-5 ircles in the oordinate Plane -5. Plan What You ll Learn To write an equation of a circle To find the center and radius of a circle... nd Wh To describe the position and range of three cellular telephone towers, as in Eample heck Skills You ll Need G for Help Find the length of each segment to the nearest tenth. 5.8... 5.8.8 - - - - New Vocabular standard form of an equation of a circle Lesson -8 - bjectives To write an equation of a circle To find the center and radius of a circle Eamples Writing the Equation of a ircle Using the enter and a Point on a ircle Graphing a ircle Given its Equation Real-World onnection Writing an Equation of a ircle Ke oncepts Theorem - You can use the Distance Formula to find an equation of a circle with center (h, k) and radius r. Let (, ) be an point on the circle. Then the radius r is the distance from (h, k) to (, ). r = "( h) ( k) r = ( - h) + ( - k) Distance Formula Square both sides. This essentiall proves the following theorem. n equation of a circle with center (h, k) and radius r is ( - h) + ( - k) = r. The equation ( - h) + ( - k) = r is in standard form. You ma also call it the standard equation of a circle. r (h, k) (, ) Math ackground The equation of a circle is used etensivel in analtic geometr and trigonometr. The equation of a unit circle with center at the origin leads to the fundamental trigonometric relationship sin + cos =, whose importance in finding formulas for derivatives and integrals in calculus cannot be understated. More Math ackground: p. 660D Lesson Planning and See p. 660E for a list of the resources that support this lesson. For: ircle ctivit Use: Interactive Tetbook, -5 Quick heck Writing the Equation of a ircle Write the standard equation of the circle with center (5, -) and radius 7. ( - h) + ( - k) = r Use standard form. ( - 5) + [ - (-)] = 7 Substitute (5, -) for (h, k), and 7 for r. ( - 5) + ( + ) = 9 Simplif. Write the standard equation of each circle. a. center (, 5); radius 6 ( ± ) ± ( ± ) b. center (-, -); radius! ( ) ± ( 5) 6 Lesson -5 ircles in the oordinate Plane 695 ell Ringer Practice heck Skills You ll Need For intervention, direct students to: Finding Distance on the oordinate Plane Lesson -8: Eample Etra Skills, Word Problems, Proof Practice, h. Special Needs L For Eample, have students with motor difficulties identif the center and radius and have partners actuall graph the circle. elow Level L Displa the graphs of the circles in Eamples and. Then have students check that several points on the graphs satisf the equations. learning stle: verbal learning stle: visual 695
. Teach Guided Instruction Teaching Tip Students ma be overwhelmed b the five variables used to derive the standard form of an equation of a circle. Instead of using radius r and center (h, k), ou ma want to specif natural-number values for r, h, and k before introducing the theorem. Error Prevention Review subtraction of negative numbers to help students simplif correctl after substituting a negative coordinate of a center in the standard equation. - - - -6 (, ) (, -) Quick heck If ou know the center of a circle and a point on the circle, ou can write the standard equation of the circle. Using the enter and a Point on a ircle Write the standard equation of the circle with center (, -) that passes through the point (, ). r = "( h) ( k) Use the Distance Formula to find r. = "( ) ( ()) Substitute (, ) for (h, k), and (, ) for (, ). = " 5 =!6 Simplif. ( - h) + ( - k) = r Use standard form. ( - ) + - (-) D =!6 Substitute (, ) for (h, k), and!6 for r. ( - ) + ( + ) = 6 Simplif. Write the standard equation of the circle with center (, ) that passes through the point (-, ). ( ) ± ( ) Finding the enter and Radius of a ircle Point out that r, h, and k replace d,, and in the Distance Formula d = "( ) ( ) because of their special meanings in a circle and that the standard equation cannot be found without first finding the radius. dditional Eamples Write the standard equation of a circle with center ( 8, 0) and radius "5. ( ± 8) ± 5 Write the standard equation of a circle with center (5, 8) that passes through the point ( 5, ). ( 5) ± ( 8) 8. center: (, ); radius: 0 6 8 (, ) 8 Quick heck If ou know the standard equation of a circle, ou can describe the circle b naming its center and radius. Then ou can use this information to graph the circle. Graphing a ircle Given its Equation Find the center and radius of the circle with equation ( - 7) + ( + ) = 6. Then graph the circle. ( - 7) + ( + ) = 6 ( - 7) + ( - (-)) = 8 Use standard form. c c c h k r The center is (7, -) and the radius is 8. 