles intercept. They will apply the relationships involving tangents and secants in finding the lengths of segments of some geometric figures. Moreover, the students will be given opportunities to formulate and solve real-life problems involving tangents and secants of a circle. Lesson 2B of this module is about the different geometric relationships involving tangent and secant segments. The students will apply these geometric relationships in finding the lengths of segments formed by tangents and secants. To demonstrate their understanding of the lesson, the students will make a design of a real-life object where tangent and secant segments are illustrated or applied, then formulate and solve problems out of this design. In all the lessons, the students are given the opportunity to use their prior knowledge and skills in learning circles. They are also given varied activities to process the knowledge and skills learned and further deepen and transfer their understanding of the different lessons. As an introduction to the main lesson, show the students the pictures below, then ask them the questions that follow: 124 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 Have you imagined yourself pushing a cart or riding a bus having wheels that are not round? Do you think you can move heavy objects from one place to another easily or travel distant places as fast as you can? What difficulty do you think would you experience without circles? Have you ever thought of the importance of circles in the field of transportation, industries, sports, navigation, carpentry, and in your daily life? Entice the students to find out the answers to these questions and to determine the vast applications of circles through this module. Objectives: After the learners have gone through the lessons contained in this module, they are expected to: 1. identify and describe terms related to circles; 2. use the relationship among chords, arcs, central angles, and inscribed angles of circles; 3. find the area of segments and sectors of circles; 4. find the lengths of arcs of circles; 5. use two-column proofs in proving theorems related to chords, arcs, central angles, and inscribed angles of circles; 6. identify the tangents and secants of circles; 7. formulate and solve problems involving chords, arcs, central angles, and inscribed angles of circles; 8. use two-column proofs in proving theorems related to tangents and secants of circles; and 9. formulate and solve problems involving tangents and secants of circles. PRE-ASSESSMENT: Check students’ prior knowledge, skills, and understanding of mathematics concepts related to circles. Assessing these will facilitate teaching and students’ understanding of the lessons in this module. 125 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 Key Part I 11. A Part II (Use the rubric to rate students’ 1. B 12. A works/outputs) 2. A 13. B 3. D 14. A 1. 24.67 m 4. D 15. A 2. 27.38 km 5. C 16. A 6. C 17. A Part III (Use the rubric to rate students’ 7. C 18. C works/outputs) 8. B 19. B 9. A 20. C10. D DEPED COPYLEARNING GOALS AND TARGETS: Students are expected to demonstrate understanding of key conceptsof circles, formulate real-life problems involving these concepts, and solvethese using a variety of strategies. They are also expected to investigatemathematical relationships in various situations involving circles.Lesson 1A: Chords, arcs, and Central anglesWhat to Know Assess students’ knowledge of the different mathematics conceptspreviously studied and their skills in performing mathematical operations.Assessing these will facilitate teaching and students’ understanding of chords,arcs, and central angles. Tell them that as they go through this lesson, theyhave to think of this important question: “How do the relationships amongchords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions?”Ask the students to identify, name, and describe the terms related to circlesby doing Activity 1. Let them explain how they arrived at their answers. Also,ask them to describe and differentiate these terms. 126 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 1: Know My Terms and Conditions…Answer Key1. AN , AJ , AE 5. JL , JN , EN , EL2. EJ 6. LEN , LJE , ENL , JLN , LNJ3. EL , EJ4. JNE , JLE 7. JAN , NAE 8. LEJ , JENQuestions: a. Recall the definition of the terms related to circles.DEPED COPYTerms related to circle Description1. radius It is a segment drawn from the center of2. diameter the circle to any point on the circle.3. chord It is a segment whose endpoints are on the4. semicircle circle and it passes through the center of5. minor arc the circle. It is the longest chord.6. major arc7. central angle It is a segment joining any two points on8. inscribed angle the circle. It is an arc measuring one-half of the circumference of a circle. It is an arc of a circle that measures less than a semicircle. It is an arc of a circle that measures greater than a semicircle. It is an angle whose vertex is at the center of the circle and with two radii as its sides. It is an angle whose vertex is on a circle and whose sides contain chords of the circle. 127 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 COPYAnswer Key b. 1. A radius is half the measure of the diameter. 2. A diameter is twice the measure of the radius and it is the longest chord. 3. A chord is a segment joining any two points on the circle. 4. A semicircle is an arc measuring one-half the circumference of a circle. 5. A minor arc is an arc of a circle that measures less than the semicircle. 6. A major arc is an arc of a circle that measures greater than the semicircle. 7. A central angle is an angle whose vertex is the center of the circle and with two radii as its sides. 8. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. Show the students the right triangles with different measures of sidesand let them find the missing side. Give focus on the mathematics conceptsor principles applied to find the unknown side particularly the Pythagoreantheorem.Activity 2: What is my missing side? Answer Key 1. c  10 units 2. c  17.49 units 3. c  12.73 units 4. a  12units 5. b  4 units 6. b  12.12units Questions: a. Using the equation a2  b2  c2 . b. Pythagorean theorem 128 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.

