Math Grade 10

Using the Distance Formula in Proving Geometric Properties Many geometric properties can be proven by using a coordinate plane.A proof that uses figures on a coordinate plane to prove geometric propertiesis called a coordinate proof. To prove geometric properties using the methods of coordinategeometry, consider the following guidelines for placing figures on a coordinateplane.1. Use the origin as vertex or center of a figure.2. Place at least one side of a polygon on an axis.3. If possible, keep the figure within the first quadrant.4. Use coordinates that make computations simple and easy. Sometimes, using coordinates that are multiples of two would make the computation easier.DEPED COPYIn some coordinate proofs, the Distance Formula is applied.Example: Prove that the diagonals of a rectangle areSolution: congruent using the methods of coordinate geometry. AB Given: DC ABCD with diagonals AC and BD Prove: AC  BD To prove: ABCD on a coordinate plane. 1. Place BC AD 239 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

2. Label the coordinates as shown below. B(0, b) C(a, b) A(0, 0) D(a, 0)DEPED COPYa. Find the distance between A and C. Given: A(0,0) and C(a, b) AC  a  02  b  02 AC  a 2  b 2 b. Find the distance between B and D. Given: B(0, b) and D(a, 0) BD  a  02  0  b 2 BD  a 2  b 2 Since AC  a 2  b 2 and BD  a 2  b 2 , then AC  BD by substitution. Therefore, AC  BD . The diagonals of a rectangle are congruent. 240 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Learn more about the http://www.regentsprep.org/Regents/math/Distance Formula, the geometry/GCG2/indexGCG2.htmMidpoint Formula, and theCoordinate Proof through http://www.cliffsnotes.com/math/geometry/the WEB. You may open the coordinate-geometry/midpoint-formulafollowing links. http://www.regentsprep.org/Regents/math/ geometry/GCG3/indexGCG3.htm http://www.cliffsnotes.com/math/geometry/ coordinate-geometry/distance-formula http://www.regentsprep.org/Regents/math/ geometry/GCG4/indexGCG4.htmDEPED COPY Your goal in this section is to apply the key concepts of the distanceformula including the midpoint formula and the coordinate proof. Use themathematical ideas and the examples presented in the preceding sectionto perform the given activities.Activity 5:Find the distance between each pair of points on the coordinate plane.Answer the questions that follow.1. M(2, –3) and N(10, –3) 6. C(–3, 2) and D(9, 7)2. P(3, –7) and Q(3, 8) 7. S(–4, –2) and T(1, 7)3. C(–4, 3) and D(7, 6) 8. K(3, –3) and L(–3, 7)4. A(2, 3) and B(14, 8) 9. E(7, 1) and F(–6, 5)5. X(–3, 9) and Y(2, 5) 10. R(4, 7) and S(–6, –1)Questions: a. How do you find the distance between points that are aligned horizontally? vertically? b. If two points are not aligned horizontally or vertically, how would you determine the distance between them? 241 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Were you able to use the distance formula in finding the distancebetween each pair of points on the coordinate plane? In the next activity,you will be using the midpoint formula in determining the coordinates ofthe midpoint of the segment whose endpoints are given.Activity 6:Find the coordinates of the midpoint of the segment whose endpoints aregiven below. Explain how you arrived at your answers.1. A(6, 8) and B(12,10) 6. M(–9, 15) and N(–7, 3)DEPED COPY2. C(5, 11) and D(9, 5) 7. Q(0, 8) and R(–10, 0)3. K(–3, 2) and L(11, 6) 8. D(12, 5) and E(3, 10)4. R(–2, 8) and S(10, –6) 9. X(–7, 11) and Y(–9, 3)5. P(–5, –1) and Q(8, 6) 10. P(–3, 10) and T(–7, –2) Was it easy for you to determine the coordinates of the midpoint ofeach segment? I am sure it was. You need this skill in proving geometricrelationships using coordinate proof, and in solving real-life problemsinvolving the use of the midpoint formula.Activity 7:Plot each set of points on the coordinate plane. Then connect the consecutivepoints by a line segment to form the figure. Answer the questions that follow.1. A(6, 11), B(1, 2), C(11, 2) 6. L(–4, 4), O(3, 9), V(8, 2), E(1, –3)2. G(5, 14), O(–3, 8), T(17, –2) 7. S(–1, 5), O(9, –1), N(6, –6), G(–4, 0)3. F(–2, 6), U(–2, –3), N(7, 6) 8. W(–2, 6), I(9, 6), N(11, –2), D(–4, –2)4. L(–2, 8), I(5, 8), K(5, 1), E(–2, 1) 9. B(1, 6), E(13, 7), A(7, –2), T(–5, –3)5. D(–4, 6), A(8, 6), T(8, –2), 10. C(4, 12), A(9, 9), R(7, 4), E(1, 4), E(–4, –2) S(–1, –9) 242 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Questions: a. How do you describe each figure formed? Which figure is a triangle? quadrilateral? pentagon? b. Which among the triangles formed is isosceles? right? c. How do you know that the triangle is isosceles? right? d. Which among the quadrilaterals formed is a square? rectangle? parallelogram? trapezoid? e. How do you know that the quadrilateral formed is a square? rectangle? parallelogram? trapezoid? Did you find the activity interesting? Were you able to identify and describe each figure? In the next activity, you will be using the different properties of geometric figures in determining the missing coordinates.DEPED COPYActivity 8:Name the missing coordinates in terms of the given variables. Answer thequestions that follow.1. COME is a parallelogram. . ∆RST is a right triangle with right y RTS . V is the midpoint of RS .C(b, c) O(?, ?) y R(0, 2b) V(?, ?)E(0, 0) M(a, 0) x T(0, 0) S(2a, 0) x 243 All rights reserved. No part of this material may be reproduced or 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. ∆MTC is an isosceles triangle 4. WISE is an isosceles trapezoid. and V is the midpoint of CT. y y W(?, ?) I(b, c) M(?, b)C(0, 0) V(?, ?) T(6a, x E(-a, 0) S(a, 0) x 0) DEPED COPY5. ABCDEF is a regular hexagon. 6. TOPS is a square. O(0, d) y y C(–a, d) D(?, ?)B(–b, c) E(?, ?) T(–a, b) P(?, ?) A(?, ?) F(a, 0) x S(?, ?) xQuestions: a. How did you determine the missing coordinates in each figure? b. Which guided you in determining the missing coordinates in each figure? c. In which figure are the missing coordinates difficult to determine? Why? d. Compare your answers with those of your classmates. Do you have the same answers? Explain. 244 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY How was the activity you have just done? Was it easy for you to determine the missing coordinates? It was easy for sure! In this section, the discussion was about the distance formula, the midpoint formula, and the use of coordinate proof. Now that you know the important ideas about this topic, you can now move on to the next section and deepen your understanding of these concepts. Your goal in this section is to think deeper and test further your understanding of the distance formula and the midpoint formula. You will also write proofs using coordinate geometry. After doing the following activities, you should be able to answer this important question: How does the distance formula facilitate finding solutions to real-life problems and making wise decisions. Activity 9: Answer the following. 1. The coordinates of the endpoints of ST are (-2, 3) and (3, y), respectively. Suppose the distance between S and T is 13 units. What value/s of y would satisfy the given condition? Justify your answer. 2. The length of MN  15 units. Suppose the coordinates of M are (9, –7) and the coordinates of N are (x, 2). a. What is the value of x if N lies on the first quadrant? second quadrant? Explain your answer. b. What are the coordinates of the midpoint of MN if N lies in the second quadrant? Explain your answer. 3. The midpoint of CS has coordinates (2, –1). If the coordinates of C are (11, 2), what are the coordinates of S? Explain your answer. 