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Math 10 Unit 2 LM

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Math 10 Unit 2 LM

  1. 1. D EPED C O PY 10 Mathematics Department of Education Republic of the Philippines This book was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. We encourage teachers and other education stakeholders to email their feedback, comments, and recommendations to the Department of Education at action@deped.gov.ph. We value your feedback and recommendations. Learner’s Module Unit 2 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. VISIT DEPED TAMBAYAN http://richardrrr.blogspot.com/ 1. Center of top breaking headlines and current events related to Department of Education. 2. Offers free K-12 Materials you can use and share
  2. 2. D EPED C O PY Mathematics – Grade 10 Learner’s Module First Edition 2015 Republic Act 8293, section 176 states that: No copyright shall subsist in any work of the Government of the Philippines. However, prior approval of the government agency or office wherein the work is created shall be necessary for exploitation of such work for profit. Such agency or office may, among other things, impose as a condition the payment of royalties. Borrowed materials (i.e., songs, stories, poems, pictures, photos, brand names, trademarks, etc.) included in this book are owned by their respective copyright holders. DepEd is represented by the Filipinas Copyright Licensing Society (FILCOLS), Inc. in seeking permission to use these materials from their respective copyright owners. All means have been exhausted in seeking permission to use these materials. The publisher and authors do not represent nor claim ownership over them. Only institution and companies which have entered an agreement with FILCOLS and only within the agreed framework may copy this Learner’s Module. Those who have not entered in an agreement with FILCOLS must, if they wish to copy, contact the publisher and authors directly. Authors and publishers may email or contact FILCOLS at filcols@gmail.com or (02) 439-2204, respectively. Published by the Department of Education Secretary: Br. Armin A. Luistro FSC Undersecretary: Dina S. Ocampo, PhD Printed in the Philippines by REX Book Store Department of Education-Instructional Materials Council Secretariat (DepEd-IMCS) Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600 Telefax: (02) 634-1054, 634-1072 E-mail Address: imcsetd@yahoo.com Development Team of the Learner’s Module Consultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, and Rosemarievic Villena-Diaz, PhD Authors: Melvin M. Callanta, Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, Mirla S. Esparrago, Elino S. Garcia, Aries N. Magnaye, Fernando B. Orines, Rowena S. Perez, and Concepcion S. Ternida Editor: Maxima J. Acelajado, PhD Reviewers: Maria Alva Q. Aberin, PhD, Maxima J. Acelajado, PhD, Carlene P. Arceo, PhD, Rene R. Belecina, PhD, Dolores P. Borja, Agnes D. Garciano, Phd, Ma. Corazon P. Loja, Roger T. Nocom, Rowena S. Requidan, and Jones A. Tudlong, PhD Illustrator: Cyrell T. Navarro Layout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. Pagulayan Management and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr., Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr. 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. 3. D EPED C O PY Introduction This material is written in support of the K to 12 Basic Education Program to ensure attainment of standards expected of students. In the design of this Grade 10 materials, it underwent different processes - development by writers composed of classroom teachers, school heads, supervisors, specialists from the Department and other institutions; validation by experts, academicians, and practitioners; revision; content review and language editing by members of Quality Circle Reviewers; and finalization with the guidance of the consultants. There are eight (8) modules in this material. Module 1 – Sequences Module 2 – Polynomials and Polynomial Equations Module 3 – Polynomial Functions Module 4 – Circles Module 5 – Plane Coordinate Geometry Module 6 – Permutations and Combinations Module 7 – Probability of Compound Events Module 8 – Measures of Position With the different activities provided in every module, may you find this material engaging and challenging as it develops your critical-thinking and problem-solving skills. 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. 4. D EPED C O PY Unit 2 Module 3: Polynomial Functions............................................................ 99 Lessons and Coverage........................................................................ 100 Module Map......................................................................................... 100 Pre-Assessment .................................................................................. 101 Learning Goals and Targets ................................................................ 105 Activity 1.................................................................................... 106 Activity 2.................................................................................... 107 Activity 3.................................................................................... 108 Activity 4.................................................................................... 108 Activity 5.................................................................................... 110 Activity 6.................................................................................... 111 Activity 7.................................................................................... 112 Activity 8.................................................................................... 115 Activity 9.................................................................................... 115 Activity 10.................................................................................. 118 Activity 11.................................................................................. 119 Activity 12.................................................................................. 121 Activity 13.................................................................................. 122 Activity 14.................................................................................. 123 Summary/Synthesis/Generalization........................................................... 125 Glossary of Terms ...................................................................................... 125 References Used in this Module................................................................. 126 Module 4: Circles .................................................................................... 127 Lessons and Coverage........................................................................ 127 Module Map......................................................................................... 128 Pre-Assessment .................................................................................. 129 Learning Goals and Targets ................................................................ 134 Lesson 1A: Chords, Arcs, and Central Angles.......................................... 135 Activity 1.................................................................................... 135 Activity 2.................................................................................... 137 Activity 3.................................................................................... 138 Activity 4.................................................................................... 139 Activity 5.................................................................................... 150 Activity 6.................................................................................... 151 Activity 7.................................................................................... 151 Activity 8.................................................................................... 152 Activity 9.................................................................................... 152 Activity 10.................................................................................. 155 Activity 11.................................................................................. 155 Activity 12.................................................................................. 157 Activity 13.................................................................................. 159 Table of Contents 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. 5. D EPED C O PY Summary/Synthesis/Generalization ...........................................................160 Lesson 1B: Arcs and Inscribed Angles.......................................................161 Activity 1 ....................................................................................161 Activity 2 ....................................................................................162 Activity 3 ....................................................................................163 Activity 4 ....................................................................................164 Activity 5 ....................................................................................167 Activity 6 ....................................................................................168 Activity 7 ....................................................................................169 Activity 8 ....................................................................................170 Activity 9 ....................................................................................172 Activity 10 ..................................................................................174 Activity 11 ..................................................................................175 Activity 12 ..................................................................................176 Summary/Synthesis/Generalization ...........................................................177 Lesson 2A: Tangents and Secants of a Circle ............................................178 Activity 1 ....................................................................................178 Activity 2 ....................................................................................179 Activity 3 ....................................................................................180 Activity 4 ....................................................................................188 Activity 5 ....................................................................................189 Activity 6 ....................................................................................192 Activity 7 ....................................................................................194 Activity 8 ....................................................................................197 Summary/Synthesis/Generalization ...........................................................198 Lesson 2B: Tangent and Secant Segments.................................................199 Activity 1 ....................................................................................199 Activity 2 ....................................................................................200 Activity 3 ....................................................................................200 Activity 4 ....................................................................................201 Activity 5 ....................................................................................204 Activity 6 ....................................................................................205 Activity 7 ....................................................................................206 Activity 8 ....................................................................................207 Activity 9 ....................................................................................208 Activity 10 ..................................................................................210 Summary/Synthesis/Generalization ...........................................................211 Glossary of Terms.......................................................................................212 List of Theorems and Postulates on Circles...............................................213 References and Website Links Used in this Module..................................215 Module 5: Plane Coordinate Geometry ...............................................221 Lessons and Coverage ........................................................................222 Module Map .........................................................................................222 Pre-Assessment...................................................................................223 Learning Goals and Targets.................................................................228 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  6. 6. D EPED C O PY Lesson 1: The Distance Formula, the Midpoint Formula, and the Coordinate Proof .......................................................... 229 Activity 1.................................................................................... 229 Activity 2.................................................................................... 230 Activity 3.................................................................................... 231 Activity 4.................................................................................... 232 Activity 5.................................................................................... 241 Activity 6.................................................................................... 242 Activity 7.................................................................................... 242 Activity 8.................................................................................... 243 Activity 9.................................................................................... 245 Activity 10.................................................................................. 248 Activity 11.................................................................................. 250 Summary/Synthesis/Generalization........................................................... 251 Lesson 2: The Equation of a Circle............................................................ 252 Activity 1.................................................................................... 252 Activity 2.................................................................................... 253 Activity 3.................................................................................... 254 Activity 4.................................................................................... 263 Activity 5.................................................................................... 265 Activity 6.................................................................................... 265 Activity 7.................................................................................... 266 Activity 8.................................................................................... 267 Activity 9.................................................................................... 267 Activity 10.................................................................................. 269 Summary/Synthesis/Generalization........................................................... 270 Glossary of Terms ...................................................................................... 270 References and Website Links Used in this Module ................................. 271 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  7. 7. D EPED C O PY 99 I. INTRODUCTION You are now in Grade 10, your last year in junior high school. In this level and in the higher levels of your education, you might ask the question: What are math problems and solutions for? An incoming college student may ask, “How can designers and manufacturers make boxes having the largest volume with the least cost?” And anybody may ask: In what other fields are the mathematical concepts like functions used? How are these concepts applied? Look at the mosaic picture below. Can you see some mathematical representations here? Give some. As you go through this module, you are expected to define and illustrate polynomial functions, draw the graphs of polynomial functions and solve problems involving polynomial functions. The ultimate goal of this module is for you to answer these questions: How are polynomial functions related to other fields of study? How are these used in solving real-life problems and in decision making? All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  8. 8. D EPED C O PY 100 II. LESSON AND COVERAGE This is a one-lesson module. In this module, you will learn to:  illustrate polynomial functions  graph polynomial functions  solve problems involving polynomial functions Solutions of Problems Involving Polynomial Functions Graphs of Polynomial Functions Illustrations of Polynomial Functions The Polynomial Functions 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.
