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LESSON 4.1 Name Circles Class 4.1 Date Circles Essential Question: What is the standard form for the equation of a circle, and what does the standard form tell you about the circle? Common Core Math Standards The student is expected to: COMMON CORE Resource Locker A-CED.A.3 Recall that a circle is the set of points in a plane that are a fixed distance, called the radius, from a given point, called the center. Mathematical Practices COMMON CORE Deriving the Standard-Form Equation of a Circle Explore Represent constraints by equations or inequalities, ... and interpret solutions as viable or nonviable options in a modeling context. Also A-CED.A.2, G-GPE.A.1, G-GPE.B.4 A MP.7 Using Structure Language Objective Work with a partner to match graphs of circles to their equations in standard form. The coordinate plane shows a circle with center C(h, k) and radius r. P(x, y) is an arbitrary point on the circle but is not directly above or below or to the left or right of C. A(x, k) is a point with the same x-coordinate as P and the same y-coordinate as C. Explain why △CAP is a right triangle. y Since point A has the same x-coordinate as point P, segment P (x, y) r C (h, k) A (x, k) PA is a vertical segment. Since point A has the same x y-coordinate as point C, segment CA is a horizontal segment. ENGAGE This means that segments PA and CA are perpendicular, Essential Question: What is the standard form for the equation of a circle, and what does the standard form tell you about the circle? triangle. PREVIEW: LESSON PERFORMANCE TASK View the online Engage. Discuss the photo and the generally circular nature of radio-signal reception strength. Then preview the Lesson Performance Task. B © Houghton Mifflin Harcourt Publishing Company Possible answer: The standard form for the equation 2 2 of a circle is (x - h) + (y - k) = r 2, which tells you that the center is (h, k) and the radius is r. which means that ∠CAP is a right angle and △CAP is a right Identify the lengths of the sides of △CAP. Remember that point P is arbitrary, so you cannot rely upon the diagram to know whether the x-coordinate of P is greater than or less than h or whether the y-coordinate of P is greater than or less than k, so you must use absolute value for the lengths of the legs of △CAP. Also, remember that the length of the hypotenuse of △CAP is just the radius of the circle. ⎜ The length of segment AP is ⎜ C The length of segment AC is x-h The length of segment CP is r ⎟. ⎟. y-k . Apply the Pythagorean Theorem to △CAP to obtain an equation of the circle. (x - h ) + (y - k ) = 2 Module 4 2 2 r be ges must EDIT--Chan DO NOT Key=NL-A;CA-A Correction Lesson 1 159 gh “File info” made throu Date Class Name Circles HARDCOVER does and what ion of a circle, for the equat Resource ard form the circle? or Locker is the stand ns as viable tell you about ion: What ard form interpret solutio the stand lities, ... and G-GPE.B.4 or inequa G-GPE.A.1, by equation A-CED.A.2, constraints n Represent context. Also A-CED.A.3 a modeling m Equatio options in dard-For nonviable Stan the point, Deriving , from a given the radius le Explore called Circ ce, distan of a are a fixed a plane that points in the set of a circle is y Recall that radius r. P (x, y) center. C(h, k) and called the with center above 4.1 Quest Essential COMMON CORE A2_MNLESE385894_U2M04L1.indd 159 directly shows a circle but is not r inate plane the circle with the same The coord ry point on k) is a point n why an arbitra of C. A(x, P(x, y) is as C. Explai left or right (x, k) or to the y-coordinate C (h, k) A or below the same as P and segment x-coordinate le. point P, a right triang rdinate as △CAP is same x-coo same A has the A has the Since point ent. Since point segm ent. ontal al segm CA is a horiz PA is a vertic segment ndicular, point C, as te perpe CA are y-coordina a right ents PA and △CAP is s that segm angle and This mean is a right s that ∠CAP which mean ry, so you P is arbitra triangle. r than that point P is greate . Remember dinate of of △CAP k, so you the sides er the x-coor less than lengths of know wheth r than or ber that the Identify the P is greate diagram to of the . Also, remem dinate upon cannot rely er the y-coor s of the legs of △CAP h or wheth the circle. for the length or less than radius of absolute value of △CAP is just the must use enuse the hypot length of h . x AC is of segment The length y-k . AP is of segment The length r . CP is circle. of segment on of the The length an equati to obtain to △CAP Theorem 2 Pythagorean 2 Apply the 2 = r k + yx- h Turn to Lesson 4.1 in the hardcover edition. x ⎜ ⎜ ⎟ ⎟ © Houghto n Mifflin Harcour t Publishin y g Compan ) ( ( ) Lesson 1 159 Module 4 159 Lesson 4.1 L1.indd 4_U2M04 SE38589 A2_MNLE 159 3/27/14 3:27 PM 3/27/14 5:58 PM Reflect 1. EXPLORE Discussion Why isn’t absolute value used in the equation of the circle? Since squaring removes any negative signs just as absolute value does, there’s no need to Deriving the Standard-Form Equation of a Circle take absolute value before squaring. 