245 All rights reserved. No part of this material may be reproduced or 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. A tracking device attached to a kidnap victim prior to his abduction indicates that he is located at a point whose coordinates are (8, 10). In the tracking device, each unit on the grid is equivalent to 10 kilometers. How far is the tracker from the kidnap victim if he is located at a point whose coordinates are (1, 3)?5. The diagram below shows the coordinates of the location of the houses ofLuisa and Grace. yLuisaDEPED COPY(-7, 4) Grace (11, 1) x Luisa says that the distance of her house from Grace’s house can be determined by evaluating the expression 11  72  1 42 . Grace does not agree with Luisa. She says that the expression  7 112  4 12 gives the distance between their houses. Who do you think is correct? Justify your answer.6. A study shed will be constructed midway between two school buildings. On a school map drawn on a coordinate plane, the coordinates of the first building are (10, 30) and the coordinates of the second building are (170, 110). a. Why do you think the study shed will be constructed midway between the two school buildings? b. What are the coordinates of the point where the study shed will be constructed? c. If each unit on the coordinate plane is equivalent to 2 m, what is the distance between the two buildings? How far would the study shed be from the first building? second building? Explain your answer. 246 All rights reserved. No part of this material may be reproduced or 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. A Global Positioning System (GPS) device shows that car A travelling at a speed of 60 kph is located at a point whose coordinates are (100, 90). Behind car A is car B, travelling in the same direction at a speed of 80 kph, that is located at a ypoint whose coordinates are (20, 30). Car ADEPED COPYCar B xa. What is the distance between the two cars?b. After how many hours will the two cars be at the same point?8. Carmela claims that the triangle on the coordinate plane y shown on the right is an equilateral triangle. Do you B(0, a) agree with Carmela? Justify your answer. A(–a, 0) C(a, 0)x9. Fa,d , Ac,d, Sc,b, and T a,b are distinct points on the coordinate plane. a. Is FS  AT ? Justify your answer. b. What figure will be formed when you connect consecutive points by a line segment? Describe the figure. 247 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

How was the activity you have just performed? Did you gain betterunderstanding of the lesson? Were you able to use the mathematicsconcepts learned in solving problems? Were you able to realize theimportance of the lesson in the real world? I am sure you were! In thenext activity you will be using the distance formula and the coordinateproof in proving geometric relationships.Activity 10:Write a coordinate proof to prove each of the following.1. The diagonals of an isosceles trapezoid are congruent.DEPED COPYGiven: Trapezoid PQRS with PS  QR P QProve: PR  QS SR2. The median to the hypotenuse of a right triangle is half the hypotenuse.Given: ∆LGC is a right triangle with rt. LCG L and M is the midpoint of LG .Prove: MC  1 LG M 2 CG3. The segments joining the midpoints of consecutive sides of an isoscelestrapezoid form a rhombus. HP OGiven: Isosceles trapezoid HOME with HE  OM S Q P, Q, R, and S are the midpoints of the sides of the trapezoid. EProve: Quadrilateral PQRS is a rhombus. R M 248 All rights reserved. No part of this material may be reproduced or 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. The medians to the legs of an isosceles triangle are congruent.Given: Isosceles triangle ABC with AB  AC. A BT and CS are the medians.Prove: BT  CS T S C B5. If the diagonals of a parallelogram are congruent, then it is a rectangle.DEPED COPYGiven: Parallelogram ABCD A B AC  BDProve: Parallelogram ABCD is a rectangle. D C6. If a line segment joins the midpoints of two sides of a triangle, then its length is equal to one-half the length of the third side.Given: Triangle LME LProve: C and G are midpoints of C LM and EM , respectively. CG  1 LE 2 EG M In this section, the discussion was about the applications of thedistance formula, the midpoint formula, and the use of coordinate proofs. What new realizations do you have about the distance formula, themidpoint formula, and the coordinate proof? In what situations can youuse the formulas discussed in this section? Now that you have a deeper understanding of the topic, you areready to do the tasks in the next section. 249 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Your goal in this section is to apply your learning to real-lifesituations. You will be given a practical task which will demonstrate yourunderstanding of the distance formula, the midpoint formula, and the useof coordinate proofs.Activity 11:Perform the following activities. Use the rubric provided to rate your work. 1. Have a copy of the map of your municipality, city, or province then make a sketch of it on a coordinate plane. Indicate on the sketch some important landmarks, then determine their coordinates. Explain why the landmarks you have indicated are significant in your community. Write also a paragraph explaining how you selected the coordinates of these important landmarks. 2. Using the coordinates assigned to the different landmarks in item #1, formulate then solve problems involving the distance formula, midpoint formula, and the coordinate proof. DEPED COPYRubric for the Sketch of a MapScore Descriptors 4 The sketch of the map is accurately made, presentable, and appropriate. 3 2 The sketch of the map is accurately made and appropriate but 1 not presentable. The sketch of the map is not accurately made but appropriate. The sketch of the map is not accurately made and not appropriate.Rubric for the Explanation of the Significance of the LandmarksScore Descriptors 4 The explanations are clear and coherent and the significance of 3 all the landmarks are justified. 2 1 The explanations are clear and coherent but the significance of the landmarks are not well justified. The explanations are not so clear and coherent and the significance of the landmarks are not well justified. The explanations are not clear and coherent and the significance of the landmarks are not justified. 250 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Rubric on Problems Formulated and SolvedScore Descriptors 6 Poses a more complex problem with 2 or more correct possible solutions and communicates ideas unmistakably, shows in-depth 5 comprehension of the pertinent concepts and/or processes, and 4 provides explanations wherever appropriate 3 2 Poses a more complex problem and finishes all significant parts 1 of the solution and communicates ideas unmistakably, shows in- depth comprehension of the pertinent concepts and/or processes Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension Poses a problem but demonstrates minor comprehension, not being able to develop an approach.DEPED COPYSource: D.O. #73, s. 2012 In this section, your task was to make a sketch of a map on acoordinate plane and determine the coordinates of some importantlandmarks. Then using the coordinates assigned to the different landmarks,you were asked to formulate, then, solve problems involving the distanceformula and the midpoint formula. How did you find the performance task? How did the task help yourealize the importance of the topic in real life?SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about the distance formula, the midpoint formula, andcoordinate proofs and their applications in real life. The lesson provided youwith opportunities to find the distance between two points or places, provegeometric relationships using the distance formula, and formulate and solvereal-life problems. Your understanding of this lesson and other previously learnedmathematics concepts and principles will facilitate your learning of the next lesson,Equation of a Circle. 