10 Circles - MR. HUANG
1010.110.210.310.410.510.610.7CirclesLines and Segments That Intersect CirclesFinding Arc MeasuresUsing ChordsInscribed Angles and PolygonsAngle Relationships in CirclesSegment Relationships in CirclesCircles in the Coordinate PlaneSEE the Big IdeaSeismograph (p.(p 578)(p. 573)573))Saturn (p.Car (p.(p 550)DartboardDartbboardd (p.(p. 543)3)Bicyclel Chainh(p 535)535))Bicycle(p.hs geo pe 10co.indd 5261/19/15 2:25 PM
Maintaining Mathematical ProficiencyMultiplying BinomialsExample 1 Find the product (x 3)(2x 1).FirstOuterInnerLast(x 3)(2x 1) x(2x) x( 1) 3(2x) (3)( 1)FOIL Method 2x2 ( x) 6x ( 3)Multiply. 2x2 5x 3Simplify.The product is 2x2 5x 3.Find the product.1. (x 7)(x 4)2. (a 1)(a 5)3. (q 9)(3q 4)4. (2v 7)(5v 1)5. (4h 3)(2 h)6. (8 6b)(5 3b)Solving Quadratic Equations by Completing the SquareExample 2 Solve x2 8x 3 0 by completing the square.x2 8x 3 0Write original equation.x2 8x 3Add 3 to each side.( )2Complete the square by adding —82 , or 42, to each side.x2 8x 42 3 42(x 4)2 19Write the left side as a square of a binomial.—x 4 19Take the square root of each side.—x 4 19Subtract 4 from each side.——The solutions are x 4 19 0.36 and x 4 19 8.36.Solve the equation by completing the square. Round youranswer to the nearest hundredth, if necessary.7. x2 2x 59. w2 8w 911. k2 4k 7 08. r2 10r 710. p2 10p 4 012. z2 2z 113. ABSTRACT REASONING Write an expression that represents the product of two consecutivepositive odd integers. Explain your reasoning.Dynamic Solutions available at BigIdeasMath.comhs geo pe 10co.indd 5275271/19/15 2:25 PM
MathematicalPracticesMathematically proficient students make sense of problems anddo not give up when faced with challenges.Analyzing Relationships of CirclesCore ConceptCircles and Tangent CirclesA circle is the set of all points in a plane that areequidistant from a given point called the centerof the circle. A circle with center D is called“circle D” and can be written as D.Coplanar circles that intersect in one point arecalled tangent circles.Dcircle D, or DSRT R and S are tangent circles. S and T are tangent circles.Relationships of Circles and Tangent Circlesa. Each circle at the right consists of points that are 3 units from thecenter. What is the greatest distance from any point on A to anypoint on B?ABb. Three circles, C, D, and E, consist of points that are3 units from their centers. The centers C, D, and E of the circlesare collinear, C is tangent to D, and D is tangent to E.What is the distance from C to E?SOLUTIONAa. Because the points on each circle are 3 units from the center,the greatest distance from any point on A to any point on B is 3 3 3 9 units.b. Because C, D, and E are collinear, C is tangentto D, and D is tangent to E, the circlesare as shown. So, the distance from C to Eis 3 3 6 units.3CD3B33E3Monitoring ProgressLet A, B, and C consist of points that are 3 units from the centers.1. Draw C so that it passes through points A and B in the figureat the right. Explain your reasoning.2. Draw A, B, and C so that each is tangent to the other two.ABDraw a larger circle, D, that is tangent to each of the otherthree circles. Is the distance from point D to a point on D less than, greater than, or equal to 6? Explain.528Chapter 10hs geo pe 10co.indd 528Circles1/19/15 2:25 PM
10.1Lines and Segments ThatIntersect CirclesEssential QuestionWhat are the definitions of the lines andsegments that intersect a circle?Lines and Line Segments That Intersect CirclesngtachordtenWork with a partner. The drawing atthe right shows five lines or segments thatintersect a circle. Use the relationships shownto write a definition for each type of line orsegment. Then use the Internet or some otherresource to verify your ecaRadius:Diameter:Using String to Draw a CircleWork with a partner. Use two pencils, a piece of string, and a piece of paper.a. Tie the two ends of the piece of string loosely around the two pencils.b. Anchor one