2. Discussion Why does the equation of the circle also apply to the cases in which P has the same x-coordinate as C or the same y-coordinate as C so that △CAP doesn’t exist? If P has the same x-coordinate as C, then P’s y-coordinate must be either k + r or k - r. INTEGRATE TECHNOLOGY So,(x - h) + (y - k) = (h - h) + ((k ± r) - k) = 02 + (±r) = r 2, and the equation 2 2 2 2 2 Students have the option of completing the Explore activity either in the book or online. of the circle is still satisfied. Similarly, if P has the same y-coordinate as C, then P’s x-coordinate must be either h + r or h - r. So, (x - h) + (y - k) = ((h ± r) - h) 2 2 2 + (k - k) = (±r) + 0 2 = r 2, and the equation of the circle is still satisfied. 2 2 QUESTIONING STRATEGIES Writing the Equation of a Circle Explain 1 Why does point A have coordinates (x, k)? It is below P(x, y), which has x-coordinate x, and on the same horizontal line as C(h, k), which has y-coordinate k. The standard-form equation of a circle with center C(h, k) and radius r is (x - h) + (y - k) = r 2. If you solve this ___ 2 2 equation for r, you obtain the equation r = √(x -h) + (y - k) , which gives you a means for finding the radius of a 2 2 circle when the center and a point P(x, y) on the circle are known. Example 1 Write the equation of the circle. The circle with center C(-3, 2) and radius r = 4 Why is it necessary to use absolute value signs when representing the length of the legs of the right triangle? Since P could be any point on the circle, absolute value signs are used to make sure the length of each leg is a positive number. Substitute -3 for h, 2 for k, and 4 for r into the general equation and simplify. (x - (-3)) 2 + (y - 2) = 4 2 2 (x + 3) 2 + (y - 2) 2 = 16 The circle with center C(-4, -3) and containing the point P(2, 5) r = CP = = = © Houghton Mifflin Harcourt Publishing Company Step 1 Find the radius. ――――――――――――― 2 2 ) ( √( √( ) ( ) 2 - (-4) + ――――――― 2 2 ) 5 - (-3) 6 + 8 ――――― 36 + 64 √ _ = √ 100 = 10 Step 2 Write the equation of the circle. (x - (-4)) 2 + (y - (-3)) = 10 2 EXPLAIN 1 Writing the Equation of a Circle AVOID COMMON ERRORS Some students may forget to square the radius when writing the equation. Others may take its square root. Help students to avoid making these errors by having them write r 2 above the place in the equation where they need to write the square of the radius. 2 (x + 4) 2 + (y + 3) = 100 Module 4 2 160 Lesson 1 PROFESSIONAL DEVELOPMENT A2_MNLESE385894_U2M04L1.indd 160 Math Background The equation of a circle is based on the fact that all of the points on the circle are a fixed distance from a given point. This distance can be found using the Pythagorean Theorem. In the derivation, the fixed distance, the radius, is represented by the hypotenuse of a right triangle that has one vertex at the center of the circle, and one on the circle itself. Applying the Pythagorean Theorem produces the equation of the 2 2 that taking the square root of each side of this circle, (x - h) + (y - k) = r 2. Note ―――――― 2 2 equation produces the equation √(x - h) + (y - k) = r. This equation shows that the radius is the distance between the two points. QUESTIONING STRATEGIES 5/22/14 2:22 PM Do you need to be given the coordinates of the center to write an equation of a circle? If not, what other information could you use to find the center? Explain. No. If you know the endpoints of a diameter of the circle, you can find the coordinates of the center using the Midpoint Formula. Circles 160 Your Turn INTEGRATE MATHEMATICAL PRACTICES Focus on Math Connections MP.1 Discuss with students how the graph of the Write the equation of the circle. 3. 2 (x - 1) 2 + (y - (-4)) = 2 2 2 (x - 1) 2 + (y + 4) = 4 2 equation (x - h) + (y - k) = r is a transformation of the graph of x 2 + y 2 = r 2. Have students describe the transformation and compare the two graphs. 2 The circle with center C(1, -4) and radius r = 2 2 4. The circle with center C(-2, 5) and containing the point P(-2, -1) Because points C and P have the same x-coordinate, the radius of the circle is just the absolute value of the difference of their y-coordinates, so r = |5 - (-1)| = |6| = 6. (x - (-2)) + (y - 5) = 6 2 2 (x + 2) + (y - 5) 2 = 36 EXPLAIN 2 2 Explain 2 Rewriting an Equation of a Circle to Graph the Circle Expanding the standard-form equation (x - h) + (y - k) = r results in a general second-degree 2 equation in two variables having the form x 2 + y + cx + dy + e = 0. In order to graph such an equation or an even more general equation of the form ax 2 + ay 2 + cx + dy + e = 0. you must complete the square on both x and y to put the equation in standard form and identify the circle’s center and radius. Rewriting an Equation of a Circle to Graph the Circle 2 QUESTIONING STRATEGIES Example 2 How do you know what number to add to make perfect square trinomials when converting to standard form? Take half of the coefficient of the x term, and square it. Then do the same with the coefficient of the y term. 2 2 Graph the circle after writing the equation in standard form. x 2 + y 2 - 10x + 6y + 30 = 0 Write the equation. ( x 2 + y 2 - 10x + 6y + 30 = 0 Prepare to complete the x 2 - 10x + square on x and y. Complete both squares. © Houghton Mifflin Harcourt Publishing Company Once the equation is in standard form, how do you find the diameter of the circle? The diameter is 2 times the square root of the number that represents r 2. 