251 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Start Lesson 2 of this module by relating and connecting previouslylearned mathematical concepts to the new lesson, the equation of acircle. As you go through this lesson, think of this important question:“How does the equation of a circle facilitate finding solutions to real-lifeproblems and making wise decisions?” To find the answer, perform eachactivity. If you find any difficulty in answering the exercises, seek theassistance of your teacher or peers or refer to the modules you havestudied earlier. You may check your work with your teacher.DEPED COPYActivity 1:Determine the number that must be added to make each of the following aperfect square trinomial. Then, express each as a square of a binomial.Answer the questions that follow.1. x 2  4x  _________ 6. w 2  9w  _________2. t 2  10t  _________ 7. x 2 11x  _________3. r 2 14r  _________ 8. v 2  25v  _________4. r 2  22r  _________ 9. s 2  1 s  _________5. x 2  36x  _________ 3 10. t 2  3 t  _________ 4Questions:a. How did you determine the number that must be added to each expression to produce a perfect square trinomial?b. How did you express each resulting perfect square trinomial as a square of a binomial?c. Suppose you are given a square of a binomial. How will you express it as a perfect square trinomial? Give 3 examples. 252 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Was it easy for you to determine the number that must be added to the given terms to make each a perfect square trinomial? Were you able to express a perfect square trinomial as a square of a binomial and vice- versa? Completing the square is a prerequisite to your lesson, Equation of a Circle. Do you know why? Find this out as you go through the lesson. Activity 2: Use the situation below to answer the questions that follow. An air traffic controller (the person who tells the pilot where a plane needs to go using coordinates on the grid) reported that the airport is experiencing air traffic due to the big number of flights that are scheduled to arrive. He advised the pilot of one of the airplanes to move around the airport for the meantime to give way to the other planes to land first. The air traffic controller further told the pilot to maintain its present altitude or height from the ground and its horizontal distance from the origin, point P(0, 0). Airplane Air Traffic Controller 253 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY1. Suppose the plane is located at a point whose coordinates are (30, 40) and each unit on the air traffic controller’s grid is equivalent to 1 km. How far is the plane from the air traffic controller? Explain your answer. 2. What would be the y-coordinate of the position of the plane at a particular instance if its x-coordinate is 5? 10? 15? -20? -30? Explain your answer. 3. Suppose that the pilot strictly follows the advice of the air traffic controller. Is it possible for the plane to be at a point whose x- coordinate is 60? Why? 4. How would you describe the path of the plane as it goes around the airport? What equation do you think would define this path? Were you able to describe the path of the plane and its location as it goes around the air traffic controller’s position? Were you able to determine the equation defining the path? How is the given situation related to the new lesson? You will find this out as you go through this lesson. Activity 3:Perform the following activities. Answer the questions that follow.A. On the coordinate plane below, use a compass to draw a circle with center at the origin and which passes through A(8, 0). y x 254 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY 1. How far is point A from the center of the circle? Explain how you arrived at your answer. 2. Does the circle pass through (0, 8)? How about through (–8, 0)? (0, –8)? Explain your answer. 3. Suppose another point M(–4, 6) is on the coordinate plane. Is M a point on the circle? Why? How about N(9, –2)? Explain your answer. 4. What is the radius of the circle? Explain how you arrived at your answer. 5. If a point is on the circle, how is its distance from the center related to the radius of the circle? 6. How will you find the radius of the circle whose center is at the origin? B. On the coordinate plane below, use a compass to draw a circle with center at (3, 1) and which passes through C(9, –4). y x 1. How far is point C from the center of the circle? Explain how you arrived at your answer. 2. Does the circle pass through (–2, 7)? How about through (8, 7)? (–3, –4)? Explain your answer. 3. Suppose another point M(–7, 6) is on the coordinate plane. Is M a point on the circle? Why? 4. What is the radius of the circle? Explain how you arrived at your answer. 5. How will you find the radius of the circle whose center is not at the origin? 255 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Were you able to determine if a circle passes through a givenpoint? Were you able to find the radius of a circle given the center? Whatequation do you think would relate the radius and the center of a circle?Find this out as you go through the lesson. How did you find the preceding activities? Are you ready to learnabout the equation of a circle? I am sure you are! From the activities you have done, you were able to find the squareof a binomial, a mathematics skill that is needed in understanding theequation of a circle. You were also able to find out how circles areillustrated in real life. You were also given the opportunity to find theradius of a circle and determine if a point is on the circle or not. But howdoes the equation of a circle help in solving real-life problems and inmaking wise decisions? You will find these out in the succeeding activities.Before doing these activities, read and understand first some importantnotes on the equation of a circle and the examples presented.DEPED COPYThe Standard Form of the Equation of a Circle The standard equation of a circle with center at (h, k) and a radius of runits is x  h 2   y  k 2  r 2 . The values of h and k indicate that the circleis translated h units horizontally and k units vertically from the origin.If the center of the circle is at the origin, the equation of the circle isx2 y2 r2.x  h 2   y  k 2  r 2 y Px,y  y Qx,y  x2 y2 r2 r r (h,k) x x (0,0)Circle with center at (h, k) Circle with center at the origin 256 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Example 1: The equation of a circle with center at (2, 7) and a radius of 6 units is x  22   y  72  62 or x  22   y  72  36 .Example 2: The equation of a circle with center at (–5, 3) and a radius of 12 units is x  52   y  32  122 orExample 3: x  52   y  32  144 . The equation of a circle with center at (–4, –9) and a radius of 8 units is x  42   y  92  82 orDEPED COPY x  42   y  92  64 .Example 4: The equation of a circle with center at the origin and a radius of 4 units is x 2  y 2  42 or x 2  y 2  16 .Example 5: The equation of a circle with center at the origin and a radius of 15 units is x 2  y 2  152 or x 2  y 2  225 .The General Equation of a Circle The general equation of a circle is x 2  y 2  Dx  Ey  F  0 , where D,E, and F are real numbers. This equation is obtained by expanding thestandard equation of a circle, x  h 2   y  k 2  r 2 .x  h 2   y  k 2  r 2     x 2  2hx  h 2  y 2  2ky  k 2  r 2 x 2  2hx  h 2  y 2  2ky  k 2  r 2 x 2  y 2  2hx  2ky  h 2  k 2  r 2 x 2  y 2  2hx  2ky  h 2  k 2  r 2  0If D  2h, E  2k ,and F  h 2  k 2  r 2, the equationx 2  y 2  2hx  2ky  h 2  k 2  r 2  0 becomesx 2  y 2  Dx  Ey  F  0. 257 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Example: Write the general equation of a circle with center C(4, –1) and a radius of 7 units. Then determine the values of D, E, and F. The center of the circle is at (h, k), where h = 4 and k = –1. Substitute these values in the standard form of the equation of a acircle together with the length of the radius r which is equal to 7 units. x  h 2   y  k 2  r 2  x  42   y  12  72 Simplify x  42   y  12  72 .DEPED COPY    x  42   y  12  72  x 2  8x 16  y 2  2y 1  49 x 2  8x  16  y 2  2y 1 49 x 2  y 2  8x  2y  17  49 x 2  y 2  8x  2y 17  49  0 x 2  y 2  8x  2y  32  0 Answer: x 2  y 2  8x  2y  32  0 is the general equation of the circle with center C(4, –1) and radius of 7 units. In the equation, D = –8, E = 2, and F = –32.Finding the Center and the Radius of a Circle Given the Equation The center and the radius of a circle can be found given the equation.To do this, transform the given equation to its standard formx  h 2   y  k 2  r 2 if the center of the circle is h,k  , or x 2  y 2  r 2 ifthe center of the circle is the origin. Once the center and the radius of thecircle are found, its graph can be shown on the coordinate plane. Example 1: Find the center and the radius of the circle x 2  y 2  64, and then draw its graph. 