pencil on the paper at the center of the circle. Use the other pencilto draw a circle around the anchor point while using slight pressure to keep thestring taut. Do not let the string wind around either pencil.REASONINGABSTRACTLYTo be proficient in math,you need to know andflexibly use differentproperties of operationsand objects.c. Explain how the distance between the two pencil points as you draw the circleis related to two of the lines or line segments you defined in Exploration 1.Communicate Your Answer3. What are the definitions of the lines and segments that intersect a circle?4. Of the five types of lines and segments in Exploration 1, which one is a subsetof another? Explain.5. Explain how to draw a circle with a diameter of 8 inches.Section 10.1hs geo pe 1001.indd 529Lines and Segments That Intersect Circles5291/19/15 2:31 PM
10.1 LessonWhat You Will LearnIdentify special segments and lines.Draw and identify common tangents.Core VocabulVocabularylarryUse properties of tangents.circle, p. 530center, p. 530radius, p. 530chord, p. 530diameter, p. 530secant, p. 530tangent, p. 530point of tangency, p. 530tangent circles, p. 531concentric circles, p. 531common tangent, p. 531Identifying Special Segments and LinesA circle is the set of all points in a plane that are equidistant froma given point called the center of the circle. A circle with center Pis called “circle P” and can be written as P.Pcircle P, or PCore ConceptLines and Segments That Intersect CirclesA segment whose endpoints are the center andany point on a circle is a radius.chordcenterA chord is a segment whose endpoints are ona circle. A diameter is a chord that contains thecenter of the circle.READINGThe words “radius” and“diameter” refer to lengthsas well as segments. For agiven circle, think of a radiusand a diameter as segmentsand the radius and thediameter as lengths.radiusdiameterA secant is a line that intersects a circle intwo points.A tangent is a line in the plane of a circle thatintersects the circle in exactly one point, theAB andpoint of tangency. The tangent ray ⃗— are also called tangents.the tangent segment ABsecantpoint oftangencytangent BAIdentifying Special Segments and LinesDACBGSTUDY TIPIn this book, assume that allsegments, rays, or lines thatappear to be tangent toa circle are tangents.ETell whether the line, ray, or segment is best described asa radius, chord, diameter, secant, or tangent of C.—a. AC—b. AB ⃗c. DEd. ⃖ ⃗AESOLUTION— is a radius because C is the center and A is a point on the circle.a. AC— is a diameter because it is a chord that contains the center C.b. ABc. ⃗DE is a tangent ray because it is contained in a line that intersects the circle inexactly one point.d. ⃖ ⃗AE is a secant because it is a line that intersects the circle in two points.Monitoring ProgressHelp in English and Spanish at BigIdeasMath.com— —1. In Example 1, what word best describes AG ? CB ?2. In Example 1, name a tangent and a tangent segment.530Chapter 10hs geo pe 1001.indd 530Circles1/19/15 2:31 PM
Drawing and Identifying Common TangentsCore ConceptCoplanar Circles and Common TangentsIn a plane, two circles can intersect in two points, one point, or no points.Coplanar circles that intersect in one point are called tangent circles. Coplanarcircles that have a common center are called concentric circles.no points ofintersection1 point of intersection(tangent circles)2 points ofintersectionconcentriccirclesA line or segment that is tangent to two coplanar circles is called a commontangent. A common internal tangent intersects the segment that joins the centersof the two circles. A common external tangent does not intersect the segment thatjoins the centers of the two circles.Drawing and Identifying Common TangentsTell how many common tangents the circles have and draw them. Use blue to indicatecommon external tangents and red to indicate common internal tangents.a.b.c.SOLUTIONDraw the segment that joins the centers of the two circles. Then draw the commontangents. Use blue to indicate lines that do not intersect the segment joining the centersand red to indicate lines that intersect the segment joining the centers.a. 4 common tangentsb. 3 common tangentsMonitoring Progressc. 2 common tangentsHelp in English and Spanish at BigIdeasMath.comTell how many common tangents the circles have and draw them. State whetherthe tangents are external tangents or internal tangents.3.4.Section 10.1hs geo pe 1001.indd 5315.Lines and Segments That Intersect Circles5311/19/15 2:31 PM