Provide the students with an opportunity to derive the relationshipbetween the measures of the central angle and the measure of its interceptedarc. Ask them to perform Activity 3. In this activity, students will measure theangles of the given figures using a protractor. Ask them to get the sum of theangles in the first figure as well as the sum of the central angles in the secondfigure. Ask them also to identify the intercepted arc of each central angle.Emphasize that the sum of the angles formed by the coplanar rays withcommon vertex but with no common interior points is equal to the sum of thecentral angles formed by the radii of a circle with no common interior points.Activity 3: Measure Me and You Will See…DEPED COPYAnswer Key d. 90 e. 30 1. a. 105 b. 75 c. 602. a. 105 d. 90 b. 75 e. 30 c. 603. In each figure, the angles have a common vertex.4. 360 ; 3605. 3606. 3607.Central Angle Measure Intercepted Arc1. FAB 105 FB2. BAC 75 BC3. CAD 60 CD4. EAD 90 ED5. EAF 30 EF8. 360 because the measure of the central angle is equal to the measure of its intercepted arc.9. Equal 129 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 COPYPresent a real-life situation to the students to develop theirunderstanding of arcs and central angles of circles. In this activity, ask themto find the degree measure of each arc of the wheel and also the angleformed at the hub. Ask them further the importance of the spokes of thewheel.Activity 4: Travel Safely Answer Key a. 60 ; 60 b. Evaluate students’ responses Before proceeding to the next activities, let the students give a briefsummary of what they have learned so far. Provide them with an opportunityto relate or connect their responses in the activities given to this lesson. Letthe students read and understand some important notes on chords, arcs, andcentral angles. Tell them to study carefully the examples given.What to PROCESS In this section, let the students apply the key concepts of chords, arcs,and central angles. Tell them to use the mathematical ideas and theexamples presented in the preceding section to answer the activitiesprovided. Ask the students to perform Activity 5. In this activity, the students willidentify and name arcs and central angles in the given circle and explain howthey identified them. 130 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 5: Identify and Name MeAnswer Key1. LMH (or LGH ) and LKH (or LJH ); JKM (or JLM ) and JGM (or JHM )2.DEPED COPY Minor Arcs Major Arcs JK KMJ KL KGL LM LJM MG MKG HG HKG JH JMHNote: There are many ways of naming the major arcs. The given answers are just some of those ways.3. Some Possible Answers: LAM ; MAG ; GAH ; JAH ; JAK ; LAKQuestions: a. A semicircle is an arc with measure equal to one-half of the circumference of a circle and is named by using the two endpoints and another point on the arc. A minor arc is an arc of a circle that measures less than the semicircle. It is named by using the two endpoints on the circle. A major arc is an arc of a circle that measures greater than the semicircle. It is named by using the two endpoints and another point on the arc. A central angle is an angle whose vertex is the center of the circle and with two radii as its sides. b. Yes. A circle has an infinite set of points. Therefore, a circle has many semicircles, arcs, and central angles. In activities 6, 7, and 8, ask the students to apply the differentgeometric relationships in finding the degree measure of the central angles,the arcs that the angles intercept, and the lengths of chords. Then, let themexplain how they arrived at their answers. 131 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 My Degree MeasureAnswer Key 6. 90 7. 48 1. 90 8. 150 2. 48 9. 42 3. 138 10. 132 4. 42 5. 132Activity 7: Find Me!DEPED COPYAnswer Key1. JSO and NSI ; JSN and OSI . They are vertical angles.2. a. 113b. 67c. 673. Yes. Yes. Opposite sides of rectangles are congruent.4. JO and NI ; JN and OI . The central angles that intercept the arcsare congruent.5. a. 67 d. 113b. 113 e. 180c. 67 f. 1806. NJO ; NIO; JOI ; JNI . The arcs measure 180°. Each arc or semicircle contains the endpoints of the diameter.Activity 8: Get My LengthAnswer Key1. 8 units 5. 39  6.24 units2. 2 units 6. 8 units3. 5 units 7. 2 7  5.29 units4. 39  6.24 units 8. 