258 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Solution: The equation of the circle x 2  y 2  64 has its center at the origin. Hence, it can be transformed to the form. x2 y2 r2. x 2  y 2  64  x 2  y 2  82 The center of the circle is (0, 0) and its radius is 8 units. Its graph is shown below. y r=8DEPED COPY xExample 2: Determine the center and the radius of the circle x  22   y  42  25, and draw its graph.Solution: The equation of the circle x  22   y  42  25 can be written in the form x  h 2   y  k 2  r 2 . x  22   y  42  25  x  22   y  42  52 The center of the circle is (2, 4) and its radius is 5 units. Its graph is shown below. y r=5 x 259 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Example 3: What is the center and the radius of the circle x 2  y 2  6x 10y  18  0 ? Show the graph.Solution: The equation of the circle x 2  y 2  6x 10y  18  0 is written in general form. To determine its center and radius, write the equation in the form x  h 2   y  k 2  r 2 . x 2  y 2  6x 10y  18  0  x 2  6x  y 2 10y  18 Add to both sides of the equation x 2  6x  y 2 10y  18 the square of one-half the coefficient of x and the square of one-half the coefficient of y.DEPED COPY 1 6  3 ; 32  9 1  10  5 ;  5 2  25 2 2 Simplify x 2  6x  9  y 2 10y  25  18  9  25 . x 2  6x  9  y 2 10y  25  16    x 2  6x  9  y 2 10y  25  16 Rewriting, we obtain x  32   y  52  16 or x  32   y  52  42 The center of the circle is at (3, 5) and its radius is 4 units. y r=4 x 260 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Example 4: What is the center and the radius of the circle 4x2 + 4y2 + 12x – 4y – 90 = 0? Show the graph.Solution: 4x2 + 4y2 + 12x – 4y – 90 = 0 is an equation of a circle that is written in general form. To determine its center and radius, write the equation in the form x  h 2   y  k 2  r 2 . 4x 2  4y 2  12x  4y  90  0 or 4x 2  4y 2  12x  4y  90 Divide both sides of the equation by 4. 4x 2  4y 2  12x  4y  90  4x 2  4y 2  12x  4y  90 44  x 2  y 2  3x  y  90 4 Add on both sides of the equation x 2  y 2  3x  y  90 4 the square of one-half the coefficient of x and the square of one-half the coefficient of y.DEPED COPY 13  3 ;  3 2  9 1 1   1 ;   1 2  1  2  4   4 22   22  2  Simplify x 2  3x  9  y 2  y  1  90  9  1 . 4 4 4 44  100 4 x 2  3x  9  y 2  y  1  25 44 Rewriting, we have  x  3 2   y  1 2  25 .      2   2  261 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Write the equation  x  3 2   y  1 2 25 in the form  2   2        x  h2 2 r 2, that is  x 3 2  y 1 2 52 y k      2  2     The center of the circle is at   3 , 1 and its radius is  2   2 5 units. yDEPED COPY r=5 xLearn more about the http://www.mathopenref.com/coordbasiccircle.htmlEquation of a Circle throughthe WEB. You may open the http://www.mathopenref.com/coordgeneralcircle.htmlfollowing links. https://www.khanacademy.org/math/geometry/cc- geometry-circles/equation-of-a-circle/v/equation-for- a-circle-using-the-pythagorean-theorem http://www.math-worksheet.org/using-equations-of- circles 262 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Your goal in this section is to apply the key concepts of the equationof a circle. Use the mathematical ideas and the examples presented inthe preceding section to perform the activities that follow.Activity 4:DEPED COPYDetermine the center and the radius of the circle that is defined by each of thefollowing equations. Then graph each circle on a coordinate plane (or useGeoGebra to graph each). Answer the questions that follow.1. x2  y 2 49 2.  x  52  y  62 81 yy xx 263 All rights reserved. No part of this material may be reproduced or 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. x2  y 2  100 4.  x  72  y 12  49 y y xxDEPED COPY5. x 2  y 2  8x  6y  39  06. x2  y 2 10x 16y  32  0 y y\ xxQuestions:a. How did you determine the center of each circle? How about the radius?b. How do you graph circles that are defined by equations of the formx2 y2 r2? x  h 2   y  k 2  r 2 ?x 2  y 2  Dx  Ey  F  0 ? 264 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

How was the activity? Did it challenge you? Were you able todetermine the center and the radius of the circle? I am sure you were! Inthe next activity, you will write the equation of the circle as described.Activity 5:Write the equation of each of the following circles given the center and theradius. Answer the questions that follow. Center Radius 1. origin 12 units 2. (2, 6) 9 units 3. (–7, 2) 15 units 4. (–4, –5) 5 2 units 5. (10, –8)DEPED COPY 3 3 unitsQuestions:a. How do you write the equation of a circle, given its radius, if the center is at the origin?b. How about if the center is not at (0, 0)?c. Suppose two circles have the same center. Should the equations defining these circles be the same? Why? Were you able to write the equation of the circle given its radius andits center? I know you were! In the next activity, you will write theequation of a circle from standard to general form.Activity 6:Write each equation of a circle in general form. Show your solutionscompletely.1.  x  22  y  42 36 6.  x  72  y 2 642.  x  42  y  92 144 7. x2  y  22 493.  x  62  y 12 81 8.  x  22  y 2 1004. x  82  y  72 225 9.  x  52  y  52 275. x2  y  52 36 10.  x  42  y  42 32 265 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

How did you find the activity? Were you able to write all theequations in their general form? Did the mathematics concepts andprinciples that you previously learned help you in transforming theequations? In the next activity, you will do the reverse. This time, you willtransform the equation of a circle from general to standard form, thendetermine the radius and the center of the circle.Activity 7:In numbers 1 to 6, a general equation of a circle is given. Transform theequation to standard form, then give the coordinates of the center and theradius. Answer the questions that follow.DEPED COPY1. x2  y 2  2x  8y  47 0 4. x2  y 2  8y  84 02. x2  y 2  4x  4y  28 0 5. 9x2  9y2 12x  6y  3103. x2  y 2 10x  4y  3 0 6. 4x2  4y 2  20x 12y  2 0Questions: a. How did you write each general equation of a circle to standard form? b. What mathematics concepts or principles did you apply in transforming each equation to standard form? c. Is there a shorter way of transforming each equation to standard form? Describe this way, if there is any. Were you able to write each equation of a circle from general formto standard form? Were you able find a shorter way of transforming eachequation to standard form? In this section, the discussion was about the equation of a circle, itsradius and center, and the process of transforming the equation from oneform to another. Go back to the previous section and compare your initial ideas withthe discussion. How much of your initial ideas are found in the discussion?Which ideas are different and need modification? Now that you know the important ideas about this topic, let usdeepen your understanding by moving on to the next section. 266 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Your goal in this section is to test further your understanding of theequation of a circle by solving more challenging problems involving thisconcept. After doing the following activities, you should be able to findout how the equations of circles are used in solving real-life problemsand in making decisions.Activity 8:DEPED COPYDetermine which of the following equations describe a circle and which donot. Justify your answer.1. x2  y 2  2x  8y  26 0 3. x2  y 2  6x  8y  32 02. x2  y 2  9  4x 10y 4. x2  y 2  8x 14y  65 0 How was the activity? Were you able to determine which are circlesand which are not? In the next activity, you will further deepen yourunderstanding about the equation of a circle and solve real-life problems.Activity 9:Answer the following. 