Using Properties of TangentsTheoremsTheorem 10.1 Tangent Line to Circle TheoremIn a plane, a line is tangent to a circle if and only ifthe line is perpendicular to a radius of the circle atits endpoint on the circle.PQmLine m is tangent to Qif and only if m QP.Proof Ex. 47, p. 536Theorem 10.2 External Tangent Congruence TheoremTangent segments from a common external pointare congruent.RSPTIf SR and ST are tangentsegments, then SR ST.Proof Ex. 46, p. 536Verifying a Tangent to a Circle— tangent to P?Is ST35S37T12PSOLUTIONUse the Converse of the Pythagorean Theorem (Theorem 9.2). Because 122 352 372,— PT—. So, ST— is perpendicular to a radius of P at its PTS is a right triangle and STendpoint on P.— is tangent to P.By the Tangent Line to Circle Theorem, STFinding the Radius of a CircleIn the diagram, point B is a point of tangency. Findthe radius r of C.A50 ftCrr80 ftBSOLUTION— BC—, so ABC isYou know from the Tangent Line to Circle Theorem that ABa right triangle. You can use the Pythagorean Theorem (Theorem 9.1).AC 2 BC 2 AB2(r 50)2 r 2 802r 2 100r 2500 r 2 6400100r 3900r 39Pythagorean TheoremSubstitute.Multiply.Subtract r 2 and 2500 from each side.Divide each side by 100.The radius is 39 feet.532Chapter 10hs geo pe 1001.indd 532Circles1/19/15 2:31 PM
Constructing a Tangent to a CircleGiven C and point A, construct a line tangentto C that passes through A. Use a compassand straightedge.CASOLUTIONStep 1Step 2CMStep 3ACMACBFind a midpoint—. Construct the bisectorDraw ACof the segment and label themidpoint M.MABDraw a circleConstruct M with radius MA.Label one of the points where M intersects C as point B.Construct a tangent lineDraw ⃖ ⃗AB. It is a tangentto C that passes through A.Using Properties of Tangents— is tangent to C at S, and RT— is tangent to C at T. Find the value of x.RSS28RC3x 4TSOLUTIONRS RTExternal Tangent Congruence Theorem28 3x 4Substitute.8 xSolve for x.The value of x is 8.Monitoring Progress—6. Is DE tangent to C?3CD42 ESection 10.1hs geo pe 1001.indd 533Help in English and Spanish at BigIdeasMath.com—8. Points M and N are7. ST is tangent to Q.Find the radius of Q.QrSpoints of tangency.Find the value(s) of x.Mr24x2P18TN9Lines and Segments That Intersect Circles5331/19/15 2:31 PM
10.1 ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. WRITING How are chords and secants alike? How are they different?2. WRITING Explain how you can determine from the context whether the words radius anddiameter are referring to segments or lengths.3. COMPLETE THE SENTENCE Coplanar circles that have a common center are called .4. WHICH ONE DOESN’T BELONG? Which segment does not belong with the other three?Explain your reasoning.chordradiustangentdiameterMonitoring Progress and Modeling with MathematicsIn Exercises 5–10, use the diagram. (See Example 1.)5. Name the circle.6. Name two radii.B8. Name a diameter.J9. Name a secant.— is tangent to C.In Exercises 19–22, tell whether ABExplain your reasoning. (See Example 3.)CDHG18.KA7. Name two rr14AChapter 10hs geo pe 1001.indd 5347Br61624C9ArBIn Exercises 15–18, tell whether the common tangent isinternal or external.5348B23.r16.CIn Exercises 23–26, point B is a point of tangency. Findthe radius r of C. (See Example 4.)C15.1216A13.18CA21.A1595310. Name a tangent and a point of tangency.In Exercises 11–14, copy the diagram. Tell how manycommon tangents the circles have and draw them.(See Example 2.)BrA26.B30rCCrA18Circles1/19/15 2:31 PM