  9. 9. D EPED C O PY 101 III. PRE-ASSESSMENT Part 1 Let us find out first what you already know related to the content of this module. Answer all items. Choose the letter that best answers each question. Please take note of the items/questions that you will not be able to answer correctly and revisit them as you go through this module for self-assessment. 1. What should n be if f(x) = xn defines a polynomial function? A. an integer C. any number B. a nonnegative integer D. any number except 0 2. Which of the following is an example of a polynomial function? A. 13 4 )( 3  x x xf C. 6 27)( xxxf  B. 22 3 2 3 2)( xxxf  D. 53)( 3  xxxf 3. What is the leading coefficient of the polynomial function 42)( 3  xxxf ? A. 1 C. 3 B. 2 D. 4 4. How should the polynomial function 432)( 53  xxxxf be written in standard form? A. 432)( 53  xxxxf C. 53 324)( xxxxf  B. 35 234)( xxxxf  D. 423)( 35  xxxxf 5. Which of the following could be the graph of the polynomial function 1234 23  xxxy ? A. B. C. D. x x x yy y y 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. 10. D EPED C O PY 102 6. From the choices, which polynomial function in factored form represents the given graph? 7. If you will draw the graph of 2 )2(  xxy , how will you sketch it with respect to the x-axis? A. Sketch it crossing both (-2,0) and (0,0). B. Sketch it crossing (-2,0) and tangent at (0,0). C. Sketch it tangent at (-2,0) and crossing (0,0). D. Sketch it tangent at both (-2,0) and (0,0). 8. What are the end behaviors of the graph of 432)( 53  xxxxf ? A. rises to the left and falls to the right B. falls to the left and rises to the right C. rises to both directions D. falls to both directions 9. You are asked to illustrate the sketch of 43)( 53  xxxf using its properties. Which will be your sketch? A. B. C. D. A. )1)(1)(2(  xxxy B. )2)(1)(1(  xxxy C. )1)(1)(2(  xxxxy D. )2)(1)(1(  xxxxy y x y x y x y x 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.
  11. 11. D EPED C O PY 103 10. Your classmate Linus encounters difficulties in showing a sketch of the graph of 3 2 2 3 4y x x x    6. You know that the quickest technique is the Leading Coefficient Test. You want to help Linus in his problem. What hint/clue should you give? A. The graph falls to the left and rises to the right. B. The graph rises to both left and right. C. The graph rises to the left and falls to the right. D. The graph falls to both left and right. 11. If you will be asked to choose from -2, 2, 3, and 4, what values for a and n will you consider so that y = axn could define the graph below? 12. A car manufacturer determines that its profit, P, in thousands of pesos, can be modeled by the function P(x) = 0.00125x4 + x – 3, where x represents the number of cars sold. What is the profit when x = 300? A. Php 101.25 C. Php 3,000,000.00 B. Php 1,039,500.00 D. Php 10,125,297.00 13. A demographer predicts that the population, P, of a town t years from now can be modeled by the function P(t) = 6t4 – 5t3 + 200t + 12 000. What will the population of the town be two (2) years from now? A. 12 456 C. 1 245 600 B. 124 560 D. 12 456 000 14. Consider this Revenue-Advertising Expense situation: The total revenue R (in millions of pesos) for a company is related to its advertising expense by the function  3 21 R x 600x , 0 x 400 100 000      where x is the amount spent on advertising (in ten thousands of pesos). A. a = 2 , n = 3 B. a = 3 , n = 2 C. a = - 2 , n = 4 D. a = - 2 , n = 3 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.
  12. 12. D EPED C O PY 104 Currently, the company spends Php 2,000,000.00 for advertisement. If you are the company manager, what best decision can you make with this business circumstance based on the given function with its restricted domain? A. I will increase my advertising expenses to Php 2,500,000.00 because this will give a higher revenue than what the company currently earns. B. I will decrease my advertising expenses to Php 1,500,000.00 because this will give a higher revenue than what the company currently earns. C. I will decrease my advertising expenses to Php 1,500,000.00 because lower cost means higher revenue. D. It does not matter how much I spend for advertisement, my revenue will stay the same. Part 2 Read and analyze the situation below. Then, answer the question and perform the tasks that follow. Karl Benedic, the president of Mathematics Club, proposed a project: to put up a rectangular Math Garden whose lot perimeter is 36 meters. He was soliciting suggestions from the members for feasible dimensions of the lot. Suppose you are a member of the club, what will you suggest to Karl Benedic if you want a maximum lot area? You must convince him through a mathematical solution. Consider the following guidelines: 1. Make an illustration of the lot with the needed labels. 2. Solve the problem. Hint: Consider the formulas P = 2l + 2w for perimeter and A = lw for the area of the rectangle. Use the formula for P and the given information in the problem to express A in terms of either l or w. 3. Make a second illustration that satisfies the findings in the solution made in number 2. 4. Submit your solution on a sheet of paper with recommendations. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  13. 13. D EPED C O PY 105 Rubric for Rating the Output: Score Descriptors 4 The problem is correctly modeled with a quadratic function, appropriate mathematical concepts are fully used in the solution, and the correct final answer is obtained. 3 The problem is correctly modeled with a quadratic function, appropriate mathematical concepts are partially used in the solution, and the correct final answer is obtained. 2 The problem is not properly modeled with a quadratic function, other alternative mathematical concepts are used in the solution, and the correct final answer is obtained. 1 The problem is not totally modeled with a quadratic function, a solution is presented but has incorrect final answer. The additional two (2) points will be determined from the illustrations made. One (1) point for each if properly drawn with necessary labels. IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate understanding of key concepts on polynomial functions. Furthermore, you should be able to conduct a mathematical investigation involving polynomial functions. 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. 14. D EPED C O PY 106 Start this module by recalling your knowledge on the concept of polynomial expressions. This knowledge will help you understand the formal definition of a polynomial function. Determine whether each of the following is a polynomial expression or not. Give your reasons. 1. 14x 6.  2. 3 5 4 2x x x  7. 3 2 3 3 9 2x x x   3. 2014x 8. 3 2 1x x  4. 3 1 4 4 3 7x x  9. 100 100 4 4x x   5. 3 4 5 1 2 3 2 3 4x x x   10. 1 – 16x2 Did you answer each item correctly? Do you remember when an expression is a polynomial? We defined a related concept below. A polynomial function is a function of the form            1 2 1 2 1 0 ...n n n n n n P x a x a x a x a x a , ,0na where n is a nonnegative integer, naaa ...,,, 10 are real numbers called coefficients, n n xa is the leading term, na is the leading coefficient, and 0a is the constant term. The terms of a polynomial may be written in any order. However, if they are written in decreasing powers of x, we say the polynomial function is in standard form. Other than P(x), a polynomial function may also be denoted by f(x). Sometimes, a polynomial function is represented by a set P of ordered pairs (x,y). Thus, a polynomial function can be written in different ways, like the following. Activity 1: 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.