1. The diameter of a circle is 18 units and its center is at (–3, 8). What is the equation of the circle? 2. Write an equation of the circle with a radius of 6 units and is tangent to the line y  1 at (10, 1). 3. A circle defined by the equation x  62   y  92 34 is tangent to a line at the point (9, 4). What is the equation of the line? 4. A line passes through the center of a circle and intersects it at points (2, 3) and (8, 7). What is the equation of the circle? 5. The Provincial Disaster and Risk Reduction Management Committee (PDRRMC) advised the residents living within the 10 km radius critical area to evacuate due to eminent eruption of a volcano. On the map that is drawn on a coordinate plane, the coordinates corresponding to the location of the volcano is (3, 4). 267 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYa. If each unit on the coordinate plane is equivalent to 1 km, what is the equation of the circle enclosing the critical area? b. Suppose you live at point (11, 6). Would you follow the advice of the PDRRMC? Why? c. In times of eminent disaster, what precautionary measures should you take to be safe? d. Suppose you are the leader of a two-way radio team with 15 members that is tasked to give warnings to the residents living within the critical area. Where would you position each member of the team who is tasked to inform the other members as regards the current situation and to warn the residents living within his/her assigned area? Explain your answer. 6. Cellular phone networks use towers to transmit calls to a circular area. On a grid of a province, the coordinates that correspond to the location of the towers and the radius each covers are as follows: Wise Tower is at (–5, –3) and covers a 9 km radius; Global Tower is at (3, 6) and covers a 4 km radius; and Star Tower is at (12, –3) and covers a 6 km radius. a. What equation represents the transmission boundaries of each tower? b. Which tower transmits calls to phones located at (12, 2)? (–6, –7)? (2, 8)? (1, 3)? c. If you were a cellular phone user, which cellular phone network will you subscribe to? Why? Did you find the activity challenging? Were you able to answer all the questions and problems involving the equations of circles? I am sure you were! In this section, the discussion was about your understanding of the equation of a circle and their applications in real life. What new realizations do you have about the equation of a circle? How would you connect this to real life? How would you use this in making wise decisions? Now that you have a deeper understanding of the topic, you are ready to do the tasks in the next section. 268 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Your goal in this section is to apply your learning to real-life situations.You will be given a practical task which will demonstrate your understanding ofthe equation of a circle.Activity 10:On a clean sheet of grid paper, paste some small pictures of objects such thatthey are positioned at different coordinates. Then, draw circles that contain thesepictures. Using the pictures and the circles drawn on the grid, formulate andsolve problems involving the equation of the circle, then solve them. Use therubric provided to rate your work.DEPED COPYRubric for a Scrapbook PageScore Descriptors 4 The scrapbook page is accurately made, presentable, and appropriate. 3 The scrapbook page is accurately made and appropriate. 2 The scrapbook page is not accurately made but appropriate. 1 The scrapbook page is not accurately made and not appropriate.Rubric on Problems Formulated and SolvedScore Descriptors 6 5 Poses a more complex problem with 2 or more correct possible 4 solutions and communicates ideas unmistakably, shows in-depth 3 comprehension of the pertinent concepts and/or processes, and 2 provides explanations wherever appropriate. 1 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes all significant parts of the solution and communicates ideas unmistakably, shows in-depth comprehension of the pertinent concepts and/or processes. Poses a complex problem and finishes most significant parts of the solution and communicates ideas unmistakably, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details. Poses a problem and finishes some significant parts of the solution and communicates ideas unmistakably but shows gaps on theoretical comprehension. Poses a problem but demonstrates minor comprehension, not being able to develop an approach.Source: D.O. #73, s. 2012 269 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

In this section, your task was to formulate problems involving the equationof a circle using the pictures of objects that you positioned on a grid. How did you find the performance task? How did the task help you realizethe importance of the topic in real life?SUMMARY/SYNTHESIS/GENERALIZATION This lesson was about the equations of circles and their applications inreal life. The lesson provided you with opportunities to give the equations ofcircles and use them in practical situations. Moreover, you were given thechance to formulate and solve real-life problems. Understanding this lesson andrelating it to the mathematics concepts and principles that you have previouslylearned is essential in any further work in mathematics.DEPED COPYGLOSSARY OF TERMSCoordinate Proof – a proof that uses figures on a coordinate plane to provegeometric relationshipsDistance Formula – an equation that can be used to find the distancebetween any pair of points on the coordinate plane. The distance formula is        d  x 2  x1 2  y 2  y1 2 or PQ  x 2  x1 2  y 2  y1 2 , if P x1, y1 and Q x 2, y 2 are points on a coordinate plane.Horizontal Distance (between two points) – the absolute value of thedifference of the x-coordinates of two pointsMidpoint – a point on a line segment and divides the same segment into twoequal partsMidpoint Formula – a formula that can be used to find the coordinates of themidpoint of a line segment on the coordinate plane. The midpoint of   P  x1  x2 , y1  y2 x1, y 1 and Q x 2, y 2 is  2 2 .  The General Equation of a Circle – the equation of a circle obtained byexpanding x  h 2   y  k 2  r 2 . The general equation of a circle isx 2  y 2  Dx  Ey  F  0 , where D, E, and F are real numbers. 270 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYThe Standard Equation of a Circle – the equation that defines a circle with center at (h, k) and a radius of r units. It is given by x  h 2   y  k 2  r 2 . Vertical Distance (between two points) – the absolute value of the difference of the y-coordinates of two points DepEd Instructional Materials that can be used as additional resources: 1. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third Year Mathematics. Plane Coordinate Geometry. Module 20: Distance and Midpoint Formulae 2. Basic Education Assistance for Mindanao (BEAM) Learning Guide, Third Year Mathematics. Plane Coordinate Geometry. Module 22: Equation of a Circle 3. Distance Learning Module (DLM) 3, Module 3: Plane Coordinate Geometry. 4. EASE Modules Year III, Module 2: Plane Coordinate Geometry REFERENCES AND WEBSITE LINKS USED IN THIS MODULE: References: Bass, Laurie E., Randall I. Charles, Basia Hall, Art Johnson, and Dan Kennedy. Texas Geometry. Pearson Prentice Hall, Boston, Massachusetts 02116, 2008. Bass, Laurie E., Basia Rinesmith Hall, Art Johnson, and Dorothy F. Wood. Prentice Hall Geometry Tools for a Changing World. Prentice-Hall, Inc., NJ, USA, 1998. Boyd, Cummins, Malloy, Carter, and Flores. Glencoe McGraw-Hill Geometry. The McGraw-Hill Companies, Inc., USA, 2008. Callanta, Melvin M. Infinity, Worktext in Mathematics III. EUREKA Scholastic Publishing, Inc., Makati City, 2012. Chapin, Illingworth, Landau, Masingila and McCracken. Prentice Hall Middle Grades Math, Tools for Success, Prentice-Hall, Inc., Upper Saddle River, New Jersey, 1997. 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.