2 ) + (y + 6y + ) = -30 + 2 + (x 2 - 10x + 25) + (y 2 + 6y + 9) = -30 + 25 + 9 (x - 5) 2 + (y + 3) 2 = 4 Factor and simplify. _ The center of the circle is C(5, -3), and the radius is r = √4 = 2. Graph the circle. y 0 -2 x 2 4 6 -4 -6 Module 4 161 Lesson 1 COLLABORATIVE LEARNING A2_MNLESE385894_U2M04L1.indd 161 Peer-to-Peer Activity Have students work in pairs. Instruct each student in each pair to write an equation of a circle in standard form. Have them graph the circles, keeping their graphs hidden from their partners. Have them also convert their equations to the general form ax 2 + by 2 + cx + dy + e = 0 by expanding and combining like terms. Instruct students to exchange the general forms of their equations, write each other’s equations in standard form, and draw the graph. Have students compare their work. 161 Lesson 4.1 3/28/14 12:26 PM B 4x 2 + 4y 2 + 8x - 16y + 11 = 0 AVOID COMMON ERRORS 4x 2 + 4y 2 + 8x - 16y + 11 = 0 Write the equation. When adding the numbers to the constant term to maintain the equality, students may forget to multiply the number added to complete each square by the leading coefficient that was factored from the variable terms. Help them to avoid this error by encouraging them to circle the leading coefficients that have been factored out in the process of completing the square. 4(x 2 + 2x) + 4(y 2 - 4y) + 11 = 0 Factor 4 from the x terms and the y terms. ( ) + 4(y 4 x 2 + 2x + Prepare to complete the square on x and y. ( - 4y + ) = -11 + 4( ) + 4( ) ) + 4(y - 4y + 4 ) = -11 + 4( 1 ) + 4( 4 ) 4(x + 1 ) + 4(y - 2 ) = 9 (x + 1 ) + (y - 2 ) = _ 4 x 2 + 2x + 1 Complete both squares. 2 2 2 Factor and simplify. 2 2 2 9 4 Divide both sides by 4. ( ) The center is C -1 , 2 Graph the circle. _ , and the radius is r = 4 √ _9 = _ 2 . INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 Have students compare how the 3 4 y 2 completing-the-square process is used to write quadratic functions in vertex form with how it is used to write a circle in standard form. Have them describe both the similarities and the differences. x -4 -2 0 2 4 -2 -4 Your Turn Graph the circle after writing the equation in standard form. 5. y 2 -6 -4 -2 (x 2 + 4x) + (y 2 + 6y) = -4 2 (x + 4x + 4) + (y 2 + 6y + 9) = -4 + 4 + 9 (x + 2) 2 + (y + 3) = 9 2 6. ― -8 9x + 9y − 54x − 72y + 209 = 0 6 9x 2 + 9y 2 - 54x - 72y + 209 = 0 9(x 2 - 6x) + 9(y 2 - 8y) = -209 9(x - 6x + 9) + 9(y - 8y + 16) = -209 + 9(9) + 9(16) (x - 3) 2 + (y - 4) 2 16 =_ 9 The center is C(3, 4), and the radius is r = Module 4 ―― 16 _ 4 = . √_ 9 3 162 y 4 2 2 2 9(x - 3) + 9(y - 4) = 16 -2 -6 2 2 x 2 -4 The center is C(-2, -3), and the radius is r = √9 = 3. 2 0 © Houghton Mifflin Harcourt Publishing Company x 2 + y + 4x + 6y + 4 = 0 x 2 + y 2 + 4x + 6y + 4 = 0 2 x -2 0 2 4 6 -2 Lesson 1 DIFFERENTIATE INSTRUCTION A2_MNLESE385894_U2M04L1.indd 162 Kinesthetic Experience 3/24/14 10:44 PM Prepare a length of string with a marker attached at one end. Display a coordinate plane. With a tack, tape, or some other means, attach the non-marker end of the string to an arbitrary point on the plane and draw a circle. Ask students to explain how they could use a point on the circle and the center to find the length of the string. Help them to see that they can construct a right triangle whose hypotenuse is the length of the string, and apply the Pythagorean Theorem to determine the length of the string. Circles 162 EXPLAIN 3 A circle in a coordinate plane divides the plane into two regions: points inside the circle and 2 2 points outside the circle. Points inside the circle satisfy the inequality (x - h) + (y - k) < r 2, 2 2 2 while points outside the circle satisfy the inequality (x - h) + (y - k) > r . Solving a Real-World Problem Involving a Circle Example 3 QUESTIONING STRATEGIES If you know the equation of a circle, how can you determine whether a given point lies inside, outside, or on the circle? You can substitute the coordinates of the point for x and y in the equation of the circle, and see whether the value 2 2 of (x - h) + (y - k) is less than, greater than, or equal to the value of r 2. If it is less than r 2, the point lies inside the circle; if greater than, the point lies outside the circle; and if equal to, the point lies on the circle. inside the circle satisfy the inequality (x - h)2 + (y - k)2 < r 2. Students should recognize that a point inside the circle would lie on a circle whose radius would be shorter than r, and whose 2 2 equation would be (x - h) + (y - k) = r 1 2, with 2 2 r 1 < r. Thus, r 1 2 < r 2 and (x - h) + (y - k) < r 2. Write an inequality representing the given situation, and draw a circle to solve the problem. The table lists the locations of the homes of five friends along with the locations of their favorite pizza restaurant and the school they attend. The friends are deciding where to have a pizza party based on the fact that the restaurant offers free delivery to locations within a 3-mile radius of the restaurant. At which homes should the friends hold their pizza party to get free delivery? Place Location Alonzo’s home A(3, 2) Barbara’s home B(2, 4) Constance’s home C(-2, 3) Dion’s home D(0, -1) Eli’s home E(1, -4) Pizza restaurant (-1, 1) School (1, -2) Write the equation of the circle with center (-1, 1) and radius 3. © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Jose Luis Pelaez/Corbis INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Ask students to explain why points that are Solving a Real-World Problem Involving a Circle Explain 3 (x - (-1)) 2 + (y - 1) = 3 , or (x + 1) + (y - 1) = 9 2 2 2 2 C The inequality (x + 1) + (y - 1) < 9 represents the situation. Plot the points from the table and graph the circle. 