4 7  10.58 unitsNote: Evaluate students’ explanations. Provide the students opportunity to develop their skills in writing proofs.Ask them to complete the proof of a theorem involving the diameter, chord,and arc of a circle by doing Activity 9. If needed, guide the students as theycomplete the proof of the theorem. 132 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 9: Make Me Complete!Problem: To prove that 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. The proof has two parts. EGiven: ES is a diameter of U and N G perpendicular to chord GN at I. IProve: 1. NI  GI U2. EN  EGDEPED COPY3. NS  GS SAnswer KeyProof of Part 1: We will show that ES bisects GN and the minor arc GN. Statements Reasons1. U with diameter ES and chord Given GN ; ES  GN Definition of perpendicular lines2. GIU and NIU are right angles. Right angles are congruent.3. GIU  NIU4. UG  UN Radii of the same circle are congruent. 133 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: Reasons Statements Reflexive/Identity Property 5. UI  UI HyL Theorem 6. GIU  NIU Corresponding parts of congruent 7. GI  NI triangles are congruent (CPCTC). 8. ES bisects GN . Definition of segment bisector 9. GUI  NUI From 6, CPCTC E, I, U are collinear.10. GUI and GUE are the same angles. From 9, 10, definition of congruent NUI and NUE are the same angles angles.11. mGUE  mNUEDEPED COPY12. mEG  mGUE Degree measure of an arc mEN  mNUE From 11, 12, substitution13. mEN  mEG14. mGUS  mNUS From 11, definition of supplementary angles, angles that are supplementary to congruent angles are congruent.15. mGS  mGUS Degree measure of an arc mNS  mNUS From 14, 15, substitution16. mNS  mGS Definition of arc bisector17. ES bisects GN . 134 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Given: ES is a diameter of U; ES bisects GN N Eat I and the minor arc GN. I G U SDEPED COPYAnswer KeyProof of Part 2: We will show that ES  GN . Statements Reasons1. U with diameter ES , ES Given bisects GN at I and the minor arc GN. Definition of bisector2. GI  NI GE  NE Reflexive/Identity Property3. UI  UI4. UG  UN Radii of the same circle are congruent.5. GIU  NIU SSS Postulate6. UIG  UIN CPCTC7. UIG and UIN are right Angles which form a linear pair and are congruent are right angles. angles. Definition of perpendicular lines8. IU  GN9. ES  GN IU is on ESCombining Parts 1 and 2, the theorem is proven. Have the students apply the knowledge and skills they have learnedabout arc length, segment, and sector of a circle. Ask the students to performActivity 10 and Activity 11. 135 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 10: Find My Arc LengthAnswer Key1. 3.925 units2. 5.23 units or 5.23 units3. 7.85 units4. 10.46 units or 10.47 units5. 8.29 unitsQuestions:a. The area of each shaded region was determined by using theDEPED COPY A l proportion 360  2r where A = degree measure of the arc, l = length of the arc, r = radius of the circle. Use the formula for finding the area of a segment and the area of a triangle. A lb. The proportion 360  2r , area of a segment and the area of a triangle were used and so with substitution and the division property.Activity 11: Find This Part!Answer Key 1. 9  cm2 or 28.26 cm2 2. 18  cm2 or 56.52 cm2 3. 52.77 cm2 4. 9.31 cm2 5. 59.04 cm2 6. 40 cm2Questions:a. The area of the sector is equal to the product of the ratio measureof the arc and the area of the circle. 360 Subtract the area of the triangle from the area of the sector.b. Area of a circle, area of a triangle, ratio, equilateral triangle, and regular pentagon 136 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 Ask the students to take a closer look at some aspects of thegeometric concepts contained in this lesson. Provide them opportunities tothink deeply and test further their understanding of the lesson by doingActivity 12. In this activity, the students will solve problems involving chords,arcs, central angles, area of a segment and a sector, and arc length of acircle.Activity 12: More Circles Please …DEPED COPYAnswer Key1. a. 72b. 3.768 cmc. regular pentagon2. Yes. There are two pairs of congruent central angles/vertical anglesformed and they intercept congruent arcs.3. a. Yes. because the arcs are intercepted by the same central angle.b. No. Even if the two circles have the same central angles, thelengths of their intercepted arcs are not equal because the 2circles have different radii.4. 