DEPED COPYCifarelli, Victor, et al. cK-12 Geometry, Flexbook Next Generation Textbooks, Creative Commons Attribution-Share Alike, USA, 2009.Clemens, Stanley R., Phares G. O’Daffer, Thomas J. Cooney, and John A. Dossey. Addison-Wesley Geometry. Addison-Wesley Publishing Company, Inc., USA, 1990.Clements, Douglas H., Kenneth W. Jones, Lois Gordon Moseley, and Linda Schulman. Math in my World, McGraw-Hill Division, Farmington, New York, 1999.Department of Education. K to 12 Curriculum Guide Mathematics, Department of Education, Philippines, 2012.Gantert, Ann Xavier. AMSCO’s Geometry. AMSCO School Publications, Inc., NY, USA, 2008.Renfro, Freddie L. Addison-Wesley Geometry Teacher’s Edition. Addison- Wesley Publishing Company, Inc., USA, 1992.Rich, Barnett and Christopher Thomas. Schaum’s Outlines Geometry Fourth Edition. The McGraw-Hill Companies, Inc., USA, 2009.Smith, Stanley A., Charles W. Nelson, Roberta K. Koss, Mervin L. Keedy, and Marvin L. Bittinger. Addison-Wesley Informal Geometry. Addison- Wesley Publishing Company, Inc., USA, 1992.Wilson, Patricia S., et al. Mathematics, Applications and Connections, Course I, Glencoe Division of Macmillan/McGraw-Hill Publishing Company, Westerville, Ohio, 1993.Website Links as References and Sources of Learning Activities:CliffsNotes. Midpoint Formula. (2013). Retrieved June 29, 2014, fromhttp://www.cliffsnotes.com/math/geometry/coordinate-geometry/midpoint-formulaCliffsNotes. Distance Formula. (2013). Retrieved June 29, 2014, fromhttp://www.cliffsnotes.com/math/geometry/coordinate-geometry/distance-formulaMath Open Reference. Basic Equation of a Circle (Center at 0,0). (2009).Retrieved June 29, 2014, from http://www.mathopenref.com/coordbasiccircle.html 272 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYMath Open Reference. Equation of a Circle, General Form (Center anywhere). (2009). Retrieved June 29, 2014, from http://www.mathopenref.com/coordgeneralcircle.html Math-worksheet.org. Using equations of circles. (2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/using-equations-of-circles Math-worksheet.org. Writing equations of circles. (2014). Retrieved June 29, 2014, from http://www.math-worksheet.org/writing-equations-of-circles Roberts, Donna. Oswego City School District Regents exam Prep Center. Geometry Lesson Page. Midpoint of a Line Segment. (2012). Retrieved June 29, 2014, from http://www.regentsprep.org/Regents/ math/geometry/GCG2/ Lmidpoint.htm Roberts, Donna. Oswego City School District Regents exam Prep Center. Geometry Lesson Page. Midpoint of a Line Segment. (2012). Retrieved June 29, 2014, from http://www.regentsprep.org/Regents/math/geometry/GCG3/ Ldistance.htm Stapel, Elizabeth. \"Conics: Circles: Introduction & Drawing.\" Purplemath. Retrieved June 29, 2014, from http://www.purplemath.com/modules/ circle.htm Website Links for Videos: Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved June 29, 2014, from https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the- pythagorean-theorem Khan Academy. Completing the square to write equation in standard form of a circle. Retrieved June 29, 2014, from https://www.khanacademy.org/math/ geometry/cc-geometry-circles/equation-of-a-circle/v/completing-the-square-to- write-equation-in-standard-form-of-a-circle Khan Academy. Equation for a circle using the Pythagorean Theorem. Retrieved June 29, 2014, from https://www.khanacademy.org/math/geometry/ cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the- pythagorean-theorem 273 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYKhan Academy. Equation for a circle using the Pythagorean Theorem.Retrieved June 29, 2014, from https://www.khanacademy.org/math/geometry/cc-geometry-circles/equation-of-a-circle/v/equation-for-a-circle-using-the-pythagorean-theoremUkmathsteacher. Core 1 – Coordinate Geometry (3) – Midpoint and distanceformula and Length of Line Segment. Retrieved June 29, 2014, fromhttp://www.youtube.com/watch?v=qTliFzj4wucVividMaths.com. Distance Formula. Retrieved June 29, 2014, fromhttp://www.youtube.com/watch?v=QPIWrQyeuYwWebsite Links for Images:asiatravel.com. Pangasinan Map. Retrieved June 29, 2014, fromhttp://www.asiatravel.com/philippines/pangasinan/pangasinanmap.jpgDownTheRoad.org. Pictures of, Chengdu to Kangding, China Photo, Images,Picture from. (2005). Retrieved June 29, 2014, fromhttp://www.downtheroad.org/Asia/Photo/9Sichuan_China_Image/3Chengdu_Kangding_China.htmHugh Odom Vertical Consultants. eleven40 theme on Genesis Framework ·WordPress. Cell Tower Development – How Are Cell Tower LocationsSelected? Retrieved June 29, 2014, from http://blog.thebrokerlist.com/cell-tower-development-how-are-cell-tower-locations-selected/LiveViewGPS, Inc. GPS Tracking PT-10 Series. (2014). Retrieved June29, 2014, from http://www.liveviewgps.com/gps+tracking+device+pt-10+series.htmlSloan, Chris. Current \"1991\" Air Traffic Control Tower at Amsterdam SchipholAirport – 2012. (2012). Retrieved June 29, 2014, fromhttp://airchive.com/html/airplanes-and-airports/amsterdam-schipol-airport-the-netherlands-/current-1991-air-traffic-control-tower-at-amsterdam-schiphol-airport-2012-/25510wordfromthewell.com. Your Mind is Like an Airplane. (2012). Retrieved June29, 2014, from http://wordfromthewell.com/2012/11/14/your-mind-is-like-an-airplane/ 274 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY 10 Mathematics Learner’s Module Unit 3 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 [email protected]. We value your feedback and recommendations. Department of Education Republic of the Philippines All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Mathematics – Grade 10Learner’s ModuleFirst Edition 2015 Republic Act 8293, section 176 states that: No copyright shall subsist in any workof the Government of the Philippines. However, prior approval of the government agency oroffice 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 ofroyalties. 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 seekingpermission to use these materials from their respective copyright owners. All means havebeen exhausted in seeking permission to use these materials. The publisher and authors donot represent nor claim ownership over them. Only institution and companies which have entered an agreement with FILCOLS andonly within the agreed framework may copy this Learner’s Module. Those who have notentered in an agreement with FILCOLS must, if they wish to copy, contact the publisher andauthors directly. Authors and publishers may email or contact FILCOLS at [email protected] or (02)439-2204, respectively.Published by the Department of EducationSecretary: Br. Armin A. Luistro FSCUndersecretary: Dina S. Ocampo, PhDDEPED COPY Development Team of the Learner’s ModuleConsultants: Soledad A. Ulep, PhD, Debbie Marie B. Verzosa, PhD, andRosemarievic Villena-Diaz, PhDAuthors: 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. TernidaEditor: Maxima J. Acelajado, PhDReviewers: 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, PhDIllustrator: Cyrell T. NavarroLayout Artists: Aro R. Rara and Ronwaldo Victor Ma. A. PagulayanManagement and Specialists: Jocelyn DR Andaya, Jose D. Tuguinayo Jr.,Elizabeth G. Catao, Maribel S. Perez, and Nicanor M. San Gabriel Jr.Printed in the Philippines by REX Book StoreDepartment of Education-Instructional Materials Council Secretariat (DepEd-IMCS)Office Address: 5th Floor Mabini Building, DepEd Complex Meralco Avenue, Pasig City Philippines 1600Telefax: (02) 634-1054, 634-1072E-mail Address: [email protected] All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY 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.