2 2 Restaurant -4 The points inside the circle satisfy the inequality. So, the friends should hold their pizza party at either Constance’s home or Dion’s home to get free delivery. -2 4 y B A 2 x 0 D -2 2 4 School -4 E In order for a student to ride the bus to school, the student must live more than 2 miles from the school. Which of the five friends are eligible to ride the bus? ( ( )) 2 ( 1) ( 2) 4 ( 1) ( 2) Write the equation of the circle with center ( x- 1 x- ) 2 2 + y - -2 2 2 = 2 . 2 + y+ = 2 The inequality x - ( 1 , -2 ) and radius + y+ 2 > 4 represents the situation. Use the coordinate grid in Part A to graph the circle. The points outside the circle satisfy the inequality. So, Alonzo, Barbara, and Constance are eligible to ride the bus. Module 4 163 Lesson 1 LANGUAGE SUPPORT A2_MNLESE385894_U2M04L1.indd 163 Communicate Math Have students work in pairs. Provide each pair of students with different graphs of circles and, on separate note cards or sheets of paper, the equations for those graphs. The first student chooses a graph and decides which equation goes with it, then explains why they are a match. The second student repeats the procedure using another graph and equation. 163 Lesson 4.1 3/28/14 12:30 PM Reflect 7. ELABORATE For Part B, how do you know that point E isn’t outside the circle? The coordinates of point E are (1, -4). Substituting 1 for x and -4 for y in (x - 1) + (y + 2) gives (1 - 1) + (-4 + 2) = 0 2 + (-2) = 4, so the coordinates of 2 2 2 2 2 QUESTIONING STRATEGIES E satisfy the equation of the circle, which means that E is on the circle and not outside it. How is the equation of a circle related to the equation a 2 + b 2 = c 2 from the Pythagorean Theorem? The radius is c, and the lengths of the legs of the right triangle that has the radius as its hypotenuse are a and b. Your Turn Write an inequality representing the given situation, and draw a circle to solve the problem. 8. Sasha delivers newspapers to subscribers that live within a 4-block radius of her house. Sasha’s house is located at point (0, -1). Points A, B, C, D, and E represent the houses of some of the subscribers to the newspaper. To which houses does Sasha deliver newspapers? 2 2 (x - 0) + (y - (-1)) = 4 2 2 -4 x 2 + (y + 1) = 16 2 -2 D The inequality x + (y + 1) < 16 represents the situation. A B x 0 2 -2 Sasha -4 2 2 y C 4 INTEGRATE MATHEMATICAL PRACTICES Focus on Reasoning MP.2 Discuss with students why, in the equation E The points inside the circle satisfy the inequality x 2 + (y + 1) <16. 2 So, Sasha delivers to the houses located at points B, D, and E. ax2 + by 2 + cx + dy + e = 0, a and b must be equal for the equation to be that of a circle. Focus their attention on the steps involved in converting the equation to the standard form of a circle, and on the roles of a and b in the conversion. Elaborate 9. Describe the process for deriving the equation of a circle given the coordinates of its center and its radius. First, choose an arbitrary point P on the circle. Next, find a third point A that forms a right triangle with points C and P. Then, use the coordinates of the three points to find the lengths of segments CA and PA. (The length of segment CP is the circle’s radius.) Finally, use the Pythagorean Theorem to write an equation of the circle. 10. What must you do with the equation ax + ay + cx + dy + e = 0 in order to graph it? 2 2 standard-form equation you can then identify the circle’s center and radius, which you can then use to graph the circle. 11. What do the inequalities (x - h) + (y - k) < r and (x - h) + (y - k) > r represent? 2 2 2 2 2 2 2 The inequality (x - h) + (y - k) < r represents points inside the circle with 2 2 equation (x - h) + (y - k) = r , and the inequality (x - h) + (y - k) > r 2 2 2 2 2 2 represents points outside the circle. SUMMARIZE THE LESSON © Houghton Mifflin Harcourt Publishing Company Complete the square on x and y to write the equation in standard form. From the How can you write the equation of a circle? You can use the coordinates of the center for h and k, and the radius for r, in the 2 2 equation (x - h) + (y - k) = r 2. 12. Essential Question Check-In What information must you know or determine in order to write an equation of a circle in standard form? You must know the center of the circle and its radius to write an equation of the circle in standard form. If only the center and a point on the circle are known, you can determine the radius from those two points. Module 4 A2_MNLESE385894_U2M04L1.indd 164 164 Lesson 1 3/27/14 5:55 PM Circles 164 Evaluate: Homework and Practice EVALUATE • Online Homework • Hints and Help • Extra Practice Write the equation of the circle. 