CONSTRUCTION In Exercises 27 and 28, construct Cwith the given radius and point A outside of C. Thenconstruct a line tangent to C that passes through A.27. r 2 in.37. USING STRUCTURE Each side of quadrilateralTVWX is tangent to Y. Find the perimeter ofthe quadrilateral.28. r 4.5 cm1.2 TIn Exercises 29–32, points B and D are points oftangency. Find the value(s) of x. (See Example 5.)29.B2x 730.B4.5VX3x 103.3YA8.3AC5x 831.32.2x2 4A—C2x 5BA3x2 2x 7D33. ERROR ANALYSIS Describe and correct the error in— is tangent to Z.determining whether XY 60Z11Y61X—38. LOGIC In C, radii CA and CB are perpendicular.DC22WDDB3.17x 6CBecause 112 602 612, XYZ is a— is tangent to Z.right triangle. So, XY⃖ ⃗ are tangent to C.⃖ ⃗BD and AD—, CB—, BD⃖ ⃗, and ⃖ ⃗a. Sketch C, CAAD.b. What type of quadrilateral is CADB? Explainyour reasoning.39. MAKING AN ARGUMENT Two bike paths are tangentto an approximately circular pond. Your class isbuilding a nature trail that begins at the intersection Bof the bike paths and runs between the bike paths andover a bridge through the center P of the pond. Yourclassmate uses the Converse of the Angle BisectorTheorem (Theorem 6.4) to conclude that the trail mustbisect the angle formed by the bike paths. Is yourclassmate correct? Explain your reasoning.EP34. ERROR ANALYSIS Describe and correct the error infinding the radius of T. BMU39TS40. MODELING WITH MATHEMATICS A bicycle chain36V392 362 152So, the radius is 15.— is a common tangent ofis pulled tightly so that MNthe gears. Find the distance between the centers ofthe gears.17.6 in.35. ABSTRACT REASONING For a point outside of acircle, how many lines exist tangent to the circle thatpass through the point? How many such lines existfor a point on the circle? inside the circle? Explainyour reasoning.1.8 in.MLN4.3 in.P41. WRITING Explain why the diameter of a circle is thelongest chord of the circle.36. CRITICAL THINKING When will two lines tangent tothe same circle not intersect? Justify your answer.Section 10.1hs geo pe 1001.indd 535Lines and Segments That Intersect Circles5351/19/15 2:31 PM
42. HOW DO YOU SEE IT? In the figure, ⃗PA is tangent46. PROVING A THEOREM Prove the External Tangentto the dime, ⃗PC is tangent to the quarter, and ⃗PB is acommon internal tangent. How do you know that— PB— PC—?PACongruence Theorem (Theorem 10.2).RPPST— and ST— are tangent to P.Given SR— ST—Prove SRCAB47. PROVING A THEOREM Use the diagram to prove eachpart of the biconditional in the Tangent Line to CircleTheorem (Theorem 10.1).—43. PROOF In the diagram, RS is a common internalAC RCtangent to A and B. Prove that — —.BC SCQRAmBCa. Prove indirectly that if a line is tangent to a circle,then it is perpendicular to a radius. (Hint: If you—, thenassume line m is not perpendicular to QPthe perpendicular segment from point Q to line mmust intersect line m at some other point R.)Given Line m is tangent to Q at point P.—Prove m QPS44. THOUGHT PROVOKING A polygon is circumscribedabout a circle when every side of the polygon istangent to the circle. In the diagram, quadrilateralABCD is circumscribed about Q. Is it always truethat AB CD AD BC? Justify your answer.YDXBWAb. Prove indirectly that if a line is perpendicular toa radius at its endpoint, then the line is tangent tothe circle.—Given m QPProve Line m is tangent to Q.CQZ48. REASONING In the diagram, AB AC 12, BC 8,and all three segments are tangent to P. What is theradius of P? Justify your answer.45. MATHEMATICAL CONNECTIONS Find the values of xBand y. Justify your answer.P4y 1 RQ2x 5x 8Tx 6PDPEFCSMaintaining Mathematical ProficiencyAReviewing what you learned in previous grades and lessonsFind the indicated measure. (Section 1.2 and Section 1.5)49. m JKM50. ABJ15 L10AB7C28 K536Chapter 10hs geo pe 1001.indd 536MCircles1/19/15 2:31 PM