8 To graph the circle, place the compass point at the center (7, -) and draw a circle with radius 8. Find the center and radius of the circle with equation ( - ) + ( - ) = 00. Then graph the circle. See left. -8 - You can use equations of circles to model real-world situations. 696 hapter ircles 696 dvanced Learners L fter Eample, challenge students to graph the inequalities ( - 7) + ( + ) 6 and ( - 7) + ( + ) 6. learning stle: verbal English Language Learners ELL For Theorem -, make sure students understand the and are variables and h, k, and r are constants. sk: Which of these variables and constants can have negative values? all ecept r learning stle: verbal
Real-World onnection To blend into the landscape, cellular phone towers can be disguised as trees. Quick heck Real-World onnection ommunications When ou make a call on a cellular phone, a tower receives the call. In the diagram, the centers of circles and are locations of cellular telephone towers. a. The equation ( - 6) + ( - 0) = 00 0 6 models the position and range of Tower. Describe the position and range of Tower. ( - 6) + ( - 0) = 0 is in standard form. It shows that 8 8 8 6 0 tower is located at (6, 0) and has a range of 0 units. 8 b. new tower is to be built at with range indicated in the graph. Write an equation that describes the position and range of this tower. has center (, 0) and radius 0. Substitute these into the standard equation. ( - h) + ( - k) = r ( - ) + ( - 0) = 0 Substitute. ( - ) + ( - 0) = 00 This is an equation for Tower. Write an equation that describes the position and range of Tower. ( 0) ± ( 0) ± Guided Instruction sk: How can ou tell quickl that Tower s circle has radius 0? ount the units from the center along a horizontal or vertical radius. dditional Eamples Find the center and radius of the circle with equation ( + ) + ( ) = 5. Then graph the circle. 8 6 (, ); r 5 0 EXERISES For more eercises, see Etra Skill, Word Problem, and Proof Practice. Practice and Problem Solving G Practice b Eample for Help Eample (page 695) Eample (page 696) Eample (page 696) Write the standard equation of each circle. 9. See margin.. center (, -8); r = 9. center (0, ); r = 7. center (0.,.); r = 0.. center (5, -); r = 5. center (-6, ); r = 8 6. center (-9, -); r =!5 7. center (0, 0); r = 8. center (-, 0); r = 9. center (-, -); r = Write the standard equation of the circle with the given center that passes through the given point. 0 5. See margin. 0. center (-, 6); point (-, 0). center (, ); point (0, 6). center (7, -); point (, -6). center (-0, -5); point (-5, 5). center (6, 5); point (0, 0) 5. center (-, -); point (-, 0) 6. See back of book. Find the center and radius of the circle with the given equation. Then graph the circle. 6. ( + 7) + ( - 5) = 6 7. ( - ) + ( + 8) = 00 8. ( + ) + ( - ) = 5 9. + = 6 0. ( - 0.) + = 0.0. ( + 5) + ( + ) = 8 diagram locates a radio tower at (6, ) on a coordinate grid where each unit represents mi. The radio signal s range is 80 mi. Find an equation that describes the position and range of the tower. ( 6) ± ( ± ) 600 Dail Notetaking Guide -5 L Dail Notetaking Guide -5 dapted Instruction L losure The point (, ) is on a circle with center (5, ). Find the standard equation and diameter of the circle. Leave our answer in simplest radical form. ( 5) ± ( ± ) 7; d " Lesson -5 ircles in the oordinate Plane 697. ( ) ± ( ± 8) 8. ± ( ) 9. ( 0.) ± (.) 0.6. ( 5) ± ( ± ) 5. ( ± 6) ± ( ) 6 6. ( ± 9) ± ( ± ) 5 7. ± 6 8. ( ± ) ± 9 9. ( ± ) ± ( ± ) 0. ( ± ) ± ( 6) 6. ( ) ± ( ) 7. ( 7) ± ( ± ) 5. ( ± 0) ± ( ± 5) 5. ( 6) ± ( 5) 6 5. ( ± ) ± ( ± ) 5 697