60. (Evaluate students’ explanations. They are expected to use the A lproportion 360  2r to support their explanations.)5. Draw two chords on the garden and a perpendicular bisector toeach of the chords. The intersection of the perpendicular bisectorsto the chord is the center of the circular garden.6. a. Education, because it has the highest budget which isPhp12,000.00Savings & Utilities, because they have the lowest budget whichis Php4,500.00b. Education. It should be given the greater allocation because it isa very good investment.c. Education – 120Food – 90Utilities – 45Savings – 45Other expenses – 60d. Get the percentage for each item by dividing the allotted budgetby the monthly income, then multiply it by 360. 137 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.

e. Sector Arc Length Item 654.16 cm2 52.3 cm 490.625 cm2 Education 245.3125 cm2 39.25 cm 245.3125 cm2 19.625 cm Food 19.625 cm Utilities 327.083 cm2 Savings 26.16 cm Other expenses 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 chords, arcs, andcentral angles. Refer to the Assessment Map.What to TRANSFER Give the students opportunities to demonstrate their understanding ofcircles by doing a practical task. Let them perform Activity 13. You can askthe students to work individually or in group. In this activity, the students willname 5 objects or cite 5 situations in real life where chords, arcs, and centralangles of a circle are illustrated. Then, instruct them to formulate and solveproblems out of these objects or situations. Also, ask them to make a circlegraph showing the different school fees that students like them have to payvoluntarily like Parents-Teachers Association fee, miscellaneous fee, schoolpaper fee, Supreme Student Government fee, and other fees. Ask them toexplain how they applied their knowledge of central angles and arcs of circlein preparing the graph. Then, using the circle graph that they made, ask themto formulate and solve at least two problems involving arcs, central angles,and sectors of a circle.DEPED COPYActivity 13: My Real World Answer Key Evaluate students’ product. You may use the rubric provided. 138 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 COPYSummary/Synthesis/Generalization: This lesson was about chords, arcs and central angles of a circle, area of a segment and a sector, and arc length of a circle. In this lesson, the students determined the relationship between the measures of the central angle and its intercepted arc. They were also given the opportunity to apply the different geometric relationships among chords, arcs, and central angles in solving problems, complete the proof of a theorem related to these concepts, find the area of a segment and the sector of a circle, and determine the length of an arc. Moreover, the students were asked to name objects and cite real-life situations where chords, arcs, and central angles of a circle are illustrated and the relationships among these concepts are applied. Lesson 1B: Arcs and Inscribed Angles What to KNOW Let the students relate and connect previously learned mathematics concepts to the new lesson, arcs and inscribed angles. As they go through this lesson, tell them to think of this important question: “How do geometric relationships involving arcs and inscribed angles facilitate solving real-life problems and making decisions?” Start the lesson by asking the students to perform Activity 1. In this activity, the students will identify in a given figure the angles and their intercepted arcs. The students should be able to explain how they identified and named these angles and intercepted arcs. 139 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 1: My Angles and Intercepted ArcsAnswer Key Angles Arc That the Angle Intercepts MSC MC CSD CD MSD MD MGC MC DGC CD MGD MDDEPED COPY1. Determine the chords having a common endpoint on the circle. The chords are the sides of the angle and the common endpoint on the circle is the vertex.Determine two radii of the circle. The two radii are the sides of the angle and the center of the circle is the vertex.Determine the arc that lies in the interior of the angle with endpoints on the same angle.2. There are 6 angles and there are also 6 arcs that these angles intercept.3. An angle intercepts an arc if a point on one side of the angle is an endpoint of the arc. Give the students opportunity to determine the relationship betweenthe measure of an inscribed angle and the measure of its intercepted arc byperforming Activity 2. The students should be able to realize in this activitythat the measure of an angle inscribed in a circle is one-half the measure ofits intercepted arc (or the measure of the intercepted arc is twice the measureof the inscribed angle). 