  15. 15. D EPED C O PY 107 f ( x )          1 2 1 2 1 0 ...n n n n n n a x a x a x a x a or 1 2 1 2 1 0 ...n n n n n n y a x a x a x a x a          Polynomials may also be written in factored form and as a product of irreducible factors, that is, a factor that can no longer be factored using coefficients that are real numbers. Here are some examples. a. y = x4 + 2x3 – x2 + 14x – 56 in factored form is y = (x2 + 7)(x – 2)(x + 4) b. y = x4 + 2x3 – 13x2 – 10x in factored form is y = x(x – 5)(x + 1)(x + 2) c. y = 6x3 + 45x2 + 66x – 45 in factored form is y = 3(2x – 1)(x + 3)(x + 5) d. f(x) = x3 + x2 + 18 in factored form is f(x) = (x2 – 2x + 6)(x + 3) e. f(x) = 2x3 + 5x2 + 7x – 5 in factored form is f(x) = (x2 + 3x + 5)(2x – 1) Consider the given polynomial functions and fill in the table below. Polynomial Function Polynomial Function in Standard Form Degree Leading Coefficient Constant Term 1. f ( x ) = 2 – 11x + 2x2 2. f ( x )   3 2 5 15 3 3 x x 3. y = x (x2 – 5) 4.   3 3y x x x   5. 2 )1)(1)(4(  xxxy After doing this activity, it is expected that the definition of a polynomial function and the concepts associated with it become clear to you. Do the next activity so that your skills will be honed as you give more examples of polynomial functions. Activity 2: 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.
  16. 16. D EPED C O PY 108 Use all the numbers in the box once as coefficients or exponents to form as many polynomial functions of x as you can. Write your polynomial functions in standard form. 1 –2 4 7 2 6 1  3 How many polynomial functions were you able to give? Classify each according to its degree. Also, identify the leading coefficient and the constant term. In this section, you need to revisit the lessons and your knowledge on evaluating polynomials, factoring polynomials, solving polynomial equations, and graphing by point-plotting. Your knowledge of these topics will help you sketch the graph of polynomial functions manually. You may also use graphing utilities/tools in order to have a clearer view and a more convenient way of describing the features of the graph. Also, you will focus on polynomial functions of degree 3 and higher, since graphing linear and quadratic functions were already taught in previous grade levels. Learning to graph polynomial functions requires your appreciation of its behavior and other properties. Factor each polynomial completely using any method. Enjoy working with your seatmate using the Think-Pair-Share strategy. 1. (x – 1) (x2 – 5x + 6) 2. (x2 + x – 6) (x2 – 6x + 9) 3. (2x2 – 5x + 3) (x – 3) 4. x3 + 3x2 – 4x – 12 5. 2x4 + 7x3 – 4x2 – 27x – 18 Did you get the answers correctly? What method(s) did you use? Now, do the same with polynomial functions. Write each of the following polynomial functions in factored form: Activity 4: Activity 3: All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  17. 17. D EPED C O PY 109 6. xxxy 1223  7. 164  xy 8. 68482 234  xxxxy 9. xxxy 910 35  10. 1827472 234  xxxxy The preceding task is very important for you since it has something to do with the x-intercepts of a graph. These are the x-values when y = 0, thus, the point(s) where the graph intersects the x-axis can be determined. To recall the relationship between factors and x-intercepts, consider these examples: a. Find the intercepts of 64 23  xxxy . Solution: To find the x-intercept/s, set y = 0. Use the factored form. That is, y = x3 – 4x2 + x + 6 y = (x + 1)(x – 2)(x – 3) Factor completely. 0 = (x + 1)(x – 2)(x – 3) Equate y to 0. x + 1 = 0 x = –1 or x – 2 = 0 x = 2 or x – 3 = 0 x = 3 The x-intercepts are –1, 2, and 3. This means the graph will pass through (-1, 0), (2, 0), and (3, 0). Finding the y-intercept is more straightforward. Simply set x = 0 in the given polynomial. That is, y = x3 – 4x2 + x + 6 y = 03 – 4(0)2 + 0 + 6 y = 6 The y-intercept is 6. This means the graph will also pass through (0,6). Equate each factor to 0 to determine x. 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. 18. D EPED C O PY 110 b. Find the intercepts of xxxxy 66 234  Solution: For the x-intercept(s), find x when y = 0. Use the factored form. That is, y = x4 + 6x3 – x2 – 6x y = x(x + 6)(x + 1)(x – 1) 0 = x(x + 6)(x + 1)(x – 1) x=0 or x+6 =0 x=–6 or x+1 =0 x=–1 or x–1=0 x=1 The x-intercepts are -6, -1, 0, and 1. This means the graph will pass through (-6,0), (-1,0), (0,0), and (1,0). Again, finding the y-intercept simply requires us to set x = 0 in the given polynomial. That is, 4 3 2 6 6y x x x x    y  (04 )  6(03 ) (02 ) 6(0) 0y  The y-intercept is 0. This means the graph will pass also through (0,0). You have been provided illustrative examples of solving for the x- and y- intercepts, an important step in graphing a polynomial function. Remember, these intercepts are used to determine the points where the graph intersects or touches the x-axis and the y-axis. But these points are not sufficient to draw the graph of polynomial functions. Enjoy as you learn by performing the next activities. Determine the intercepts of the graphs of the following polynomial functions: 1. y = x3 + x2 – 12x 2. y = (x – 2)(x – 1)(x + 3) 3. y = 2x4 + 8x3 + 4x2 – 8x – 6 4. y = –x4 + 16 5. y = x5 + 10x3 – 9x You have learned how to find the intercepts of a polynomial function. You will discover more properties as you go through the next activities. Activity 5: Equate each factor to 0 to determine x. Factor completely. Equate y to 0. 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.
  19. 19. D EPED C O PY 111 Work with your friends. Determine the x-intercept/s and the y-intercept of each given polynomial function. To obtain other points on the graph, find the value of y that corresponds to each value of x in the table. 1. y = (x + 4)(x + 2)(x – 1)(x – 3) x-intercepts: __ __ __ __ y-intercept: __ x -5 -3 0 2 4 y List all your answers above as ordered pairs. 2. y = –(x + 5)(2x + 3)(x – 2)(x – 4) x-intercepts: __ __ __ __ y-intercept: __ x -6 -4 -0.5 3 5 y List all your answers above as ordered pairs. y = –x(x + 6)(3x – 4) x-intercepts: __ __ __ y-intercept: __ x -7 -3 1 2 y List all your answers above as ordered pairs. 3. y = x2 (x + 3)(x + 1)(x – 1)(x – 3) x-intercepts: __ __ __ __ __ y-intercept: __ x -4 -2 -0.5 0.5 2 4 y List all your answers above as ordered pairs In this activity, you evaluated a function at given values of x. Notice that some of the given x-values are less than the least x-intercept, some are between two x-intercepts, and some are greater than the greatest x  intercept. For example, in number 1, the x-intercepts are -4, -2, 1, and 3. The value -5 is used as x-value less than -4; -3, 0, and 2 are between two x-intercepts; and 4 is used as x-value greater than 3. Why do you think we should consider them? Activity 6: 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.
  20. 20. D EPED C O PY 112 In the next activity, you will describe the behavior of the graph of a polynomial function relative to the x-axis. Given the polynomial function y = (x + 4)(x + 2)(x – 1)(x – 3), complete the table below. Answer the questions that follow. Value of x Value of y Relation of y value to 0: y > 0, y = 0, or y < 0? Location of the point (x, y): above the x-axis, on the x-axis, or below the x-axis? -5 144 0y above the x-axis -4 -3 -2 0 y = 0 on the x - axis 0 1 2 3 4 Questions: 1. At what point(s) does the graph pass through the x-axis? 2. If 4x , what can you say about the graph? 3. If 24  x , what can you say about the graph? 4. If 12  x , what can you say about the graph? 5. If 31  x , what can you say about the graph? 6. If 3x , what can you say about the graph? Now, this table may be transformed into a simpler one that will instantly help you in locating the curve. We call this the table of signs. The roots of the polynomial function y = (x + 4)(x + 2)(x – 1)(x – 3) are x = –4, –2, 1, and 3. These are the only values of x where the graph will cross the x-axis. These roots partition the number line into intervals. Test values are then chosen from within each interval. Activity 7: 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.