10.2 Finding Arc MeasuresEssential QuestionHow are circular arcs measured?A central angle of a circle is an angle whose vertex is the center of the circle.A circular arc is a portion of a circle, as shown below. The measure of a circulararc is the measure of its central angle.If m AOB 180 , then the circular arc is called a minor arc and is denoted by AB .Acircular arcB59 central angleOmAB 59 Measuring Circular ArcsWork with a partner. Use dynamic geometry software to find the measureBC . Verify your answers using trigonometry.of a.PointsA(0, 0)B(5, 0)C(4, 3)64C20 6 4 2A040 2 6 6PointsA(0, 0)B(4, 3)C(3, 4)CB 2d.46PointsA(0, 0)B(4, 3)C( 4, 3)B2 6644C02B26A0A0 40To be proficient inmath, you need to usetechnological tools toexplore and deepen yourunderstanding of concepts. 4 42USING TOOLSSTRATEGICALLY 6 24 4266C4B2PointsA(0, 0)B(5, 0)C(3, 4)6 2c. 6b. 4 2A0 2 2 4 4 6 6246Communicate Your Answer2. How are circular arcs measured?3. Use dynamic geometry software to draw a circular arc with the given measure.a. 30 c. 60 b. 45 d. 90 Section 10.2hs geo pe 1002.indd 537Finding Arc Measures5371/19/15 2:32 PM
10.2 LessonWhat You Will LearnFind arc measures.Identify congruent arcs.Core VocabulVocabularylarrycentral angle, p. 538minor arc, p. 538major arc, p. 538semicircle, p. 538measure of a minor arc, p. 538measure of a major arc, p. 538adjacent arcs, p. 539congruent circles, p. 540congruent arcs, p. 540similar arcs, p. 541Prove circles are similar.Finding Arc MeasuresA central angle of a circle is an angle whose vertex is the center of the circle. In thediagram, ACB is a central angle of C.If m ACB is less than 180 , then the points on C that lie in the interior of ACBform a minor arc with endpoints A and B. The points on C that do not lie on theminor arc AB form a major arc with endpoints A and B. A semicircle is an arc withendpoints that are the endpoints of a diameter.Aminor arc ABBCDmajor arc ADBMinor arcs are named by their endpoints. The minor arc associated with ACB isnamed AB . Major arcs and semicircles are named by their endpoints and a point onthe arc. The major arc associated with ACB can be named ADB .STUDY TIPThe measure of a minorarc is less than 180 . Themeasure of a major arc isgreater than 180 .Core ConceptMeasuring ArcsThe measure of a minor arc is the measure ofAB is read asits central angle. The expression m “the measure of arc AB.”The measure of the entire circle is 360 . Themeasure of a major arc is the difference of 360 and the measure of the related minor arc. Themeasure of a semicircle is 180 .AC50 mAB 50 BDm ADB 360 50 310 Finding Measures of Arcs— is a diameter.Find the measure of each arc of P, where RTa. RSRb. RTSc. RSTP110 TSSOLUTIONa. RS is a minor arc, so m RS m RPS 110 .b. RTS is a major arc, so m RTS 360 110 250 .— is a diameter, so c. RTRST is a semicircle, and m RST 180 .538Chapter 10hs geo pe 1002.indd 538Circles1/19/15 2:32 PM
Two arcs of the same circle are adjacent arcs when they intersect at exactly one point.You can add the measures of two adjacent arcs.PostulatePostulate 10.1 Arc Addition PostulateAThe measure of an arc formed by two adjacent arcsis the sum of the measures of the two arcs.BCmABC mAB mBCUsing the Arc Addition PostulateFind the measure of each arc. a. GE b. GEFG c. GFH40 RSOLUTIONa. m GE m GH m HE 40 80 120 80 110 b. m GEF m GE m EF 120 110 230 EF 360 m GEF 360 230 130 c. mGFFinding Measures of ArcsWhom Would You Rather Meet?CA recent survey asked teenagers whetherthey would rather meet a famous musician,athlete, actor, inventor, or other person. Thecircle graph shows the results. Find theindicated arc measures.a. m ACMusician:1088b. m ACDc. m ADCd. m EBDSOLUTIONa. m AC m AB m BCBInventor:298A 137 d.m EBD 360 m ED 360 137 360 61 223 299 Help in English and Spanish at BigIdeasMath.comIdentify the given arc as a major arc, minor arc,or semicircle. Then find the measure of the arc.4.E 220 Monitoring Progress1.Other:618 137 83 m ADC 360 m AC TQ QSActor:7982.5. QRT TS3.6. TQR RSTT120 60 80 SSection 10.2hs geo pe 1002.indd 539Db. m ACD m AC m CD 29 108 c.Athlete:838Finding Arc MeasuresQR5391/19/15 2:32 PM