140 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 2: Inscribe Me! 2. Answer Key Possible Responses 1.DEPED COPY3. mWEL  60 ;4. mLDW  30 mLW  60 The measure of the central angle is equal to the measure of its intercepted arc.5. An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle.6. The measure of LDW is one-half the measure of LW . 141 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Answer Key 7. Draw other inscribed angles of the circle. Determine the measures of these angles and the degree measures of their respective intercepted arcs. (Check students’ drawings.) The measure of an inscribed angle is one-half the degree measure of its intercepted arc. 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). Activity 3 is related to Activity 2. In this activity, the students willdetermine the relationship that exists when an inscribed angle intercepts asemicircle. They should be able to find out that the measure of an inscribedangle that intercepts a semicircle is 90°.Activity 3: Intercept Me so I Won’t Fall! Answer Key 1. 2. 3. 4.DEPED COPY5. a. mMOT  90 b. mMUT  90 c. mMNT  90The measures of the three angles are equal. Each angle measures 90°.The measure of an inscribed angle intercepting a semicircle is 90°.The measures of inscribed angles intercepting the same arc are equal. 142 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Develop students’ understanding of the lesson by relating it to a real- life situation. Ask them to determine the mathematics concepts that they can apply to solve the problem presented in Activity 4. Activity 4: One, Two,…, Say Cheese! Answer Key 1. 80° 40° New location where Janel could photograph the entire house with the telephoto lens 2. Relationship between the central angle or inscribed angle and the arc that the angle intercepts. 3. Go farther from the house until the entire house is seen on the eye piece or on the LCD screen viewer of the camera. 143 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Before proceeding to the next section of this lesson, let the studentsgive a brief summary of the activities done. Provide them with an opportunityto relate or connect their responses in the activities given to their new lesson,Arcs and Inscribed Angles. Let the students read and understand someimportant notes on the different geometric relationships involving arcs andinscribed angles and let them study carefully the examples given.What to PROCESS Give the students opportunities to use the different geometricrelationships involving arcs and inscribed angles, and the examplespresented in the preceding section to perform the succeeding activities. Ask the students to perform Activities 5, 6, and 7. In these activities,they will identify the inscribed angles and their intercepted arcs, and apply thetheorems pertaining to these geometric concepts and other mathematicsconcepts in finding their degree measures. Provide the students opportunitiesto explain their answers. DEPED COPYActivity 5: Inscribe, Intercept, then MeasureAnswer Key1. LCA, LCE , ACE , ALC , CAE , CAL , LAE , and AEC2. a. CAL b. ACE c. LCE and LAE d. ALC and AEC3. a. m1  28 d. m4  56 g. m7  28 b. m2  62 e. m5  124 h. m8  62 c. m3  62 f. m6  56 i. m9  624. a. mCL  52 c. mAE  52 b. mAC  128 d. mLE  128 144 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 6: Half, Equal or Twice As?Answer Key1. BAC  BDC and ACD  ABD . If inscribed angles intercept the same arc, then the angles are congruent.2. mCD  1083. mACB  484. c. mDCA  38 a. x  7 d. mAD  76 b. mABD  38DEPED COPY5. c. mBC  52 a. x  5 d. mBAC  26 b. mBDC  26Activity 7: Encircle Me! 4. a. mTIA  105 Answer Key b. mFAI  82 1. 5. a. mOA  150 a. mTM  116 b. mOG  50 b. mMA  64 c. mGOA  80 c. mAE  116 d. mGAO  25 d. mMEA  32 e. mTAM  58 2. a. mCAR  65 b. mACR  57.5 c. mARC  57.5 d. mAC  115 e. mAR  115 3. a. mRDM  35 b. mDRM  55 c. mDMR  90 d. mDM  110 e. mRD  180 145 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