  21. 21. D EPED C O PY 113 The table of signs and the rough sketch of the graph of this function can now be constructed, as shown below. The Table of Signs Intervals 4x   4 2x    2 1x   1 3x  3x  Test value -5 -3 0 2 4 4x  – + + + + 2x  – – + + + 1x  – – – + + 3x  – – – – + )3)(1)(2)(4(  xxxxy + – + – + Position of the curve relative to the x-axis above below above below above The Graph of )3)(1)(2)(4(  xxxxy We can now use the information from the table of signs to construct a possible graph of the function. At this level, though, we cannot determine the turning points of the graph, we can only be certain that the graph is correct with respect to intervals where the graph is above, below, or on the x-axis. The arrow heads at both ends of the graph signify that the graph indefinitely goes upward. All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
  22. 22. D EPED C O PY 114 Here is another example: Sketch the graph of )43)(2()(  xxxxf Roots of f(x): -2, 0, 3 4 Table of Signs: Intervals 2x   2 0x   4 0 3 x  4 3 x  Test value -3 -1 1 2 –x + + – – x + 2 – + + + 3x – 4 – – – + f(x) = –x(x + 2)(3x – 4) + – + – Position of the curve relative to the x-axis above below above below Graph: In this activity, you learned how to sketch the graph of polynomial functions using the intercepts, some points, and the position of the curves determined from the table of signs. The procedures described are applicable when the polynomial function is in factored form. Otherwise, you need to express first a polynomial in factored form. Try this in the next activity. 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.
  23. 23. D EPED C O PY 115 For each of the following functions, give (a) the x-intercept(s) (b) the intervals obtained when the x-intercepts are used to partition the number line (c) the table of signs (d) a sketch of the graph 1. y = (2x + 3)(x – 1)(x – 4) 2. y = –x3 + 2x2 + 11x – 2 3. y = x4 – 26x2 + 25 4. y = –x4 – 5x3 + 3x2 + 13x – 10 5. y = x2 (x + 3)(x + 1)4 (x – 1)3 Post your answer/output for a walk-through. For each of these polynomial functions, answer the following: a. What happens to the graph as x decreases without bound? b. For which interval(s) is the graph (i) above and (ii) below the x-axis? c. What happens to the graph as x increases without bound? d. What is the leading term of the polynomial function? e. What are the leading coefficient and the degree of the function? Now, the big question for you is: Do the leading coefficient and degree affect the behavior of its graph? You will answer this after an investigation in the next activity. After sketching manually the graphs of the five functions given in Activity 8, you will now be shown polynomial functions and their corresponding graphs. Study each figure and answer the questions that follow. Summarize your answers using a table similar to the one provided. Activity 9: Activity 8: 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.
  24. 24. D EPED C O PY 116 Case 1 The graph on the right is defined by y = 2x3 – 7x2 – 7x + 12 or, in factored form, y = (2x + 3) (x – 1) (x – 4). Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? Case 2 The graph on the right is defined by 473 2345  xxxxy or, in factored form, 22 )2)(1()1(  xxxy . Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? Case 3 The graph on the right is defined by xxxy 67 24  or, in factored form, )2)(1)(3(  xxxxy . Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? x y x y x y 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.
  25. 25. D EPED C O PY 117 Case 4 The graph on the right is defined by 2414132 234  xxxxy or, in factored form, )4)(2)(1)(3(  xxxxy . Questions: a. Is the leading coefficient a positive or a negative number? b. Is the polynomial of even degree or odd degree? c. Observe the end behaviors of the graph on both sides. Is it rising or falling to the left or to the right? Now, complete this table. In the last column, draw a possible graph for the function, showing how the function behaves. (You do not need to place your graph on the xy-plane). The first one is done for you. Sample Polynomial Function Leading Coefficient: 0n or 0n Degree: Even or Odd Behavior of the Graph: Rising or Falling Possible Sketch Left- hand Right- hand 1. 3 2 2 7 7 12y x x x    0n odd falling rising 2. 5 4 3 2 3 7 4y x x x x     3. 4 2 7 6y x x x   4. 4 3 2 2 13 14 24y x x x x     Summarize your findings from the four cases above. What do you observe if: 1. the degree of the polynomial is odd and the leading coefficient is positive? 2. the degree of the polynomial is odd and the leading coefficient is negative? 3. the degree of the polynomial is even and the leading coefficient is positive? 4. the degree of the polynomial is even and the leading coefficient is negative? x y 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.
  26. 26. D EPED C O PY 118 Congratulations! You have now illustrated The Leading Coefficient Test. You should have realized that this test can help you determine the end behaviors of the graph of a polynomial function as x increases or decreases without bound. Recall that you have already learned two properties of the graph of polynomial functions; namely, the intercepts which can be obtained from the Rational Root Theorem, and the end behaviors which can be identified using the Leading Coefficient Test. Another helpful strategy is to determine whether the graph crosses or is tangent to the x-axis at each x-intercept. This strategy involves the concept of multiplicity of a zero of a polynomial function. Multiplicity tells how many times a particular number is a zero or root for the given polynomial. The next activity will help you understand the relationship between multiplicity of a root and whether a graph crosses or is tangent to the x-axis. Given the function )2()1()1()2( 432  xxxxy and its graph, complete the table below, then answer the questions that follow. Root or Zero Multiplicity Characteristic of Multiplicity: Odd or Even Behavior of Graph Relative to x-axis at this Root: Crosses or Is Tangent to -2 -1 1 2 Activity 10: x y 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.
  27. 27. D EPED C O PY 119 Questions: a. What do you notice about the graph when it passes through a root of even multiplicity? b. What do you notice about the graph when it passes through a root of odd multiplicity? This activity extends what you learned when using a table of signs to graph a polynomial function. When the graph crosses the x-axis, it means the graph changes from positive to negative or vice versa. But if the graph is tangent to the x-axis, it means that the graph is either positive on both sides of the root, or negative on both sides of the root. In the next activity, you will consider the number of turning points of the graph of a polynomial function. The turning points of a graph occur when the function changes from decreasing to increasing or from increasing to decreasing values. Complete the table below. Then answer the questions that follow. Polynomial Function Sketch Degree Number of Turning Points 1. 4 xy  2. 152 24  xxy 3. 5 xy  Activity 11: x y x y x y 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.
  28. 28. D EPED C O PY 120 Polynomial Function Sketch Degree Number of Turning Points 4. 1235  xxxy 5. xxxy 45 35  Questions: a. What do you notice about the number of turning points of the quartic functions (numbers 1 and 2)? How about of quintic functions (numbers 3 to 5)? b. From the given examples, do you think it is possible for the degree of a function to be less than the number of turning points? c. State the relation of the number of turning points of a function with its degree n. In this section, you have encountered important concepts that can help you graph polynomial functions. Notice that the graph of a polynomial function is continuous, smooth, and has rounded turns. Further, the number of turning points in the graph of a polynomial is strictly less than the degree of the polynomial. Use what you have learned as you perform the activities in the succeeding sections. y x x y 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.
  29. 29. D EPED C O PY 121 The goal of this section is to help you think critically and creatively as you apply the techniques in graphing polynomial functions. Also, this section aims to provide opportunities to solve real-life problems involving polynomial functions. For each given polynomial function, describe or determine the following, then sketch the graph. You may need a calculator in some computations. a. leading term b. end behaviors c. x-intercepts points on the x-axis d. multiplicity of roots e. y-intercept point on the y-axis f. number of turning points g. sketch 1. )52()1)(3( 2  xxxy 2. 322 )2()1)(5(  xxxy 3. 422 23  xxxy 4. )32)(7( 22  xxxy 5. 2861832 234  xxxxy In this activity, you were given the opportunity to sketch the graph of polynomial functions. Were you able to apply all the necessary concepts and properties in graphing each function? The next activity will let you see the connections of these mathematics concepts to real life. Activity 12: 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.
  30. 30. D EPED C O PY 122 Work in groups. Apply the concepts of polynomial functions to answer the questions in each problem. Use a calculator when needed. 1. Look at the pictures below. What do these tell us? Filipinos need to take the problem of deforestation seriously. The table below shows the forest cover of the Philippines in relation to its total land area of approximately 30 million hectares. Year 1900 1920 1960 1970 1987 1998 Forest Cover (%) 70 60 40 34 23.7 22.2 Source: Environmental Science for Social Change, Decline of the Philippine Forest A cubic polynomial that best models the data is given by 3 2 26 3500 391 300 69 717000 ; 0 98 1 000 000 x x x y x      where y is the percent forest cover x years from 1900. 10 20 30 40 50 60 70 80 90 x -10 10 20 30 40 50 60 70 80 y O Activity 13: 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.