DEPED COPY Table of Contents Unit 3 Module 6: Permutations and Combinations ....................................... 275 Lessons and Coverage........................................................................ 276 Module Map......................................................................................... 277 Pre-Assessment .................................................................................. 278 Learning Goals and Targets ................................................................ 282 Lesson 1: Permutations .............................................................................. 283 Activity 1 .................................................................................... 283 Activity 2 .................................................................................... 284 Activity 3 .................................................................................... 286 Activity 4 .................................................................................... 286 Activity 5 .................................................................................... 295 Activity 6 .................................................................................... 296 Activity 7 .................................................................................... 297 Activity 8 .................................................................................... 298 Activity 9 .................................................................................... 299 Activity 10 .................................................................................. 299 Summary/Synthesis/Generalization........................................................... 300 Lesson 2: Combinations.............................................................................. 301 Activity 1 .................................................................................... 301 Activity 2 .................................................................................... 303 Activity 3 .................................................................................... 304 Activity 4 .................................................................................... 310 Activity 5 .................................................................................... 311 Activity 6 .................................................................................... 311 Activity 7 .................................................................................... 312 Activity 8 .................................................................................... 314 Activity 9 .................................................................................... 315 Activity 10 .................................................................................. 315 Summary/Synthesis/Generalization ........................................................... 316 Glossary of Terms ....................................................................................... 317 References and Website Links Used in this Module.................................. 317 Module 7: Probability of Compound Events ...................................... 319 Lessons and Coverage ........................................................................ 320 Module Map ......................................................................................... 320 Pre-Assessment................................................................................... 321 Learning Goals and Targets ................................................................. 326 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYLesson 1: Union and Intersection of Events ............................................... 328 Activity 1 .................................................................................... 328 Activity 2 .................................................................................... 330 Activity 3 .................................................................................... 332 Activity 4 .................................................................................... 334 Activity 5 .................................................................................... 336 Activity 6 .................................................................................... 337 Activity 7 .................................................................................... 338 Activity 8 .................................................................................... 339 Activity 9 .................................................................................... 340 Summary/Synthesis/Generalization ........................................................... 340 Lesson 2: Independent and Dependent Events .......................................... 341 Activity 1 .................................................................................... 341 Activity 2 .................................................................................... 343 Activity 3 .................................................................................... 344 Activity 4 .................................................................................... 344 Activity 5 .................................................................................... 345 Activity 6 .................................................................................... 345 Summary/Synthesis/Generalization ........................................................... 345 Lesson 3: Conditional Probability.............................................................. 346 Activity 1 .................................................................................... 346 Activity 2 .................................................................................... 347 Activity 3 .................................................................................... 348 Activity 4 .................................................................................... 350 Activity 5 .................................................................................... 351 Activity 6 .................................................................................... 351 Activity 7 .................................................................................... 352 Summary/Synthesis/Generalization........................................................... 352 Glossary of Terms ...................................................................................... 353 References and Website Links Used in this Module ................................. 354 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPYI. INTRODUCTION Look at the pictures shown below. Have you ever wondered why some locks such as the one shown below have codes in them? Do you know why a shorter code is “weak,” while a longer code is a “strong” personal password in a computer account? Have you ever realized that there are several possible ways in doing most tasks or activities like planning a seating arrangement or predicting the possible outcomes of a race? Have you ever been aware that there are numerous possible choices in selecting from a set, like deciding which combination of dishes to serve in a catering service or deciding which dishes to order in a menu? Did you know that awareness of these can help you form conclusions and make wise decisions? Find out the answers to these questions and discover the wide applications of permutations and combinations through this module. 275 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

II. LESSONS AND COVERAGE In this module, you will examine and determine the number of possibleways of doing certain tasks, or selecting some objects from a set. You willlearn about these through the following lessons:Lesson 1 – PermutationsLesson 2 – CombinationsIn these lessons, you will learn to:Lesson 1DEPED COPY illustrate permutation of objects;Lesson 2  derive the formula for finding the number of permutations of n objects taken r at a time, n ≥ r ; and  solve problems involving permutations.  illustrate the combination of n objects;  differentiate permutation from combination of n objects taken r at a time, n ≥ r ;  derive the formula for finding the number of combinations of n objects taken r at a time; and  solve problems involving permutations and combinations. 276 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Here is a simple map of the lessons that will be covered in this module:COMBINATORICS PermutationsDEPED COPY Finding the Permutations of n Objects Taken r at a Time Problems Involving Permutations Finding the Combinations of n Objects Taken r at a TimeCombinations Problems Involving Combinations Problems Involving Permutations and Combinations 277 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