In Activity 8, ask the students to complete the proof of the theorem oninscribed angle and its intercepted arc. This activity would further developtheir skills in writing proofs which they need in proving other geometricrelationships.Activity 8: Complete to Prove!Problem: To prove that 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).DEPED COPYCase 1: Q xGiven: PQR inscribed in S and S PQ is a diameter. PRProve: mPQR  1 mPR 2Draw RS and let mPQR  x . 146 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Answer KeyDEPED COPY Reasons Statements Given 1. PQR inscribed in S and PQ is a diameter. Radii of a circle are congruent. 2. QS  RS Definition of isosceles triangle 3. QRS is an isosceles  . The base angles of an isosceles 4. PQR  QRS triangle are congruent. 5. mPQR  mQRS The measures of congruent angles 6. mQRS  x are equal. Transitive Property 7. mPSR  2x The measure of an exterior angle of a 8. mPSR  mPR triangle is equal to the sum of the measures of its remote interior 9. mPR  2x angles. The measure of a central angle is equal to the measure of its intercepted arc. Transitive Property10. mPR  2mPQR Substitution Multiplication Property of Equality11. mQRS  1 mPR 2What to REFLECT on and UNDERSTAND Provide the students with opportunities to think deeply and test furthertheir understanding of the lesson. Let them prove the different theorems onarcs and inscribed angles of a circle and other geometric relationships byperforming Activity 9 and Activity 10. Moreover, ask the students to solve theproblems in Activity 11 for them to realize the wide applications of the lessonin real life. 147 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 9: Prove It or Else …!Answer Key1. Case 2 Given: KLM inscribed in O.Prove: mKLM  1 mKM 2To prove: Draw diameter LN.DEPED COPYProof: Statements Reasons The measure of anmKLN  1 mKN and mMLN  1 mMN inscribed angle is one-half 2 2 the measure of its intercepted arc (Case 1).mKLN  mMLN  1 mKN  1 mMN or Addition Property 2 2 Angle Addition PostulatemKLN  mMLN  1 mKN  mMN  Arc Addition Postulate 2 SubstitutionmKLN  mMLN  mKLMmKN  mMN  mKMmKLM  1 mKM 2Answer Key1. Case 3 Given: SMC inscribed in A.Prove: mSMC  1 mSC 2To prove: Draw diameter MP. 148 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Proof: Statements Reasons The measure of anmPMS  1 mPS and mPMC  1 mPC inscribed angle is one-half 2 2 the measure of its intercepted arc (Case 1).mPMS  mSMC  mPMC ormSMC  mPMC  mPMS Angle Addition PostulatemPS  mSC  mPC or Arc Addition Postulate By SubtractionmSC  mPC  mPS SubstitutionDEPED COPYmPMC  mPMS  1 mPC  1 mPS or 2 2mPMC  mPMS  1 mPC  mPS 2mSMC  1 mSC 22. Given: In T, PR and AC are the Prove: intercepted arcs of PQR and ABC , respectively. PR  AC PQR  ABCProof: Statements Reasons GivenPR  AC Congruent arcs have equal measures.mPR  mAC The measure of an inscribedmPQR  1 mPR and angle is one-half the measure 2 of its intercepted arc.mABC  1 mAC Substitution 2 Transitive PropertymPQR  1 mAC Angles with equal measures 2 are congruent.mPQR  mABCPQR  ABC 149 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

3. Given: In C, GML intercepts Prove: semicircle GEL. GML is a right angle.Proof: Reasons Statements Given The degree measure of a GML intercepts semicircle GEL. semicircle is 180. The measure of an inscribed angle is one-half the measure of its intercepted arc. Substitution Definition of right angleDEPED COPYmGEL  180mGML  1 mGEL 2mGML  1 180 or mGML  90 2GML is a right angle.4. Given: Quadrilateral WIND is inscribed in Y .Prove: 1. W and N are supplementary. 2. I and D are supplementary.To prove: Draw WY , IY , NY , and DY .Proof: Statements Reasons The sum of the measuresmWYI  mIYN  mNYD  mDYW  360 of the central angles of a circle is 360.mWYI  mWI , mIYN  mIN ,mNYD  mND , and mDYW  mDW The measure of a central angle is equal to the measure of its intercepted arc.mWI  mIN  mND  mDW  360 SubstitutionmDNI  mDWI  360 Arc Addition Postulate 150 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Answer Key 1 Statements 1 Reasons 2 2 The measure of an inscribedmDWI  mDNI and mDNI  mDWI angle is one-half the measure of its interceptedmDWI  mDNI  1 mDNI  1 mDWI or arc. 2 2 By AdditionmDWI  mDNI  1 mDNI  mDWI DEPED COPY 2 SubstitutionmDWI  mDNI  1 360 or Definition of supplementary 2 angles The sum of the measures ofmDWI  mDNI  180 the angles of a quadrilateral is 360.W and N are supplementary. Substitution Addition PropertymW  mI  mN  mD  360 Definition of supplementary anglesmI  mD 180  360mI  mD  180I and D are supplementary.Activity 10: Prove to Me if You Can!Answer Key C1. Given: MT and AC are chords of D. Prove: and MC  AT , M D T CHM  THA . HProof A Statements Reasons 1. MT and AC are chords of Given D and MC  AT. Definition of inscribed angle 2. MCA , ATM , CMT , and CAT are inscribed angles. Inscribed angles intercepting the same arc are congruent. 