  31. 31. D EPED C O PY 123 Questions/Tasks: a. Using the graph, what is the approximate forest cover during the year 1940? b. Compare the forest cover in 1987 (as given in the table) to the forest cover given by the polynomial function. Why are these values not exactly the same? c. Do you think you can use the polynomial to predict the forest cover in the year 2100? Why or why not? 2. The members of a group of packaging designers of a gift shop are looking for a precise procedure to make an open rectangular box with a volume of 560 cubic inches from a 24-inch by 18-inch rectangular piece of material. The main problem is how to identify the side of identical squares to be cut from the four corners of the rectangular sheet so that such box can be made. Question/Task: Suppose you are chosen as the leader and you are tasked to lead in solving the problem. What will you do to meet the specifications needed for the box? Show a mathematical solution. Were you surprised that polynomial functions have real and practical uses? What do you need to solve these kinds of problems? Enjoy learning as you proceed to the next section. The goal of this section is to check if you can apply polynomial functions to real-life problems and produce a concrete object that satisfies the conditions given in the problem. Read the problem carefully and answer the questions that follow. You are designing candle-making kits. Each kit contains 25 cubic inches of candle wax and a mold for making a pyramid-shaped candle with a square base. You want the height of the candle to be 2 inches less than the edge of the base. Activity 14: 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.
  32. 32. D EPED C O PY 124 Questions/Tasks: 1. What should the dimensions of your candle mold be? Show a mathematical procedure in determining the dimensions. 2. Use a sheet of cardboard as sample material in preparing a candle mold with such dimensions. The bottom of the mold should be closed. The height of one face of the pyramid should be indicated. 3. Write your solution in one of the faces of your output (mold). Rubric for the Mathematical Solution Point Descriptor 4 The problem is correctly modeled with a polynomial function, appropriate mathematical concepts are used in the solution, and the correct final answer is obtained. 3 The problem is correctly modeled with a polynomial function, appropriate mathematical concepts are partially used in the solution, and the correct final answer is obtained. 2 The problem is not properly modeled with a polynomial function, other alternative mathematical concepts are used in the solution, and the correct final answer is obtained. 1 The problem is not properly modeled with a polynomial function, a solution is presented but the final answer is incorrect. Criteria for Rating the Output:  The mold has the needed dimensions and parts.  The mold is properly labeled with the required length of parts.  The mold is durable.  The mold is neat and presentable. Point/s to Be Given: 4 points if all items in the criteria are evident 3 points if any three of the items are evident 2 points if any two of the items are evident 1 point if any of the items is evident 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.
  33. 33. D EPED C O PY 125 SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about polynomial functions. You learned how to:  illustrate and describe polynomial functions;  show the graph of polynomial functions using the following properties: - the intercepts (x-intercept and y-intercept); - the behavior of the graph using the Leading Coefficient Test, table of signs, turning points, and multiplicity of zeros; and  solve real-life problems that can be modeled with polynomial functions. GLOSSARY OF TERMS Constant Function - a polynomial function whose degree is 0 Evaluating a Polynomial - a process of finding the value of the polynomial at a given value in its domain Intercepts of a Graph - points on the graph that have zero as either the x- coordinate or the y-coordinate Irreducible Factor - a factor that can no longer be factored using coefficients that are real numbers Leading Coefficient Test - a test that uses the leading term of the polynomial function to determine the right-hand and the left-hand behaviors of the graph Linear Function - a polynomial function whose degree is 1 Multiplicity of a Root - tells how many times a particular number is a root for the given polynomial Nonnegative Integer - zero or any positive integer Polynomial Function - a function denoted by 01 2 2 1 1 ...)( axaxaxaxaxP n n n n n n      , where n is a nonnegative integer, naaa ...,,, 10 are real numbers called coefficients but ,0na n n xa is the leading term, na is the leading coefficient, and 0a is the constant term Polynomial in Standard Form - any polynomial whose terms are arranged in decreasing powers of x Quadratic Function – a polynomial function whose degree is 2 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.
  34. 34. D EPED C O PY 126 Quartic Function – a polynomial function whose degree is 4 Quintic Function – a polynomial function whose degree is 5 Turning Point – a point where the function changes from decreasing to increasing or from increasing to decreasing values REFERENCES USED IN THIS MODULE: 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. 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.
  35. 35. D EPED C O PY 127 I. INTRODUCTION Have you imagined yourself pushing a cart or riding in 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? Find out the answers to these questions and determine the vast applications of circles through this module. II. LESSONS AND COVERAGE: In this module, you will examine the above questions when you take the following lessons: Lesson 1A – Chords, Arcs, and Central Angles Lesson 1B – Arcs and Inscribed Angles Lesson 2A – Tangents and Secants of a Circle Lesson 2B – Tangent and Secant Segments 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.
  36. 36. D EPED C O PY 128 In these lessons, you will learn to: Lesson 1A Lesson 1B  derive inductively the relations among chords, arcs, central angles, and inscribed angles;  illustrate segments and sectors of circles;  prove theorems related to chords, arcs, central angles, and inscribed angles; and  solve problems involving chords, arcs, central angles, and inscribed angles of circles. Lesson 2A Lesson 2B  illustrate tangents and secants of circles;  prove theorems on tangents and secants; and  solve problems involving tangents and secants of circles. Here is a simple map of the lessons that will be covered in this module: Circles Applications of Circles Relationships among Chords, Arcs, Central Angles, and Inscribed Angles Tangents and Secants of Circles Chords, Arcs, and Central Angles Arcs and Inscribed Angles Tangents and Secants Tangent and Secant Segments 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.
  37. 37. D EPED C O PY 129 III. PRE-ASSESSMENT Part I Find out how much you already know about the topics in this module. Choose the letter that you think best answers each of the following questions. Take note of the items that you were not able to answer correctly and find the right answer as you go through this module. 1. What is an angle whose vertex is on a circle and whose sides contain chords of the circle? A. central angle C. circumscribed angle B. inscribed angle D. intercepted angle 2. An arc of a circle measures 30°. If the radius of the circle is 5 cm, what is the length of the arc? A. 2.62 cm B. 2.3 cm C. 1.86 cm D. 1.5 cm 3. Using the figure below, which of the following is an external secant segment of M? A. CO C. NO B. TI D. NI 4. The opposite angles of a quadrilateral inscribed in a circle are _____. A. right C. complementary B. obtuse D. supplementary 5. In S at the right, what is VSIm if mVI = 140? A. 35 C. 140 B. 75 D. 230 6. What is the sum of the measures of the central angles of a circle with no common interior points? A. 120 B. 240 C. 360 D. 480 I N T E C O M 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.
  38. 38. D EPED C O PY 130 P X Y N M 7. Catherine designed a pendant. It is a regular hexagon set in a circle. Suppose the opposite vertices are connected by line segments and meet at the center of the circle. What is the measure of each angle formed at the center? A. 5.22 B. 45 C. 60 D. 72 8. If an inscribed angle of a circle intercepts a semicircle, then the angle is _________. A. acute B. right C. obtuse D. straight 9. At a given point on the circle, how many line/s can be drawn that is tangent to the circle? A. one B. two C. three D. four 10. What is the length of ZK in the figure on the right? A. 2.86 units C. 8 units B. 6 units D. 8.75 units 11. In the figure on the right, mXY = 150 and mMN = 30. What is XPYm ? A. 60 B. 90 C. 120 D. 180 12. The top view of a circular table shown on the right has a radius of 120 cm. Find the area of the smaller segment of the table (shaded region) determined by a 60 arc. A.  2400 3600 3  cm2 B. 3600 3 cm2 C. 2400 cm2 D.  14 400 3600 3  cm2 60° 120 cm 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.
  39. 39. D EPED C O PY 131 13. In O given below, what is PR if NO = 15 units and ES = 6 units? A. 28 units B. 24 units C. 12 units D. 9 units 14. A dart board has a diameter of 40 cm and is divided into 20 congruent sectors. What is the area of one of the sectors? A. 20 cm2 C. 80 cm2 B. 40 cm2 D. 800 cm2 15. Mr. Soriano wanted to plant three different colors of roses on the outer rim of a circular garden. He stretched two strings from a point external to the circle to see how the circular rim can be divided into three portions as shown in the figure below. What is the measure of minor arc AB? A. 64° B. 104° C. 168° D. 192° 16. In the figure below, SY and EY are secants. If SY = 15 cm, TY = 6 cm, and LY = 8 cm. What is the length of EY ? A. 20 cm B. 12 cm C. 11.25 cm D. 6.75 cm P E R O S N S Y T L E A B M C 20° 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.