. Practice ssignment Guide -5, 7-9, -, 6, 8 6-6, 0, 5, 7, 9-6 hallenge 6-6 Test Prep 65-69 Mied Review 70-8 Homework Quick heck To check students understanding of ke skills and concepts, go over Eercises 0,,, 7, 6. uditor Learners Eercises To reinforce the standard form, have students eplain aloud wh each equation does or does not describe a circle. Visual Learners Eercises 5 58 Have some students solve graphicall and others solve algebraicall, and then have them compare their methods and solutions. Eample (page 697) ppl Your Skills 7. ± 8. ± 9 9. ± ( ) 0. ( ) ± 9. ( ) ± ( ) 6. ( ± ) ± ( ). ( ) ± ( ) 5. ( 5) ± ( ) G for Help In Eercises 8, use the Midpoint Formula (p. 55) to find centers. Use the diagram at the right. Write an equation that describes the position and radius of each circle.. P. Q ( ± ) ± ( ) 6 ( ) ± ( ± ). ommunications The plotted location of a cellular phone tower on a coordinate grid is (-, ) and the range is 5 units. Write an equation that describes the position and range of the tower. ( ± ) ± ( ) 5 Each equation models the position and range of a tornado alert siren. Describe the position and range of each. 5. ( - 5) + ( - 7) = 8 6. ( + ) + ( - 9) = position: (5, 7); range: 9 units position: (, 9); range: units Write the standard equation of each circle. 7. See left. 7. 8. 9. 0... Write an equation of a circle with diameter. 8. See left.. (0, 0), (8, 6). (, 0), (7, 6) 5. (, ), (5, 5) 6. (-, 0), (-5, -) 7. (-, ), (0, 9) 8. (-, ), (6, -7) P Q GPS Guided Problem Solving Enrichment Reteaching dapted Practice Practice Name lass Date Practice -5 Tell whether each three-dimensional object has rotational smmetr about a line and/or reflectional smmetr in a plane..... Draw all lines of smmetr for each figure. 5. 6. 7. Judging from appearance, tell what tpe(s) of smmetr each figure has. If it has line smmetr, sketch the figure and the line(s) of smmetr. If it has rotational smmetr, state the angle of rotation. 8. 9. 0. L L L L L Smmetr 9. The unit circle has center (0, 0) and radius. Write an equation for this circle. 5. ( ) ± ( ) 8 0. ritical Thinking Describe the graph of + = r 9 0. See left. when r = 0. 6. ( ± ) ± ( ±.5) 6.5. pen-ended n graph paper, make a design that includes at least three circles. 7. ( ±.5) ± ( 5) 8.5 Write the standard equations of our circles. heck students work. 8. ( ) ± ( ± ) Determine whether each equation is an equation of a circle. If not, eplain. 9. ± 0. The graph is the point (0, 0).. ( - ) +( +) = 9. + = 9. +( - ) = 9 es See left. No; the term is not squared.. No; the and terms 5. Find the circumference and area of the circle whose equation is are not squared. ( - 9) + ( - ) = 6. Leave our answers in terms of p. 6π; 6π 6. Write an equation of a circle with area 6p and center (, 7). ( ) ± ( 7) 6 G nline Homework Help Visit: PHSchool.com Web ode: aue-05 7. What are the - and -intercepts of the line tangent to the circle GPS ( - ) + ( - ) = 5 at the point (5, 6)? -int., -int. 8. For ( - h) + ( - k) = r, show that = Î r ( h) + k, or =- Î r ( h) +k. See margin. 9.. X. X 698 hapter ircles Pearson Education, Inc. ll rights reserved.. 5. 6. Each diagram shows a figure folded along a line of smmetr. Sketch the unfolded figure. 7. 8. 9. 0.. H K K 8. ( h) ± ( k) r ( k) r ( h) k w"r ( h) w"r ( h) ± k 698
Real-World PHSchool.com For: Graphing calculator procedures Web ode: aue-0 hallenge onnection In 005, Ellen Macrthur, age 8, sailed solo around the world in 7.6 das. Graphing alculator Use a graphing calculator to graph each circle. (Hint: See Eercise 8.) View the plotting in both sequential mode and simultaneous mode. 9 5. See back of book. 9. ( - ) + ( - ) = 9 50. ( + 5) + ( - 8) = 5. circle with center (0, 0) and radius 7 5. circle with center (-6, -) and radius Find all points of intersection of each pair of graphs. Make a sketch. 5 58. See back of book. 5. + = 5. + = 7 55. + = 8 =- + 5 =- = 56. + = 0 57. ( + ) + ( - ) = 8 58. ( - ) + ( - ) = 0 = + 5 = + 8 = + 6 Graphing alculator Use a graphing calculator to convince ourself that the given line is not tangent to the circle + = 5. Eplain what ou did. 59. =-5 + 6 60. + 5 = 9 59 60. See margin. 6. Writing Eplain wh it is not possible to conclude that a line and a circle are tangent b viewing their graphs. See margin. 