Identifying Congruent ArcsTwo circles are congruent circles if and only if a rigid motion or a compositionof rigid motions maps one circle onto the other. This statement is equivalent to theCongruent Circles Theorem below.TheoremTheorem 10.3 Congruent Circles TheoremCTwo circles are congruent circles if and onlyif they have the same radius.BAD A B if and only if AC BD.Proof Ex. 35, p. 544Two arcs are congruent arcs if and only if they have the same measure and they arearcs of the same circle or of congruent circles.TheoremTheorem 10.4 Congruent Central Angles TheoremIn the same circle, or in congruent circles, two minorarcs are congruent if and only if their correspondingcentral angles are congruent.CDABE BC DE if andProof Ex. 37, p. 544only if BAC DAE.Identifying Congruent ArcsTell whether the red arcs are congruent. Explain why or why not.D Ea.CThe two circles in part (c)are congruent by theCongruent Circles Theorembecause they have thesame radius.Chapter 10hs geo pe 1002.indd 540BTc.URFQSUT95 VY95 ZXSOLUTIONSTUDY TIP54080 80 b.a. CD EF by the Congruent Central Angles Theorem because they are arcs of thesame circle and they have congruent central angles, CBD FBE.b. RS and TU have the same measure, but are not congruent because they are arcsof circles that are not congruent.c. UV YZ by the Congruent Central Angles Theorem because they are arcs ofcongruent circles and they have congruent central angles, UTV YXZ.Circles1/19/15 2:32 PM
Monitoring ProgressHelp in English and Spanish at BigIdeasMath.comTell whether the red arcs are congruent. Explain why or why not.7.AB145 N8.C145 DMP120 5120 Q4Proving Circles Are SimilarTheoremTheorem 10.5 Similar Circles TheoremAll circles are similar.Proof p. 541; Ex. 33, p. 544Similar Circles TheoremAll circles are similar.Given C with center C and radius r, D with center D and radius sProve C DCDsrFirst, translate C so that point C maps to point D. The image of C is C′ withcenter D. So, C′ and D are concentric circles.CDrssDrcircle C′ C′ is the set of all points that are r units from point D. Dilate C′ using center ofsdilation D and scale factor —.rDsDrscircle C′This dilation maps the set of all the points that are r units from point D to the set of allspoints that are —(r) s units from point D. D is the set of all points that are s unitsrfrom point D. So, this dilation maps C′ to D.Because a similarity transformation maps C to D, C D.Two arcs are similar arcs if and only if they have the same measure. All congruentarcs are similar, but not all similar arcs are congruent. For instance, in Example 4, thepairs of arcs in parts (a), (b), and (c) are similar but only the pairs of arcs in parts (a)and (c) are congruent.Section 10.2hs geo pe 1002.indd 541Finding Arc Measures5411/19/15 2:32 PM
10.2 ExercisesDynamic Solutions available at BigIdeasMath.comVocabulary and Core Concept Check1. VOCABULARY Copy and complete: If ACB and DCE are congruent central angles of C,then AB and DE are .2. WHICH ONE DOESN’T BELONG? Which circle does not belong with the other three? Explainyour reasoning.1 ft6 in.6 in.12 in.Monitoring Progress and Modeling with MathematicsIn Exercises 15 and 16, find the measure of each arc.(See Example 2.)In Exercises 3–6, name the red minor arc and findits measure. Then name the blue major arc and findits measure.3.E4.A15. a. JLB135 68 CDGK5.FCN170 CM120 JJ d. JMKc. JLMP538688798 16. a. RS6.C b. KMPd.LRb. QRS c. QSTMQ428PS QTTL17. MODELING WITH MATHEMATICS A recent surveyIn Exercises 7–14, identify the given arc as a major arc,minor arc, or semicircle. Then find the measure of thearc. (See Example 1.) 7. BC 8. DC 9. ED 10. AE 11. EABE 12. ABC 13. BACChapter 10hs geo pe 1002.indd 5421108B708 F708458658CDFavorite Type of MusicBCountry:APop:R&B: 178C668 558HHip-Hop/Rap:268G 288898Other:Folk:F 328 478Christian: EDRock:a. m AE 14. EBD542Aasked high school students their favorite type ofmusic. The results are shown in the circle graph.Find each indicated arc measure. (See Example 3.)d. m BHCb. m ACEe. m FDc. m GDCf. m FBDCircles1/19/15 2:32 PM