1. The circle with C(4, -11) and radius r = 16 (x - h) 2 + (y - k) 2 = r 2 (x - 4) 2 + (y - (-11)) = 16 2 (x - 4) 2 + (y + 11) 2 = 256 2 ASSIGNMENT GUIDE Concepts and Skills 2. Practice The circle with C(-7, -1) and radius r = 13 (x - h) 2 + (y - k) = r 2 2 Explore Deriving the Standard-Form Equation of a Circle (x - (-7)) 2 + (y - (-1)) = 13 2 2 (x + 7) 2 + (y + 1) 2 = 169 Example 1 Writing the Equation of a Circle Exercises 1–4, 21, 24, 25 Example 2 Rewriting an Equation of a Circle to Graph the Circle Exercises 5–12 Example 3 Solving a Real-World Problem Involving a Circle Exercises 13–20, 22–23 3. The circle with center C(-8, 2) and containing the point P(-1, 6) r = CP 2 2 (x - (-8)) © Houghton Mifflin Harcourt Publishing Company QUESTIONING STRATEGIES ―― ― = √2 2 2 ― + (y - 2) = ( √65 ) 2 (x + 8) 2 + (y - 2) 2 = 65 = √1 + 1 2 (x - h) 2 + (y - k) 2 = r 2 2 2 (x - 5) 2 + (y - 9) = ( √― 2) (x - 5) 2 + (y - 9) 2 = 2 In Exercises 5–12, graph the circle after writing the equation in standard form. 5. x 2 + y 2 - 2x - 8y + 13 = 0 x + y - 2x - 8y + 13 = 0 (x 2 - 2x) + (y 2 - 8y) = -13 2 (x - 2x + 1) + (y 2 - 8y + 16) = -13 + 1 + 16 (x - 1) 2 + (y - 4) 2 = 4 The center of the circle is C(1, 4), and the radius is r = √4 = 2. 2 ― Exercise A2_MNLESE385894_U2M04L1.indd 165 Lesson 4.1 6 y 2 Module 4 165 ――――――― √(4 - 5) 2 + (8 - 9) 2 ――――― 2 2 = √(-1) + (-1) = 2 (x - h) 2 + (y - k) 2 = r 2 The circle with center C(5, 9) and containing the point P(4, 8) r = CP ―――――――― √(-1 - (-8)) + (6 - 2) ――― = √7 + 4 = √――― 49 + 16 = √― 65 = How can you find the radius of a circle if you know the endpoints of a diameter of the circle? You can use the distance formula to find the length of the diameter and take half of that distance. 4. 4 2 x -2 0 2 4 -2 Lesson 1 165 Depth of Knowledge (D.O.K.) 6 COMMON CORE Mathematical Practices 1–12 1 Recall of Information MP.7 Using Structure 13–20 2 Skills/Concepts MP.4 Modeling 21 2 Skills/Concepts MP.7 Using Structure 22–23 3 Strategic Thinking MP.4 Modeling 24–25 3 Strategic Thinking MP.2 Reasoning 3/27/14 5:55 PM Graph the circle after writing the equation in standard form. 6. AVOID COMMON ERRORS x 2 + y 2 + 6x - 10y + 25 = 0 y x 2 + y 2 + 6x - 10y + 25 = 0 (x 2 + 6x) + (y 2 - 10y) = -25 (x 2 + 6x + 9) + (y 2 - 10y + 25) = -25 + 9 + 25 (x + 3) 2 + (y - 5) 2 = 9 The center of the circle is C(-3, 5), and the radius is r = √9 = 3. ― 7. 4 2 x -6 -4 -2 y x + y + 4x + 12y + 39 = 0 (x 2 + 4x) + (y 2 + 12y) = -39 2 (x + 4x + 4) + (y 2 + 12y + 36) = -39 + 4 + 36 (x + 2) 2 + (y + 6) 2 = 1 The center of the circle is C(-2, -6), and the radius is r = √1 = 1. 2 -6 -4 -2 x 2 + y 2 - 8x + 4y + 16 = 0 x 0 ― A2_MNLESE385894_U2M04L1.indd 166 166 4 6 2 4 -4 -6 6 y 4 2 x -2 0 -2 © Houghton Mifflin Harcourt Publishing Company 8x 2 + 8y 2 - 16x - 32y - 88 = 0 x 2 + y 2 - 2x - 4y - 11 = 0 (x 2 - 2x) + (y 2 - 4y) = 11 (x 2 - 2x + 1) + (y 2 - 4y + 4) = 11 + 1 + 4 (x - 1) 2 + (y - 2) 2 = 16 The center of the circle is C(1, 2), and the radius is r = √16 = 4. 2 -2 8x 2 + 8y 2 - 16x - 32y - 88 = 0 Module 4 y 2 ― 9. -2 -6 2 x + y - 8x + 4y + 16 = 0 (x 2 - 8x) + (y 2 + 4y) = -16 2 (x - 8x + 16) + (y 2 + 4y + 4) = -16 + 16 + 4 (x - 4) 2 + (y + 2) 2 = 4 The center of the circle is C(4, -2), and the radius is r = √4 = 2. 2 x 0 -4 ― 8. 0 x 2 + y 2 + 4x + 12y + 39 = 0 2 Students may forget to factor out the leading coefficients of x 2 and y 2 before completing the square. Reinforce that the coefficient of each squared term must be 1 when completing each square. 6 Lesson 1 3/27/14 5:55 PM Circles 166 10. 2x 2 + 2y 2 + 20x + 12y + 50 = 0 VISUAL CUES 2x 2 + 2y 2 + 20x + 12y + 50 = 0 x 2 + y 2 + 10x + 6y + 25 = 0 (x 2 + 10x) + (y 2 + 6y) = -25 2 (x + 10x + 25) + (y 2 + 6y + 9) = -25 + 25 + 9 (x + 5) 2 + (y + 3) 2 = 9 The center of the circle is C(-5, -3), and the radius is r = √9 = 3. Suggest that students circle the numbers being added to or subtracted from x and y, and circle the preceding addition or subtraction signs, to remind them to take the opposites of these numbers when identifying the coordinates of the center of the circle. -8 -6 -4 -2 -6 -8 11. 12x 2 + 12y 2 - 96x - 24y + 201 = 0 y 12x 2 + 12y 2 - 96x - 24y + 201 = 0 12(x 2 - 8x) + 12(y 2 - 2y) = -201 12(x 2 - 8x + 16) + 12(y 2 - 2y + 1) = -201 + 12(16) + 12(1) 12(x - 4) 2 + 12(y - 1) 2 = 3 1 (x - 4) 2 + (y - 1) 2 = _ 4 1. 1 The center of the circle is C(4, 1), and the radius is r = _ =_ 4 2 ― √ on a graphing calculator. Lead them to observe that a circle is not a function, so it cannot be entered on the Y = screen as one rule. Use an 2 2 equation such as (x + 2) + (y - 3) = 4 to show how the equation can be solved for y and entered as two functions: the top half of the circle, -2 -4 ― INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Students may ask how to graph a circle y 0 x 2 1 0 1 2 4x 3 12. 16x 2 + 16y 2 + 64x - 96y + 199 = 0 y 16x 2 + 16y 2 + 64x - 96y + 199 = 0 16(x 2 + 4x) + 16(y 2 - 6y) = -199 2 16(x + 4x + 4) + 16(y 2 - 6y + 9) = -199 + 16(4) + 16(9) 16(x + 2) 2 + 16(y - 3) 2 = 9 9 (x + 2) 2 + (y - 3) 2 = _ 16 3 9 The center of the circle is C(-2, 3), and the radius is r = __ = _. 16 4 4 - (x + 2) ), and the bottom half of the (y = 3 + √――――― ――――― circle, (y = 3 - √4 - (x + 2) ). 2 ― √ © Houghton Mifflin Harcourt Publishing Company 2 In Exercises 13–20, write an inequality representing the problem, and draw a circle to solve the problem. 13. A router for a wireless network on a floor of an office building has a range of 35 feet. The router is located at the point (30, 30). The lettered points in the coordinate diagram represent computers in the office. Which computers will be able to connect to the network through the router? (x - 30) 2 + (y - 30) 2 = 35 2 (x - 30) 2 + (y - 30) 2 = 1225 The inequality (x - 30) 2 + (y - 30) 2 ≤ 1225 represents the situation. 