6. Lines = + and = 5 cut the ring formed b circles ( - ) + ( - 5) = 6 and ( - ) + ( - 5) = 5 into four parts. Find the area of each part. about.5,.5, 9.8, 9.8 See 6. Nautical Distance The radius of Earth s equator is about 960 miles. margin. a. Write the equation of the equator with the center of Earth as the origin. b. Find the length of a arc on the equator to the nearest tenth of a mile. 69. mi c. arc along the equator is 60 nautical miles long. How man miles are in a nautical mile? Round to the nearest tenth.. mi d. Histor olumbus planned his trip to the East b going west. He thought each arc was 5 miles long. He estimated that the trip would take das. Use our answer to part (b) to find a better estimate. about das 6. Geometr in Dimensions The equation of a sphere z is similar to the equation of a circle. The equation of a sphere with center (h, j, k) and radius r is ( - h) + ( - j) + (z - k) = r. M a. M(-,, ) is the center of a sphere passing "6 T through T(0, 5, ). What is the radius of the sphere? b. Write an equation of the sphere. ( ± ) ± ( ) ± (z ) 6. ssess & Reteach Lesson Quiz. Find the center and radius of the circle with equation ( - ) + ( + ) = 9. Then graph the circle. 0 (, ); r. cellular phone tower with a range of 5 units is located on a coordinate grid at (0, 5). Write an equation that describes its position and range. ( 0) + ( 5) 65 Write the standard equation of each circle.. center (0, 6); radius " ± ( ± 6). center (, ); diameter 8 ( ) ± ( ) 8 5. center ( 9, 5); passing through ( 7, ) ( ± 9) ± ( 5) 0 Test Prep Multiple hoice 65. What is an equation of a circle with radius 6 and center (, -5)? D. ( - ) + ( + 5) = 6. ( + ) + ( - 5) = 56. ( + ) + ( - 5) = D. ( - ) + ( + 5) = 56 66. What are the coordinates of the center of the circle whose equation is ( - 9) + ( + ) =? J F. (, -) G. (-, ) H. (-9, ) J. (9, -) 67. What is the diameter of the circle with equation ( - ) + ( + ) =?... D. 6 lternative ssessment Have students graph a circle that contains a diameter with endpoints (, ) and (, 5) and then write the standard equation of the circle. Have them eplain how this can be done using a compass and the standard form of an equation of a circle. lesson quiz, PHSchool.com, Web ode: aua-05 Lesson -5 ircles in the oordinate Plane 699 59 60. Eplanations ma var. Sample: Solve the circle and line eqs. for, enter in the calc., and use the zooming feature. 6. nswers ma var. Sample: Lines can appear tangent on a graph, but ma not be. 6. a. ± 5,68,600 699
Test Prep For additional practice with a variet of test item formats: Standardized Test Prep, p. 7 Test-Taking Strategies, p. 706 Test-Taking Strategies with Transparencies Short Response Etended Response Mied Review 68. Show how to find the radius of the circle whose equation is + ( + 8) = 5. See margin. 69. The line represented b the equation = + is tangent to a circle at (6, ). The center of the circle is on the -ais. Write an equation of the circle. Show our work. See margin. G for Help Lesson - Find the value of each variable. ssume that lines that appear tangent are tangent. 70. 0 7. 8 Use this heckpoint Quiz to check students understanding of the skills and concepts of Lessons - through -5. Grab & Go heckpoint Quiz Lesson 8-6 5; 75 For the given vectors aw and cw, write the sum aw ± cw as an ordered pair. 7. aw = k-, 5l and cw = k8, 7l 6, 7. aw = k-, -l and cw = k-, 6l 5, 7. = k, l and = k, l, 75. = k9, -6l and = k, -l, 7 aw 80 cw 70 aw a 8 cw Lesson 7- Find the geometric mean of each pair of numbers in simplest radical form. 76. and 6 77. 9 and 7 9" 78. and 8 6" 79.! and!7 80.! and! "6 8. and 8 heckpoint Quiz Lessons - through -5 68. [] This equation is in the standard form of an equation of a circle. This means that r 5. Taking the sq. root of each side, r 5. Thus, the radius is 5. [] incorrect answer R incorrect eplanation 69. [] The slope of the radius through (6, ) is, so the line containing this radius is 5. Since 0,, and the center is (, 0), r "( 6) (0 ) 5. ( ) ± 5 [] appropriate methods, but with one computational error 700 8. (.5) ± ( 0.5).5 9. (.5) ± ( ) 6.5 0. ( ±.5) ± ( ± ).5 700 hapter ircles [] incorrect equation R correct equation found incorrectl [] correct equation, without work shown Find the value of each variable. ssume that lines that appear tangent are tangent.. 58. 6. 70 95 5 56 w 6 0. 5. 5.5 c 5.9 about.0 0 6. In the circle at the right, what is m F? 0 7. Writing Eplain the difference between a chord and a secant. Include a diagram. See margin. 60 E 0 D 6 The endpoints of a diameter are given. Write an equation of the circle. 8 0. See left. 8. (, ) and (0, 0) 9. (-, 5) and (9, -) 0. (-, -8) and (, 0) heckpoint Quiz 7. chord is a segment whose endpoints are on the circle. secant is a line, ra, or segment that intersects a circle at two points. D F about 5.7 is a chord D is a secant