18. ABSTRACT REASONING The circle graph shows thepercentages of students enrolled in fall sports at ahigh school. Is it possible to find the measure of eachminor arc? If so, find the measure of the arc for eachcategory shown. If not, explain why it is not possible.26. MAKING AN ARGUMENT Your friend claims thatthere is not enough information given to find the valueof x. Is your friend correct? Explain your reasoning.Ax8MHigh School Fall SportsVFootball: 20%WNone: 15%P4x8ZSoccer:30%Cross-Country: 20%YVolleyball: 15%27. ERROR ANALYSIS Describe and correct the error innaming the red arc.XIn Exercises 19–22, tell whether the red arcs arecongruent. Explain why or why not. (See Example 4.)A19.20.DSR16YZFG1808EBL(2x 30)8 JK NPNPQRm CD 70 , and m DE 20 . Find two possiblemeasures of AE .H24.PKM30. REASONING In R, m AB 60 , m BC 25 ,MATHEMATICAL CONNECTIONS In Exercises 23 and 24,find the value of x. Then find the measure of the red arc.23.J— and CD—. Find m ABACD and m AC when m AD 20 .1808T AD29. ATTENDING TO PRECISION Two diameters of P are1210Q O928WDBX92822.708Cnaming congruent arcs.N8A28. ERROR ANALYSIS Describe and correct the error in858PV MLB7081808408C21.Nx8B4x8R6x8QS31. MODELING WITH MATHEMATICS On a regulationdartboard, the outermost circle is divided into twentycongruent sections. What is the measure of each arcin this circle?PAx8C7x87x8T25. MAKING AN ARGUMENT Your friend claims thatany two arcs with the same measure are similar.Your cousin claims that any two arcs with the samemeasure are congruent. Who is correct? Explain.Section 10.2hs geo pe 1002.indd 543Finding Arc Measures5431/19/15 2:32 PM
32. MODELING WITH MATHEMATICS You can use the34. ABSTRACT REASONING Is there enough informationto tell whether C D? Explain your reasoning.time zone wheel to find the time in different locationsacross the world. For example, to find the time inTokyo when it is 4 p.m. in San Francisco, rotatethe small wheel until 4 p.m. and San Franciscoline up, as shown. Then look at Tokyo to see thatit is 9 a.m. there.DenvyCaesond harna ron eFe NondeenAzor89inkieBelfast11 12Rom10Helsb. Given A B— BD—Prove AC93HalifaxBoston5nFA.M.6 7ncNew Orleansercocisran geraho— BD—a. Given ACProve A B8kentTashw itysco t CoMaiwuK4ASaAstana540 to prove each part of the biconditional in theCongruent Circles Theorem (Theorem 10.3).P.M.6 7Bang5akok35. PROVING A THEOREM Use the diagram on nadyrNoon11 12 110C36. HOW DO YOU SEE IT? Are the circles on the target21Midnightsimilar or congruent? Explain your reasoning.a. What is the arc measure between each time zoneon the wheel?b. What is the measure of the minor arc from theTokyo zone to the Anchorage zone?c. If two locations differ by 180 on the wheel, thenit is 3 p.m. at one location when it is at theother location.33. PROVING A THEOREM Write a coordinate proof ofthe Similar Circles Theorem (Theorem 10.5).37. PROVING A THEOREM Use the diagram to prove eachGiven O with center O(0, 0) and radius r, A with center A(a, 0) and radius spart of the biconditional in the Congruent CentralAngles Theorem (Theorem 10.4).Prove O Aa. Given BAC DAE DEProve BCyOrAsCb. Given BC DEAProve BAC DAExDBE38. THOUGHT PROVOKING Write a formula for thelength of a circular arc. Justify your answer.Maintaining Mathematical ProficiencyReviewing what you learned in previous grades and lessonsFind the value of x. Tell whether the side lengths form a Pythagorean triple. (Section 9.1)39.40.17841.1313147x544Chapter 10hs geo pe 1002.indd 54442.xxx1110Circles1/19/15 2:32 PM