3. MCA  ATM and ASA Congruence Postulate CMT  CAT 4. CHM  THA 151 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Answer Key2. Given: Quadrilateral DRIV is inscribed in E. R RV is a diagonal that passes through the center of the circle. E I DV  IV Prove: RVD  RVI D V ReasonsProof: Statements 1. RV is a diagonal that passes through the center of the circle 2. RV  RV 3. DRV  VRIDEPED COPY Given Reflexive Property Inscribed angles intercepting the same arc are congruent.4. RIV and RDV are Definition of semicircle semicircles. Inscribed angle intercepting a5. RDV and RIV are right angles. semicircle measures 90°6. RVD and RVI are right Definition of right triangle triangles.7. RVD  RVI Hypotenuse-Angle Congruence Theorem3. Given: In A, SE  NE and SC  NT. CProve: CSE  TNE S T AProof: E Statements N 1. SE  NE and SC  NT Reasons 2. SE  NE and SC  NT Given 3. mSE  mNE and If two arcs are congruent, then mSC  mNT the chords joined by their respective endpoints are also 4. mSE  mSC  mEC and congruent. mEN  mNT  mET Congruent arcs have equal measures. Arc Addition Postulate 152 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Answer Key Statements Reasons Substitution1. mEC  mET Definition of Congruence2. EC  ET3. Draw chord CT . Definition of chord of a circle4. ECT  ETC Inscribed angles intercepting congruent arcs are congruent.5. CET is an isosceles triangle. Definition of isosceles triangle.6. CE  TE The legs of an isosceles triangle are7. CSE  TNE congruent. SSS Congruence PostulateDEPED COPY Activity 11: Take Me to Your Real-World! Answer Key 1. a. 72° b. 36°. The measure of an inscribed angle is one-half the measure of its intercepted arc. 2. Rectangle. In a circle, there is only one chord that can be drawn parallel and congruent to another chord in the same circle. Moreover, the diagonals of the parallelogram are also the diameters of the circle. Hence, each inscribed angle formed by the adjacent sides of the parallelogram intercepts a semicircle and measures 90°. 3. 38°. If EG is drawn, the viewing angles of Joanna, Clarissa, and Juliana intercept the same arc. Hence, the viewing angles of Joanna and Juliana measure the same as the viewing angle of Clarissa. 4. Mang Ador has to draw two inscribed angles on the circle such that each measures 90°. Then, connect the other endpoints of the sides of each angle to form the diameter. The point of intersection of the two diameters is the center of the circle. 5. a. PQR is a right triangle. b. The length of RS is the geometric mean of the lengths of PS and QS . c. PS = 6 in.; QS = 2 in.; RS = 2 3 in. d. RT  4 3 in. and MN  4 3 in. 153 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYBefore 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 arcs and inscribedangles. Refer to the Assessment Map.What to TRANSFER Give the students opportunities to demonstrate their understanding ofthe geometric relationships involving arcs and inscribed angles. In Activity 12,ask the students to make a design of a stage where a special event will beheld. Tell them to include in the design some circular objects that illustrate theuse of inscribed angles and arcs of a circle, and explain how they appliedthese concepts in preparing the design. Then, ask them to formulate andsolve problems out of the design they made. You can ask the students towork individually or in groups.Activity 12: How special is the event? Answer Key Evaluate students’ product. You may use the given rubric.Summary/Synthesis/Generalization: This lesson was about arcs and inscribed angles of a circle. In thislesson, the students were given the opportunity to determine the geometricrelationships that exist among arcs and inscribed angles of a circle, applythese in solving problems, and prove related theorems. Moreover, they weregiven the chance to formulate and solve real-life problems involving thesegeometric concepts out of the product they were asked to come up with as ademonstration of their understanding of the lesson. 