  40. 40. D EPED C O PY 132 17. In C below, mAB = 60 and its radius is 6 cm. What is the area of the shaded region in terms of pi ( )? A. 6  cm 2 C. 10  cm 2 B. 8  cm 2 D. 12  cm 2 18. In the circle below, what is the measure of SAY if DSY is a semicircle and ?70SADm A. 20 C. 110 B. 70 D. 150 19. Quadrilateral SMIL is inscribed in E. If 78m SMI and m 95MSL  , find MILm . A. 78 C. 95 B. 85 D. 102 20. In M on the right, what is BROm if 60BMOm ? A. 120 C. 30 B. 60 D. 15 A D S Y C B A 6 cm 60° M I L 78° S 95° E B M O R 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.
  41. 41. D EPED C O PY 133 Part II Solve each of the following problems. Show your complete solutions. 2. A bicycle chain fits tightly around two gears. What is the distance between the centers of the gears if the radii of the bigger and smaller gears are 9.3 inches and 2.4 inches, respectively, and the portion of the chain tangent to the two gears is 26.5 inches long? Rubric for Problem Solving Score Descriptors 4 Used an appropriate strategy to come up with the correct solution and arrived at a correct answer. 3 Used an appropriate strategy to come up with a solution, but a part of the solution led to an incorrect answer. 2 Used an appropriate strategy but came up with an entirely wrong solution that led to an incorrect answer. 1 Attempted to solve the problem but used an inappropriate strategy that led to a wrong solution. Part III Read and understand the situation below, then answer the questions and perform what is required. The committee in-charge of the Search for the Cleanest and Greenest School informed your principal that your school has been selected as a regional finalist. Being a regional finalist, your principal would like to make your school more beautiful and clean by making more gardens of different shapes. He decided that every year level will be assigned to prepare a garden of particular shape. In your grade level, he said that you will be preparing circular, semicircular, or arch-shaped gardens in front of your building. He further encouraged your grade level to add garden accessories to make the gardens more presentable and amusing. 1. Mr. Javier designed an arch made of bent iron for the top of a school’s main entrance. The 12 segments between the two concentric semicircles are each 0.8 meter long. Suppose the diameter of the inner semicircle is 4 meters. What is the total length of the bent iron used to make this arch? 4 m0.8 m 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.
  42. 42. D EPED C O PY 134 1. How will you prepare the design of the gardens? 2. What garden accessories will you use? 3. Make the designs of the gardens which will be placed in front of your grade level building. Use the different shapes that were required by your principal. 4. Illustrate every part or portion of the garden including their measurements. 5. Using the designs of the gardens made, determine all the concepts or principles related to circles. 6. Formulate problems involving these mathematics concepts or principles, then solve. Rubric for Design Score Descriptors 4 The design is accurately made, presentable, and appropriate. 3 The design is accurately made and appropriate but not presentable. 2 The design is not accurately made but appropriate. 1 The design is made but not accurate and appropriate. Rubric on Problems Formulated and Solved Score Descriptors 6 Poses a more complex problem with two or more correct possible solutions, communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes, and provides explanations wherever appropriate. 5 Poses a more complex problem and finishes all significant parts of the solution, communicates ideas unmistakably, and shows in-depth comprehension of the pertinent concepts and/or processes. 4 Poses a complex problem and finishes all significant parts of the solution, communicates ideas unmistakably, and shows in-depth comprehension of the pertinent concepts and/or processes. 3 Poses a complex problem and finishes most significant parts of the solution, communicates ideas unmistakably, and shows comprehension of major concepts although neglects or misinterprets less significant ideas or details. 2 Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension. 1 Poses a problem but demonstrates minor comprehension, not being able to develop an approach. Source: D.O. #73, s. 2012 IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate understanding of key concepts of circles and formulate real-life problems involving these concepts, and solve these using a variety of strategies. Furthermore, you should be able to investigate mathematical relationships in various situations involving circles. 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.
  43. 43. D EPED C O PY 135 J A N E L s Start Lesson 1A of this module by assessing your knowledge of the different mathematical concepts previously studied and your skills in performing mathematical operations. These knowledge and skills will help you understand circles. As you go through this lesson, think of this important question: “How do the relationships among chords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions?” To find the answer, perform each activity. If you find any difficulty in answering the exercises, seek the assistance of your teacher or peers or refer to the modules you have studied earlier. You may check your work with your teacher. Use the figure below to identify and name the following terms related to A. Then, answer the questions that follow. 1. a radius 5. a minor arc 2. a diameter 6. a major arc 3. a chord 7. 2 central angles 4. a semicircle 8. 2 inscribed angles Activity 1: 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.
  44. 44. D EPED C O PY 136 Questions: a. How did you identify and name the radius, diameter, and chord? How about the semicircle, minor arc, and major arc? inscribed angle and central angle? b. How do you describe a radius, diameter, and chord of a circle? How about the semicircle, minor arc, and major arc? inscribed angle and central angle? Write your answers in the table below. Terms Related to Circles Description 1. radius 2. diameter 3. chord 4. semicircle 5. minor arc 6. major arc 7. central angle 8. inscribed angle c. How do you differentiate among the radius, diameter, and chord of a circle? How about the semicircle, minor arc, and major arc? inscribed angle and central angle? Were you able to identify and describe the terms related to circles? Were you able to recall and differentiate them? Now that you know the important terms related to circles, let us deepen your understanding of finding the lengths of sides of right triangles. You need this mathematical skill in finding the relationships among chords, arcs, and central angles as you go through this lesson. 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.
  45. 45. D EPED C O PY 137 In each triangle below, the length of one side is unknown. Determine the length of this side. 1. 4. 2. 5. 3. 6. Questions: a. How did you find the missing side of each right triangle? b. What mathematics concepts or principles did you apply to find each missing side? In the activity you have just done, were you able to find the missing side of a right triangle? The concept used will help you as you go on with this module. Activity 2: a = 6 b = 8 c = ? b = ? a = 3 c = 5 a = 9 b = 9 c = ? a = ? b = 16 c = 20 a = 7 b = ? c = 14 a = 9 b = 15 c = ? 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.
  46. 46. D EPED C O PY 138 O Use the figures below to answer the questions that follow. Figure 1 Figure 2 1. What is the measure of each of the following angles in Figure 1? Use a protractor. a. TOP d. ROS b. POQ e. SOT c. QOR 2. In Figure 2, AF , AB , AC , AD , and AE are radii of A. What is the measure of each of the following angles? Use a protractor. a. FAB d. EAD b. BAC e. EAF c. CAD 3. How do you describe the angles in each figure? 4. What is the sum of the measures of ,TOP POQ , ,QOR ,ROS and SOT in Figure 1? How about the sum of the measures of ,FAB ,BAC ,CAD ,EAD and EAF in Figure 2? Activity 3: OT S R Q P C B F D E AO 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.
  47. 47. D EPED C O PY 139 5. In Figure 1, what is the sum of the measures of the angles formed by the coplanar rays with a common vertex but with no common interior points? 6. In Figure 2, what is the sum of the measures of the angles formed by the radii of a circle with no common interior points? 7. In Figure 2, what is the intercepted arc of ?FAB How about ?BAC CAD ? ?EAD ?EAF Complete the table below. Central Angle Measure Intercepted Arc a. FAB b. BAC c. CAD d. EAD e. EAF 8. What do you think is the sum of the measures of the intercepted arcs of ,FAB BAC , CAD , ,EAD and EAF ? Why? 9. What can you say about the sum of the measures of the central angles and the sum of the measures of their corresponding intercepted arcs? Were you able to measure the angles accurately and find the sum of their measures? Were you able to determine the relationship between the measures of the central angle and its intercepted arc? For sure you were able to do it. In the next activity, you will find out how circles are illustrated in real-life situations. Use the situation below to answer the questions that follow. Rowel is designing a mag wheel like the one shown below. He decided to put 6 spokes which divide the rim into 6 equal parts. Activity 4: 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.
  48. 48. D EPED C O PY 140 In the figure on the right, BAC is a central angle. Its sides divide A into arcs. One arc is the curve containing points B and C. The other arc is the curve containing points B, D, and C. Recall that a central angle of a circle is an angle formed by two rays whose vertex is the center of the circle. Each ray intersects the circle at a point, dividing it into arcs. Questions: a. What is the degree measure of each arc along the rim? How about each angle formed by the spokes at the hub? b. If you were to design a wheel, how many spokes will you use to divide the rim? Why? How did you find the preceding activities? Are you ready to learn about the relations among chords, arcs, and central angles of a circle? I am sure you are!!! From the activities done, you were able to recall and describe the terms related to circles. You were able to find out how circles are illustrated in real-life situations. But how do the relationships among chords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions? You will find these out in the activities in the next section. Before doing these activities, read and understand first some important notes on this lesson and the examples presented. Central Angle and Arcs Definition: Sum of Central Angles (Note: All measures of angles and arcs are in degrees.) The sum of the measures of the central angles of a circle with no common interior points is 360 degrees. In the figure, 3604m3m2m1m  . 4 3 21 B A C D central angle arc arc 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.