10.3 Using ChordsEssential QuestionWhat are two ways to determine when a chordis a diameter of a circle?Drawing DiametersLOOKING FORSTRUCTURETo be proficient in math,you need to look closelyto discern a patternor structure.Work with a partner. Use dynamic geometry software to construct a circle ofradius 5 with center at the origin. Draw a diameter that has the given point as anendpoint. Explain how you know that the chord you drew is a diameter.a. (4, 3)b. (0, 5)c. ( 3, 4)d. ( 5, 0)Writing a Conjecture about ChordsWork with a partner. Use dynamicgeometry software to construct a— of a circle A. Construct achord BCchord on the perpendicular bisector—. What do you notice? Changeof BCthe original chord and the circleseveral times. Are your resultsalways the same? Use your resultsto write a conjecture.CABA Chord Perpendicular to a Diameter—Work with a partner. Use dynamic geometry software to construct a diameter BC——of a circle A. Then construct a chord DE perpendicular to BC at point F. Find the— andlengths DF and EF. What do you notice? Change the chord perpendicular to BCthe circle several times. Do you always get the same results? Write a conjecture abouta chord that is perpendicular to a diameter of a circle.DBFAECCommunicate Your Answer4. What are two ways to determine when a chord is a diameter of a circle?Section 10.3hs geo pe 1003.indd 545Using Chords5451/19/15 2:33 PM
10.3 LessonWhat You Will LearnUse chords of circles to find lengths and arc measures.Core VocabulVocabularylarryUsing Chords of CirclesPreviouschordarcdiameterRecall that a chord is a segment with endpointson a circle. Because its endpoints lie on thecircle, any chord divides the circle into two arcs.A diameter divides a circle into two
Coplanar circles that have a common center are called concentric circles. 2 points of intersection 1 point of intersection (tangent circles) no points of intersection concentric circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of .
Two circles are congruent circles if and only if they have congruent radii. A B if AC BD . AC BD if A B. Concentric circles are coplanar circles with the same center. Two coplanar circles that intersect at exactly one point are called tangent circles . Internally tangent circles Externally tangent circles Pairs of Circles
Coplanar Circles and Common Tangents In a plane, two circles can intersect in two points, one point, or no points. Coplanar circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common tangent. A
COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points tangent circles 1 point concentric circles. . no points Slide 16 / 150
COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points tangent circles 1 point concentric circles. . no points Slide 15 / 150
COPLANAR CIRCLES are two circles in the same plane which intersect at 2 points, 1 point, or no points. Coplanar circles that intersects in 1 point are called tangent circles. Coplanar circles that have a common center are called concentric. 2 points tangent circles 1 point concentric circles. . no points Slide 15 / 149
circles that intersect in one point are called tangent circles. Coplanar circles that have a common center are called concentric circles. A line or segment that is tangent to two coplanar circles is called a common internal tangent A common external tangent two circles. Notes: intersects the segment that joins the centers of the two circles .
Coplanar circles that have a common center are called concentric circles. 2 points of intersection 1 point of intersection (tangent circles) no points of intersection concentric circles A line or segment that is tangent to two coplanar circles is called a common tangent. A common internal tangent intersects the segment that joins the centers of .
The Baldrige framework is used extensively as a foundation for internal systems, but there has been a substantial decrease in the number of manufacturing organizations applying for the award. This research study validates some of the reasons associated with that development. The Value of Using the Baldrige Performance Excellence Framework in Manufacturing Organizations Prabir Kumar .