154 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYLesson 2A: Tangents and Secants of a Circle What to KNOW Assess students’ prior mathematical knowledge and skills that are related to tangents and secants of a circle. This would facilitate teaching and guide the students in understanding the different geometric relationships involving tangents and secants of a circle. Start the lesson by asking the students to perform Activity 1. This activity would lead them to some geometric relationships involving tangents and segments drawn from the center of the circle to the point of tangency. That is, the radius of a circle that is drawn to the point of tangency is perpendicular to the tangent line and is also the shortest segment. Activity 1: Measure then Compare! Answer Key 1. Use a compass to draw S. 2. Draw line m such that it intersects S at exactly one point. Label the point of intersection as T. 3. Connect S and T by a line segment. What is TS in the figure drawn? TS is a radius of S. 155 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY4. Mark four other points on line m such that two of these points are on the left side of T and the other two points are on the right side. Label these points as M, N, P, and Q, respectively. 5. Using a protractor, find the measures of MTS , NTS , PTS, and QTS . How do the measures of the four angles compare? The four angles have equal measures. Each angle measures 90°. 6. Repeat step 2 to 5. This time, draw line n such that it intersects the circle at another point. Name this point V. The four angles, AVS , BVS , DVS , and EVS have equal measures. Each angle measures 90°. 156 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY 7. Draw MS , NS , PS , and QS . 8. Using a ruler, find the lengths of TS , MS , NS , PS , andQS . How do the lengths of the five segments compare? The lengths of the five segments, TS , MS , NS , PS , and QS are not equal. What do you think is the shortest segment from the center of a circle to the line that intersects it at exactly one point? Explain your answer. The shortest segment from the center of a circle to the line that intersects the circle at exactly one point is the segment perpendicular to the line. Whereas, the other segments become the hypotenuses of the right triangles formed. Recall that the hypotenuse is the longest side of a right triangle. 157 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYProvide the students with opportunities to investigate relationshipsamong arcs and angles formed by secants and tangents. Ask them to performActivity 2 and Activity 3. Let the students realize the following geometricrelationships:1. If two secants intersect on a circle, then the measure of the angle formed is one-half the measure of the intercepted arc. (Note: Relate this to the relationship between the measure of the inscribed angle and the measure of its intercepted arc.)2. 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.3. 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.4. 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.5. 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.6. 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. 158 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Activity 2: Investigate Me! Answer Key1. Which lines intersect circle C at two points? AD, AE, DG, How about the lines that intersect the circle at exactly one point? BG2. What are the angles having A as the vertex? DAE, EAG, DAB, DAG. There are still other angles with A as the vertex, but for the purpose of our new lesson, we consider these angles. C as the vertex? ACD, ACG, ECF, DCE D as the vertex? ADG. There are still other angles with D as the vertex but for the meantime, we only consider this. G as the vertex? AGD. There are still other angles with G as the vertex but for the meantime, we only consider this.DEPED COPY3. DABAD DCE DE DAE DE ACD AD DAG DEA ACF AF EAG EFA ECF EF ADF AF AGD AF and AD4. DAE and DCE DE DAB, DCA , and AGD AD ACF , ADF , and AGD AF5. mDAE  34.43 mACG  68.87 mEAG  90 mECF  111.14 mDAB  55.57 mDCE  68.87 mDAG  124.43 mADG  34.43 mACD  111.14 mAGD  21.13 159 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.