  49. 49. D EPED C O PY 141 M O G E Arcs of a Circle An arc is a part of a circle. The symbol for arc is . A semicircle is an arc with a measure equal to one-half the circumference of a circle. It is named by using the two endpoints and another point on the arc. Degree Measure of an Arc 1. The degree measure of a minor arc is the measure of the central angle which intercepts the arc. Example: GEO is a central angle. It intercepts E at points G and O. The measure of GO is equal to the measure of .GEO If m 118,GEO  then mGO = 118. 2. The degree measure of a major arc is equal to 360 minus the measure of the minor arc with the same endpoints. Example: If mGO = 118, then mOMG = 360 – mGO. That is, mOMG = 360 – 118 = 242. Answer: mOMG = 242 3. The degree measure of a semicircle is 180 . U A minor arc is an arc of the circle that measures less than a semicircle. It is named usually by using the two endpoints of the arc. Examples: JN, NE, and JE A major arc is an arc of a circle that measures greater than a semicircle. It is named by using the two endpoints and another point on the arc. Examples: JEN, JNE, and EJN N J E A Z O C N Example: The curve from point N to point Z is an arc. It is part of O and is named as arc NZ or NZ. Other arcs of O are CN, CZ, CZN, CNZ, and NCZ. If mCNZ is one-half the circumference of O, then it is a semicircle. 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.
  50. 50. D EPED C O PY 142 W T S E 65° 65° N M K I M A T H Congruent Circles and Congruent Arcs Congruent circles are circles with congruent radii. Example: MA is a radius of A. TH is a radius of T. If THMA  , then A T. Congruent arcs are arcs of the same circle or of congruent circles with equal measures. Example: In I, KSTM  . If I E, then NWTM  and NWKS  . Theorems on Central Angles, Arcs, and Chords 1. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent. In E below, NEOSET  Since the two central angles are congruent, the minor arcs they intercept are also congruent. Hence, NOST  . If E  I and BIGNEOSET  , then BGNOST  . 65° T 50°50° T S N O E I G B 50° . 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.
  51. 51. D EPED C O PY 143 Proof of the Theorem The proof has two parts. Part 1. Given are two congruent circles and a central angle from each circle which are congruent. The two-column proof below shows that their corresponding intercepted arcs are congruent. Given: E I BIGSET  Prove: BGST  Proof: Statements Reasons 1. E I BIGSET  1. Given 2. In E , .m SET mST  In I , .m BIG mBG  2. The degree measure of a minor arc is the measure of the central angle which intercepts the arc. 3. BIGmSETm  3. From 1, definition of congruent angles 4. mBGmST  4. From 2 & 3, substitution 5. BGST  5. From 4, definition of congruent arcs Part 2. Given are two congruent circles and intercepted arcs from each circle which are congruent. The two-column proof on the next page shows that their corresponding angles are congruent. Given: E I BGST  Prove: BIGSET  G B I ES T G B I ES T 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.
  52. 52. D EPED C O PY 144 B A H E O N C T Proof: Statements Reasons 1. E I BGST  1. Given 2. In ,E .mST m SET  In I , .mBG m BIG  2. The degree measure of a minor arc is the measure of the central angle which intercepts the arc. 3. mBGmST  3. From 1, definition of congruent arcs 4. BIGmSETm  4. From 2 & 3, substitution 5. BIGSET  5. From 4, definition of congruent angles Combining parts 1 and 2, the theorem is proven. 2. In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent. In T on the right, CHBA  . Since the two chords are congruent, then CHBA  . If T  N and OECHBA  , then OECHBA  . 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.
  53. 53. D EPED C O PY 145 B A T E N O B A T E N O Proof of the Theorem The proof has two parts. Part 1. Given two congruent circles T N and two congruent corresponding chords AB and OE , the two- column proof below shows that the corresponding minor arcs AB and OE are congruent. Given: T N OEAB  Prove: OEAB  Proof: Statements Reasons 1. T N OEAB  1. Given 2. NENOTBTA  2. Radii of the same circle or of congruent circles are congruent. 3. ONEATB  3. SSS Postulate 4. ONEATB  4. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) 5. OEAB  5. From the previous theorem, “In a circle or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.” Part 2. Given two congruent circles T and N and two congruent minor arcs AB and OE , the two-column proof on the next page shows that the corresponding chords AB and OE are congruent. Given: T N OEAB  Prove: OEAB  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.
  54. 54. D EPED C O PY 146 Proof: Statements Reasons 1. T N OEAB  1. Given 2. mOEmAB  2. Definition of congruent arcs 3. BTA and ONE are central angles. 3. Definition of central angles 4. mBABTAm  mOEONEm  4. The degree measure of a minor arc is the measure of the central angle which intercepts the arc. 5. ONEmBTAm  5. From 2, 4, substitution 6. NENOTBTA  6. Radii of the same circle or of congruent circles are congruent. 7. ONEATB  7. SAS Postulate 8. OEAB  8. Corresponding Parts of Congruent Triangles are Congruent (CPCTC) Combining parts 1 and 2, the theorem is proven. 3. In a circle, a diameter bisects a chord and an arc with the same endpoints if and only if it is perpendicular to the chord. The proof of the theorem is given as an exercise in Activity 9. N E G U I S In U on the right, ES is a diameter and GN is a chord. If GNES  , then INGI  and ENGE  . 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.
  55. 55. D EPED C O PY 147 Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. Example: Adjacent arcs are arcs with exactly one point in common. In E, LO and OV are adjacent arcs. The sum of their measures is equal to the measure of LOV. If mLO = 71 and mOV = 84, then mLOV = 71 + 84 = 155. Sector and Segment of a Circle A sector of a circle is the region bounded by an arc of the circle and the two radii to the endpoints of the arc. To find the area of a sector of a circle, get the product of the ratio 360 arctheofmeasure and the area of the circle. Example: The radius of C is 10 cm. If mAB = 60, what is the area of sector ACB? Solution: To find the area of sector ACB: a. Determine first the ratio 360 arctheofmeasure . 360 60 360  arctheofmeasure 6 1  b. Find the area (A) of the circle using the equation A = 2 r , where r is the length of the radius. A = 2 r =  2 cm10 = 2 cm100 O E V L C B A 10 cm 60° 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.
  56. 56. D EPED C O PY 148 c. Get the product of the ratio 360 arctheofmeasure and the area of the circle. Area of sector ACB =  2 cm100 6 1       = 2 cm 3 50 The area of sector ACB is 2 cm 3 50 . A segment of a circle is the region bounded by an arc and the segment joining its endpoints. Example: The shaded region in the figure below is a segment of T. It is the region bounded by PQ and PQ . In the same figure, the area of ΔPTQ =   cm5cm5 2 1 or ΔPTQ = 2 cm 2 25 . The area of the shaded segment, then, is equal to 2 cm 2 25 4 25  which is approximately 7.135 cm2 . To find the area of the shaded segment in the figure, subtract the area of triangle PTQ from the area of sector PTQ. If mPQ = 90 and the radius of the circle is 5 cm, then the area of sector PTQ is one- fourth of the area of the whole circle. That is, Area of sector PTQ =         2 5 4 1 cm =        2 cm25 4 1 = 2 cm 4 25  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.
  57. 57. D EPED C O PY 149 Arc Length The length of an arc can be determined by using the proportion 360 2   A r l , where A is the degree measure of the arc, r is the radius of the circle, and l is the arc length. In the given proportion, 360 is the degree measure of the whole circle, while 2r is the circumference. Example: An arc of a circle measures 45°. If the radius of the circle is 6 cm, what is the length of the arc? Solution: In the given problem, A = 45 and r = 6 cm. To find l, the equation 360 2   A r l can be used. Substitute the given values in the equation. 360 2   A r l  45 360 2 (6)   l  1 8 12   l 12 8   l  4.71l The length of the arc is approximately 4.71 cm. Learn more about Chords, Arcs, Central Angles, Sector, and Segment of a Circle through the WEB. You may open the following links. http://www.cliffsnotes.com/math/geometry/ circles/central-angles-and-arcs http://www.mathopenref.com/arc. html http://www.mathopenref.com/chord.html http://www.mathopenref.com/circlecentral. html http://www.mathopenref.com/arclength.html http://www.mathopenref.com/arcsector.html http://www.mathopenref.com/segment.html http://www.math-worksheet.org/arc-length-and- sector-area 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.