6 4 2 x -6 80 60 -4 -2 y 0 C B G D 40 20 0 F Router A E 20 40 x 60 The points on or inside the circle satisfy the inequality. So, the computers located at points A, B, D, E, and F will be able to connect to the network. Module 4 A2_MNLESE385894_U2M04L1.indd 167 167 Lesson 4.1 167 Lesson 1 3/28/14 12:52 PM Write an inequality representing the problem, and draw a circle to solve the problem. 14. The epicenter of an earthquake is located at the point (20, -30). The earthquake is felt up to 40 miles away. The labeled points in the coordinate diagram represent towns near the epicenter. In which towns is the earthquake felt? CONNECT VOCABULARY -20 (x - 20) 2 + (y - (-30)) 2 = 40 2 (x - 20) 2 + (y + 30) 2 = 1600 The inequality (x - 20) 2 + (y + 30) 2 ≤ 1600 represents the situation. The points inside the circle satisfy the inequality. So, the earthquake is felt in the towns located at points B, D, and F. 15. Aida’s cat has disappeared somewhere in her apartment. The last time she saw the cat, it was located at the point (30, 40). Aida knows all of the cat’s hiding places, which are indicated by the lettered points in the coordinate diagram. If she searches for the cat no farther than 25 feet from where she last saw it, which hiding places will she check? (x - 30) 2 + (y - 40) 2 = 25 2 (x - 30) 2 + (y - 40) 2 = 625 The inequality (x - 30) 2 + (y - 40) 2 ≤ 625 represents the situation. Have students label the parts for the equation of a circle in standard form, identifying the parts that indicate the coordinates of the center and the radius. y A 0 C x 60 20 40 B -20 Epicenter -40 D -60 F E y 80 E 60 B 40 D F A G 20 x C 0 20 A 8 40 60 The points on or inside the circle satisfy the inequality. So, Aida will search for the cat in its hiding places at points A, B, and D. (x - (-2)) 2 + (y - 2) 2 = 4 2 (x + 2) 2 + (y - 2) 2 = 16 The inequality (x + 2) 2 + (y - 2) 2 ≤ 16 represents the situation. The points on or inside the circle satisfy the inequality. So, the music can be heard at the campsites located at points D, E, F and H. Module 4 A2_MNLESE385894_U2M04L1 168 168 H C D -8 -4 L 4 F 0 y B E G -4 -8 x 4 8 J K © Houghton Mifflin Harcourt Publishing Company • Image Credits ©wonderlandstock/Alamy: 16. A rock concert is held in a large state park. The concert stage is located at the point (-2, 2), and the music can be heard as far as 4 miles away. The lettered points in the coordinate diagram represent campsites within the park. At which campsites can the music be heard? Lesson 1 10/17/14 5:13 PM Circles 168 17. Business When Claire started her in-home computer service and support business, she decided not to accept clients located more than 10 miles from her home. Claire’s home is located at the point (5, 0), and the lettered points in the coordinate diagram represent the homes of her prospective clients. Which prospective clients will Claire not accept? (x - 5) 2 + (y - 0) 2 = 10 2 (x - 5) 2 + y 2 = 100 2 The inequality (x - 5) + y 2 > 100 represents the situation. F 20 B y C 10 A -20 -10 0 D 10 -10 G x 20 E -20 The points outside the circle satisfy the inequality. So, Claire should not accept the prospective clients located at points B, C, E, and F. 18. Aviation An airport’s radar system detects airplanes that are in flight as far as 60 miles from the airport. The airport is located at (-20, 40). The lettered points in the coordinate diagram represent the locations of airplanes currently in flight. Which airplanes does the airport’s radar system detect? (x - (-20)) B x A -80 -40 2 y G D + (y - 40) = 60 2 (x + 20) 2 + (y - 40) 2 = 3600 2 2 The inequality (x + 20) + (y - 40) ≤ 3600 represents the situation. 2 80 J 0 -40 F 40 K C -80 80 E H © Houghton Mifflin Harcourt Publishing Company • Image Credits ©Mikael Damkier/Shutterstock The points on or inside the circle satisfy the inequality. So, the airport’s radar system detects the airplanes at points A, D, G, and J. 19. Due to a radiation leak at a nuclear power plant, the towns up to a distance of 30 miles from the plant are to be evacuated. The nuclear power plant is located at the point (-10, -10). The lettered points in the coordinate diagram represent the towns in the area. Which towns are in the evacuation zone? + (y - (-10)) = 30 2 (x + 10) 2 + (y + 10) 2 = 900 2 2 The inequality (x + 10) + (y + 10) ≤ 900 represents the situation. The points on or inside the circle satisfy the inequality. So, the towns located at points B and E are in the evacuation zone. (x - (-10)) Module 4 A2_MNLESE385894_U2M04L1 169 169 Lesson 4.1 2 2 169 D 40 20 -40 -20 0 B -20 C -40 y A E x 20 40 F Lesson 1 10/17/14 5:14 PM 20. Bats that live in a cave at point (-10, 0) have a feeding range of 40 miles. The lettered points in the coordinate diagram represent towns near the cave. In which towns are bats from the cave not likely to be observed? Write an inequality representing the problem, and draw a circle to solve the problem. (x - (-10)) 2 + (y - 0) = 40 2 2 (x + 10) 2 + y 2 = 1600 2 The inequality (x + 10) + y 2 > 1600 represents the situation. 