III. PRE-ASSESSMENT Part I Find out how much you already know about the topics in this module. Choose the letter that you think best answers the question. Please answer all the items. 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. Choosing a subset of a set is an example of ___.A. combination C. integrationB. differentiation D. permutation2. Which of the following situations or activities involve permutation? A. matching shirts and pants B. forming different triangles out of 5 points on a plane, no three of which are collinear C. assigning telephone numbers to subscribers D. forming a committee from the members of a clubDEPED COPY3. The product of a positive integer n and all the positive integers lessthan it is _____. C. n - factorsA. powers of n D. n factorialB. multiples of n4. Two different arrangements of objects where some of them areidentical are called _______.A. distinguishable permutations C. circular permutationsB. unique combinations D. circular combinations5. How many different 4-digit even numbers can be formed from the digits1, 3, 5, 6, 8, and 9 if no repetition of digits is allowed?A. 1 680 B. 840 C. 420 D. 1206. In how many ways can 8 people be seated around a circular table iftwo of them insist on sitting beside each other?A. 360 B. 720 C. 1440 D. 50407. Find the number of distinguishable permutations of the letters of theword PASS.A. 4 B. 12 C. 36 D. 1448. Ms. Santos asked Renz to draw all the diagonals of a certain polygonon the blackboard. Renz was able to draw 27 diagonals which histeacher declared correct. What was the given polygon?A. pentagon C. nonagonB. hexagon D. decagon 278 All rights reserved. No part of this material may be reproduced or 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. Ms. De Leon wants to produce different sets of test questions for heressay test. If she plans to do this by putting together 3 out of 5questions she prepared, how many different sets of questions couldshe construct?A. 10 B. 20 C. 60 D. 8010. If P(9, r) = 3024, what is r ?A. 2 B. 4 C. 5 D. 611. In a town fiesta singing competition with 12 contestants, in how manyways can the organizer arrange the first three singers?A. 132 B. 990 C. 1320 D. 1716DEPED COPY12. What is P(8, 5)?B. 336 C. 1400 D. 6720 A. 5613. If P(n, 4) = 5040, then n = ____.A. 12 B. 10 C. 9 D. 814. Given x = P(n, n) and y = P(n, n – 1), what can be concluded about xand y?A. x > y B. x < y C. x = y D. x = – y15. Find the number of distinguishable permutations of the letters of theword EDUCATED.A. 1680 B. 10 080 C. 20 160 D. 40 32016. If a combination lock must contain 5 different digits, in how many wayscan a code be formed from the digits 0 to 9?A. 15 120 B. 30 240 C. 151 200 D. 1 000 00017. In how many ways can 4 men and 3 women arrange themselves in arow for picture taking if the men and women must stand in alternatepositions?A. 5040 B. 720 C. 144 D. 3018. In a room, there are 10 chairs in a row. In how many ways can 5students be seated in consecutive chairs?A. 720 B. 600 C. 252 D. 12019. Which of the following situations does NOT illustrate combination? A. Selecting 2 songs from 10 choices for an audition piece B. Fixing the schedule of a group of students who must take exactly 8 subjects C. Enumerating the subsets of a set D. Identifying the lines formed by connecting some given points on a plane 279 All rights reserved. No part of this material may be reproduced or 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. If w = C(5, 2), x = C(5, 3), y = C(5, 4), and z = C(5, 5), and we aregiven 5 points on a plane of which no three are collinear, whichexpression gives the total number of polygons that can be drawn?A. x + y C. x + y + zB. w + x + y D. w + x + y + z21. C(n, n) = ____. B. r C. 1 D. cannot be A. n determined22. If C(n, r) = 35, which of the following are possible values of n and r? A. n = 6, r = 4 B. n = 7, r = 3 C. n = 8, r = 3 D. n = 9, r = 223. If C(n, 4) = 126, what is n?DEPED COPYA. 11 B. 10 C. 9 D. 724. If C(12, r) = 792, which of the following is a possible value of r?A. 8 B. 7 C. 6 D. 425. A caterer offers 3 kinds of soup, 7 kinds of main dish, 4 kinds ofvegetable dish, and 4 kinds of dessert. In how many possible ways cana caterer form a meal consisting of 1 soup, 2 main dishes, 1 vegetabledish, and 2 desserts?A. 140 B. 336 C. 672 D. 151226. In how many ways can a committee of 7 students be chosen from 9juniors and 9 seniors if there must be 4 seniors in the committee?A. 10 584 B. 1764 C. 210 D. 8427. Jane wants to solve a system of equations through elimination bycombining any two equations. The number of equations she has isequal to the number of variables. She realizes that she has 10 possibleways to start her solution. How many equations does she have?A. 6 B. 5 C. 4 D. 328. There are 11 different food items in a buffet. A customer is asked to get a certain number of items. If the customer has 462 possible ways as a result, which of the following did he possibly do? A. Choose 4 out of the 11 items B. Choose 6 out of the 11 items C. Choose 8 out of the 11 items D. Choose 7 out of the 11 items 280 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Part II Read and understand the situation below, and then answer the question or perform what is required. Suppose you had graduated from high school but did not have enough money to continue your college education. You decided to work temporarily and save for your schooling. You applied at Mr. Aquino’s restaurant and were hired. After a few days, you noticed that the restaurant business was not doing very well, and Mr. Aquino asked for your opinion. What you noticed was that there was no variety in the food being served in the restaurant. 1. Prepare a list of different choices of food that may be served (soup, meat/chicken dishes, fish, vegetables, fruits, desserts, beverages). Consider health and nutritional values. 2. What mathematical concepts are utilized in this given situation? 3. Formulate two problems involving the mathematical concepts in the situation. 4. Write the equation(s) or expressions that describe the situation. 5. Solve the equations and problems formulated. 6. Present a sample menu for the day and explain the reason for your choice. In view of the fact that there will be several possible combinations, explain also why you should prepare certain dishes more often or less frequently. 281 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

Rubric on Problems Formulated and SolvedScore Descriptors 6 Poses a more complex problem with two or more correct 5 possible solutions and communicates ideas accurately, 4 shows in-depth comprehension of the pertinent concepts 3 and/or processes, and provides explanations wherever 2 appropriate. 1 Poses a more complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes Poses a complex problem and finishes all significant parts of the solution and communicates ideas accurately, shows in-depth comprehension of the pertinent concepts and/or processes Poses a complex problem and finishes most significant parts of the solution and communicates ideas accurately, shows comprehension of major concepts although neglects or misinterprets less significant ideas or details Poses a problem and finishes some significant parts of the solution and communicates ideas accurately, but shows gaps on theoretical comprehension Poses a problem but demonstrates little comprehension, not being able to develop an approachDEPED COPYSource: D.O. #73, s. 2012IV. LEARNING GOALS AND TARGETS After going through this module, you should be able to demonstrate understanding of key concepts of combinatorics, particularly permutations and combinations. Also, you should be able to use precise counting techniques in formulating conclusions and making decisions. 282 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.

DEPED COPY Start Lesson 1 of this module by assessing your knowledge of the basic counting technique called the Fundamental Counting Principle. This knowledge and skill will help you understand permutations of objects. As you go through this lesson, keep in mind this important question: How does the concept of permutation help in forming conclusions and in making wise decisions? To be able to answer this, perform each activity that follows. Seek the assistance of your teacher and peers if you encounter any difficulty. Have your work checked by your teacher. Activity 1: A. A close friend invited Anna to her birthday party. Anna has 4 new blouses (stripes, with ruffles, long-sleeved, and sleeveless) and 3 skirts (red, pink, and black) in her closet reserved for such occasions. 1. Assuming that any skirt can be paired with any blouse, in how many ways can Anna select her outfit? List the possibilities. 2. How many blouse-and-skirt pairs are possible? 3. Show another way of finding the answer in item 1. B. Suppose you secured your bike using a combination lock. Later, you realized that you forgot the 4-digit code. You only remembered that the code contains the digits 1, 3, 4, and 7. 1. List all the possible codes out of the given digits. 2. How many possible codes are there? 3. What can you say about the list you made? 283 All rights reserved. No part of this material may be reproduced or transmitted in any form or by any means -electronic or mechanical including photocopying – without written permission from the DepEd Central Office. First Edition, 2015.