  58. 58. D EPED C O PY 150 Your goal in this section is to apply the key concepts of chords, arcs, and central angles of a circle. Use the mathematical ideas and the examples presented in the preceding section to answer the activities provided. Use A below to identify and name the following. Then, answer the questions that follow. 1. 2 semicircles in the figure 2. 4 minor arcs and their corresponding major arcs 3. 4 central angles Questions: a. How did you identify and name the semicircles? How about the minor arcs and the major arcs? central angles? b. Do you think the circle has more semicircles, arcs, and central angles? Show. Were you able to identify and name the arcs and central angles in the given circle? In the next activity, you will apply the theorems on arcs and central angles that you have learned. Activity 5: H K L M G A J 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.
  59. 59. D EPED C O PY 151 In A below, m 42,LAM m 30,HAG and KAH is a right angle. Find the following measure of an angle or an arc, and explain how you arrived at your answer. 1. LAKm 6. LKm 2. JAKm 7. JKm 3. LAJm 8. LMGm 4. JAHm 9. JHm 5. KAMm 10. KLMm In the activity you have just done, were you able to find the degree measure of the central angles and arcs? I am sure you did! In the next activity, you will apply the relationship among the chords, arcs, and central angles of a circle. In the figure, JI and ON are diameters of S. Use the figure and the given information to answer the following. 1. Which central angles are congruent? Why? 2. If 113m JSN , find: a. ISOm b. NSIm c. JSOm 3. Is INOJ  ? How about JN and OI ? Justify your answer. 4. Which minor arcs are congruent? Explain your answer. 5. If 67m JSO , find: a. JOm d. IOm b. JNm e. JOmN c. NIm f. NIOm 6. Which arcs are semicircles? Why? Activity 7: Activity 6: I J S N O H K L M G A J 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.
  60. 60. D EPED C O PY 152 Were you able to apply the relationship among the chords, arcs and central angles of a circle? In Activity 8, you will use the theorems on chords in finding the lengths of chords. In M below, BD = 3, KM = 6, and KP = 72 . Use the figure and the given information to find each measure. Explain how you arrived at your answer. 1. AM 5. DS 2. KL 6. MP 3. MD 7. AK 4. CD 8. KP Were you able to find the length of the segments? In the next activity, you will complete the proof of a theorem on central angles, arcs, and chords of a circle. Complete the proof of the following theorem. 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. Given: ES is a diameter of U and perpendicular to chord GN at I. Prove: 1. GINI  2. EGEN  3. GSNS  Activity 9: Activity 8: D M 3 C 6 72 A S P L B K N G E I S U 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.
  61. 61. D EPED C O PY 153 Proof of Part 1: Show that ES bisects GN and the minor arc GN. Statements Reasons 1. U with diameter ES and chord GN GNES  Two points determine a line. 2. GIU and NIU are right angles. Given 3. Lines that are perpendicular tm right angles. 4. UNUG  Radii of a circle are congruent. 5. UIUI  Reflexive Property of Congruence. 6. NIUGIU  7. NIGI  Corresponding parts of congruent triangles are congruent. 8. ES bisects GN . Corresponding parts of congruent triangles are congruent 9. NUIGUI  In a circle, congruent central angles intercept congruent arcs. 10. GUI and GUE are the same angles. NUI and NUE are the same angles. Two angles that form a linear pair are supplementary. 11. NUEmGUEm  Supplements of congruent angles are congruent. 12. GUEmmEG  NUEmmEN  In a circle, congruent central angles intercept congruent arcs. 13. mEGmEN  14. NUSmGUSm  15. GUSmmGS  NUSmmNS  16. mGSmNS  17. ES bisects GN . ; GIU NIU   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.
  62. 62. D EPED C O PY 154 Given: ES is a diameter of U; ES bisects GN at I and the minor arc GN. Proof of Part 2: Show that GNES  . Statements Reasons 1. U with diameter ES , ES bisects GN at I and the minor arc GN. Two points determine a line. 2. NIGI  NEGE  Given 3. 4. UNUG  Radii of a circle are congruent. 5. NIUGIU  Reflexive Property of Congruence. 6. UINUIG  7. UIG and UIN are right angles. Corresponding parts of congruent triangles are congruent. 8. GNIU  9. GNES  Was the activity interesting? Were you able to complete the proof? You will do more of this in the succeeding lessons. Now, use the ideas you have learned in this lesson to find the arc length of a circle. S N GI U E UI UI S 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.
  63. 63. D EPED C O PY 155 The radius of O below is 5 units. Find the length of each of the following arcs given the degree measure. Answer the questions that follow. 1. mPV = 45; length of PV = ________ 2. mPQ = 60; length of PQ = ________ 3. mQR = 90; length of QR = ________ 4. mRTS = 120;length of RTS = ________ 5. mQRT = 95; length of QRT = ________ Questions: a. How did you find the length of each arc? b. What mathematics concepts or principles did you apply to find the length of each arc? Were you able to find the arc length of each circle? Now, find the area of the shaded region of each circle. Use the knowledge learned about segment and sector of a circle in finding each area. Find the area of the shaded region of each circle. Answer the questions that follow. 1. 2. 3. Activity 11: Activity 10: BC A R Q S X Z Y6 cm 90° 12 cm 45° 8 cm 135° T S P Q R Or = 5 V 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.
  64. 64. D EPED C O PY 156 4. 5. 6. Questions: a. How did you find the area of each shaded region? b. What mathematics concepts or principles did you apply to find the area of the shaded region? Explain how you applied these concepts. How was the activity you have just done? Was it easy for you to find the area of segments and sectors of circles? It was easy for sure! In this section, the discussion was about the relationship among chords, arcs, and central angles of circles, arc length, segment and sector of a circle, and the application of these concepts in solving problems. Go back to the previous section and compare your initial ideas with the discussion. How much of your initial ideas are found in the discussion? Now that you know the important ideas about this topic, let us go deeper by moving on to the next section. T X S 4 cm A R E B W M5 cm 100° J S O 6 cm Y 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.
  65. 65. D EPED C O PY 157 Your goal in this section is to take a closer look at some aspects of the topic. You are going to think deeper and test further your understanding of circles. After doing the following activities, you should be able to answer this important question: “How do the relationships among chords, arcs, and central angles of a circle facilitate finding solutions to real-life problems and making decisions?” Answer the following questions. 1. Five points on a circle separate the circle into five congruent arcs. a. What is the degree measure of each arc? b. If the radius of the circle is 3 cm, what is the length of each arc? c. Suppose the points are connected consecutively with line segments. How do you describe the figure formed? 2. Do you agree that if two lines intersect at the center of a circle, then the lines intercept two pairs of congruent arcs? Explain your answer. 3. In the two concentric circles on the right, CON intercepts CN and RW. a. Are the degree measures of CN and RW equal? Why? b. Are the lengths of the two arcs equal? Explain your answer. 4. The length of an arc of a circle is 6.28 cm. If the circumference of the circle is 37.68 cm, what is the degree measure of the arc? Explain how you arrived at your answer. 5. Activity 12: O R W N C Mr. Lopez would like to place a fountain in his circular garden on the right. He wants the pipe, where the water will pass through, to be located at the center of the garden. Mr. Lopez does not know where it is. Suppose you were asked by Mr. Lopez to find the center of the garden, how would you do it? 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.
  66. 66. D EPED C O PY 158 6. The monthly income of the Soriano family is Php36,000.00. They spend Php9,000.00 for food, Php12,000.00 for education, Php4,500.00 for utilities, and Php6,000.00 for other expenses. The remaining amount is for their savings. This information is shown in the circle graph below. a. Which item is allotted with the highest budget? How about the least? Explain. b. If you were to budget your family’s monthly income, which item would you give the greater allocation? Why? c. In the circle graph, what is the measure of the central angle corresponding to each item? d. How is the measure of the central angle corresponding to each item determined? e. Suppose the radius of the circle graph is 25 cm. What is the area of each sector in the circle graph? How about the length of the arc of each sector? In this section, the discussion was about your understanding of chords, arcs, central angles, area of a segment and a sector, and arc length of a circle including their real-life applications. What new realizations do you have about the lesson? How would you connect this to real life? Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. Soriano Family’s Monthly Expenses All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means - electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.
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Math 10 Unit 2 LM

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