40 C B -40 -20 C(9, -11); r = 13 B. x + y - 18x + 22y + 33 = 0 C(9, 11); r = 15 C. 25x 2 + 25y 2 - 450x - 550y - 575 = 0 C(-9, -11); r = 15 2 2 0 -20 D The points outside the circle satisfy the inequality. So, bats from the cave are not likely to be observed in the towns located at points A and D. 21. Match the equations to the center and radius of the circle each represents. Show your work. A. x 2 + y 2 + 18x + 22y - 23 = 0 20 y A G 20 F x 40 E -40 C(-9, 11); r = 13 D. 25x + 25y + 450x - 550y + 825 = 0 x 2 + y 2 + 18x + 22y - 23 = 0 A (x 2 + 18x) + (y 2 + 22y) = 23 2 2 (x 2 + 18x + 81) + (y 2 + 22y + 121) = 23 + 81 + 121 (x + 9) 2 + (y + 11) 2 = 225 ―― The center of the circle is C(-9, -11), and the radius is r = √225 = 15. x 2 + y 2 - 18x + 22y + 33 = 0 (x 2 - 18x) + (y 2 + 22y) = -33 2 (x - 18x + 81) + (y 2 + 22y + 121) = -33 + 81 + 121 (x - 9) 2 + (y + 11) 2 = 169 The center of the circle is C(9, -11), and the radius is r = √169 = 13. B © Houghton Mifflin Harcourt Publishing Company ―― 25x 2 + 25y 2 - 450x - 550y - 575 = 0 x 2 + y 2 - 18x - 22y - 23 = 0 (x 2 - 18x) + (y 2 - 22y) = 23 2 (x - 18x + 81) + (y 2 - 22y + 121) = 23 + 81 + 121 (x - 9) 2 + (y - 11) 2 = 225 The center of the circle is C(9, 11), and the radius is r = √225 = 15. C ―― 25x 2 + 25y 2 + 450x - 550y + 825 = 0 x 2 + y 2 + 18x - 22y = -33 2 (x + 18x) + (y 2 - 22y) = -33 2 (x + 18x + 81) + (y 2 - 22y + 121) = -33 + 81 + 121 (x + 9) 2 + (y - 11) 2 = 169 The center of the circle is (-9, 11), and the radius is r = √169 = 13. D ―― Answers: B, C, A, D Module 4 A2_MNLESE385894_U2M04L1.indd 170 170 Lesson 1 5/22/14 2:23 PM Circles 170 H.O.T. Focus on Higher Order Thinking 22. Multi-Step A garden sprinkler waters the plants in a garden within a 12-foot spray radius. The sprinkler is located at the point (5, -10). The lettered points in the coordinate diagram represent the plants. Use the diagram for parts a–c. a. Write an inequality that represents the region that does not get water from the sprinkler. Then draw a circle and use it to identify the plants that do not get water from the sprinkler. 20 y E 10 A G C -30 -20 -10 Circle 3 B 0 -10 10 F D -20 (x - 5) + (y - (-10)) = 12 (x - 5) 2 + (y + 10) 2 = 144 2 2 The inequality (x - 5) + (y + 10) > 144 represents the situation. 2 2 2 Circle 2 20 x 30 Circle 1 The points outside the circle satisfy the inequality. So, the plants located at points A, B, C, E, and G do not get water from the sprinkler. b. Suppose a second sprinkler with the same spray radius is placed at the point (10, 10). Write a system of inequalities that represents the region that does not get water from either sprinkler. Then draw a second circle and use it to identify the plants that do not get water from either sprinkler. (x - 10) 2 + (y - 10) 2 = 12 2 (x - 10) 2 + (y - 10) 2 = 144 2 2 The system of inequalities (x - 5) + (y + 10) > 144 2 2 and (x - 10) + (y - 10) > 144 represents the situation. The points outside both circles satisfy the system of inequalities. So, the plants located at points A, B, and C would not get water from either sprinkler. © Houghton Mifflin Harcourt Publishing Company c. Locate the sprinkler at the point (-15, 0). 2 (x - (-15)) + (y - 0) 2 = 12 2 (x + 15) 2 + y 2 = 144 2 2 The system of inequalities (x - 5) + (y + 10) > 144 and (x - 10) 2 + (y - 10) 2 > 144 and (x + 15) 2 + y 2 > 144 represents the situation. The points outside all three circles satisfy the system of inequalities. So, there are no plants that would not get watered by any sprinkler. Module 4 A2_MNLESE385894_U2M04L1.indd 171 171 Lesson 4.1 Where would you place a third sprinkler with the same spray radius so all the plants get water from a sprinkler? Write a system of inequalities that represents the region that does not get water from any of the sprinklers. Then draw a third circle to show that every plant receives water from a sprinkler. 171 Lesson 1 5/22/14 2:23 PM 23. Represent Real-World Situations The orbit of the planet Venus is nearly circular. An astronomer develops a model for the orbit in which the Sun has coordinates S(0, 0), the circular orbit of Venus passes through V(41, 53), and each unit of the coordinate plane represents 1 million miles. Write an equation for the orbit of Venus. How far is Venus from the sun? Since the center of the orbit is the Sun, the radius of the orbit is SV. r = SV = √(41 - 0) 2 + (53 - 0) 2 = √41 2 + 53 2 = √1681 + 2809 = √4490 ≈ 67 ――――――― ―――― ――――― ―― So, the equation of the orbit is x 2 + y 2 = 67 2, or x 2 + y 2 = 4489, and Venus is approximately 67 million miles from the Sun. 24. Draw Conclusions The unit circle is defined as the circle with radius 1 centered at the origin. A Pythagorean triple is an ordered triple of three positive integers, (a, b, c), that satisfy the relationship a 2 + b 2 = c 2. An example of a Pythagorean triple is (3, 4, 5). In parts a–d, you will draw conclusions about Pythagorean triples. a. Write the equation of the unit circle. (x - h) + (y - k) = r 2 2 2 (x - 0)2 + (y - 0)2 = 1 2 © Houghton Mifflin Harcourt Publishing Company• Image Credits: ©Digital Vision/Getty Images x2 + y2 = 1 b. Use the Pythagorean triple (3, 4, 5) and the symmetry of a circle to identify the coordinates of two points on the part of the unit circle that lies in Quadrant I. Explain your reasoning. () () 2 2 32 52 42 Dividing both sides of 32 + 42 = 52 by 52 gives __ + __ = __ , or __35 + __45 = 1, 52 52 52 ( ) ( ) so the points __35 , __45 and __45 , __35 are on the unit circle in Quadrant I. Module 4 A2_MNLESE385894_U2M04L1.indd 172 172 Lesson 1 5/22/14 2:23 PM Circles 172 c. PEERTOPEER DISCUSSION Ask students to discuss with a partner how they can tell by inspecting a circle in the form ax 2 + ay 2 + cx + dy + e = 0 whether the center of the circle lies on either the x- or y-axis. The circle lies on the x-axis if d = 0. It lies on the y-axis if c = 0. It lies on both axes (at the origin) if both c = 0 and d = 0. Use your answer from part b and the symmetry of a circle to identify the coordinates of six other points on the unit circle. This time, the points should be in Quadrants II, III, and IV. Reflecting the points (__35 , __45 ) and (__45 , __35 ) across the y-axis gives the points Reflecting the points (__35 , __45 ) and (__45 , __35 ) across the x-axis gives the points (-__35 , __45 ) and (-__45 , __35 ). ( 3 __ , - __45 5 ) and (__ 4 3 , -__ 5 5 ). ( ) 3 4 4 __ __ __ (- 5 , - 5 ) and (- 5 , -__35 ). Reflecting the points __35 , __54 and (__45 , __35 ) across both axes gives the points d. Find a different Pythagorean triple and use it to identify the coordinates of eight points on the unit circle. Answers will vary. Sample answer: The Pythagorean triple (5, 12, 13) 5 ___ 12 ___ , 5 , - ___ , 12 , ( ) (___ 13 13 ) ( 13 13 ) 5 5 12 ___ 12 12 12 5 5 12 , 5 , ___ , - ___ , ___ , - ___ , -___ , - ___ , and (- ___ , - ___ . (-___ 13 13 ) ( 13 13 ) ( 13 13 ) ( 13 13 ) 13 13 ) JOURNAL 5 __ generates these eight points: __ , 12 , 13 13 Have students describe how they can determine whether a point P lies on a circle if they know the radius of the circle and the coordinates of the center of the circle. 25. Make a Conjecture In a two-dimensional plane, coordinates are given by ordered pairs of the form (x, y). You can generalize coordinates to three-dimensional space by using ordered pairs of the form (x, y, z) where the coordinate z is used to indicate displacement above or below the xy-plane. Generalize the standard-form equation of a circle to find the general equation of a sphere. Explain your reasoning. Let the center of the sphere be C(h, k, j), the radius be r, and an arbitrary point on the sphere be P(x, y, z). The plane z = j includes the points C(h, k, j) and (P’ )(x, y, j), which is the perpendicular projection of P(x, y, z) onto the plane. Because C and P’ are both in the plane z = j, © Houghton Mifflin Harcourt Publishing Company CP’ = 2 (CP’) 2 + (P’P) 2 = (CP) 2 (x - h) +( y - k) ) + (z - j) = r ( √――――――― 2 2 2 2 2 (x - h) 2 + (y - k) 2 + (z - j) 2 = r 2 A2_MNLESE385894_U2M04L1.indd 173 Lesson 4.1 2 ∆CP’P, which is a right triangle, gives the following: Module 4 173 ――――――― + (y - k) . Applying the Pythagorean Theorem to √(x - h) 173 Lesson 1 11/12/14 9:10 PM Lesson Performance Task QUESTIONING STRATEGIES A highway that runs straight east and west passes 6 miles south of a radio tower. The broadcast range of the station is 10 miles. How do you know whether the beginning and ending points for the car are within broadcasting range? Each point is 6 units vertically below the center point and at the end of a radius 10 units from the center. There are only two such points. N a. Determine the distance along the highway that a car will be within range of the radio station’s signal. 10 miles b. Given that the car is traveling at a constant speed of 60 miles per hour, determine the amount of time the car is within range of the signal. Radio tower 6 miles a. The radius of the circle representing the broadcasting range is 10. Let the position of the radio tower be (0, 0). Then the highway passes through (0, -6) and so is represented by the line y = -6. AVOID COMMON ERRORS Students who don’t write the distance-rate-time formula may not take care in determining the amount of time that the car is within range of the signal, and thus using the reciprocal value, dividing 60 by 16 instead of the reverse. Remind them to structure their calculations and use formulas instead of just operating on numbers. Write the equation of the circle representing the range of the radio station’s signal. (x - 0) 2 + (y - 0) 2 = 10 2 x 2 + y 2 = 100 The highway intersects the circle at points where y = -6. x 2 + (-6) = 100 x 2 + 36 = 100 x 2 = 64 x = ± √64 x = ±8 2 ― So, the highway intersects the circle at (8, -6) and (-8, -6). © Houghton Mifflin Harcourt Publishing Company The distance between the intersection points (8, -6) and (-8, -6) is 8 - (-8) = 16 miles. So, the car will be within range of the radio station’s signal for 16 miles. b. d = rt d t= _ r 16 = __ 60 4 = __ 15 4 So, the car is within range of the signal for __ hour, or 16 minutes. 15 Module 4 Lesson 1 174 EXTENSION ACTIVITY A2_MNLESE385894_U2M04L1.indd 174 Have students consider a second highway that runs parallel to the first, 2 miles south of (below) the original highway. Ask how fast a car would need to go along this highway to be in range of the radio signal for the same amount of time as the first car? The second car would be in range for 2(√10 2 - 8 2 ) = 12 miles. The 12 second car would need to travel (60) = 45 miles per hour. _ 3/27/14 5:55 PM ――― 16 Students could also research radio signals and ask whether they do have a circular range, and what conditions affect both AM and FM signals, either decreasing signal range (terrain, for example) or allowing a signal to be received a long distance from its source. Scoring Rubric 2 points: Student correctly solves the problem and explains his/her reasoning. 1 point: Student shows good understanding of the problem but does not fully solve or explain his/her reasoning. 0 points: Student does not demonstrate understanding of the problem. Circles 174