And Angles, . Points On A Perpendicular Bisector Of .

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0: Mathematics Florida Standards52Perpendicular andAngle BisectorsMAFS.912.G-C0.3.9 Prove theorems about linesand angles,. points on a perpendicular bisector ofa line segment are exactly those equidistant from thesegment's endpoints. Also MAFS.912.G-SRT.2.5MP 1. MP 3, MP 4, MP S, MP8Objective To use properties of perpendicular bisectors and angle bisectorsGetting Ready!Confused? Trydrawing a diagramto "straighten"yourself out.MATHEMATICALPRACTICESYou hang a bulletin board over yourdesk using string. The bulletin boardis crooked. When you straighten thebulletin board, what type of triangledoes the string form with the topof the board? How do you know?Visualize the vertical line along the wallthat passes through the nail. Whatrelationships exist between this lineand the top edge of the straightenedbulletin board? Explain.In the Solve It, you thought about the relationships that must exist in order for a bulletinhoard to hang straight. You will explore these relationships in this lesson.LessonEssential Understanding There is a special relationship between the points onVocabularythe perpendicular bisector of a segment and the endpoints ofthe segment.equidistantIn the diagram below on the left,Sis the perpendicular bisector of 2lB. Cp isdistance from apoint to a lineperpendicular to AB at its midpoint. In the diagram on the right, CA and CB are drawnto complete ACAD and ACBD.cy-1 ,AfB/. r1 X0,aAYoushould recognize from yourworkinChapter4thatACAD ACBD.So you canconclude that or that CA CB.Apoint is equidistant from two objects if it isthe same distance from the objects. So point Cis equidistant from points A and B.This suggests a proof of Theorem 5-2, the Perpendicular Bisector Theorem.Its converseis also true and is stated as Theorem 5-3.292Chapters Relationships Within Triangles

Theorem 5-2 Perpendicular Bisector TheoremTheoremIf a point is on theperpendicular bisectorof a segment, then it isequidistant from theendpoints of the segment.If.Then .PM AB and MA — MBPA PBPA ' M,P1"I.BA, ' B'You will prove Theorem 5-2 In Exercise 32.Converse of the Perpendicular Bisector TheoremTheoremIf.Then .If a point is equidistantfrom the endpoints ofasegment, then it is on theperpendicular bisector ofthe segment.PA PBPM AB and MA MB,PBAA" A4,1,You will prove Theorem 5-3 in Exercise 33.Problem 1Using the Perpendicular Bisector TheoremfiAlgebra What is the length of ABlHow do you know BDIs the perpendicularbisector ofBD is the perpendicular bisector of AC,so B is equidistant fromA and C.The markings in thediagram show that W isperpendicular to M atthe midpoint of /4C.BA BC4a: 6a:- 10—2x —10a: 5Perpendicular Bisector TheoremSubstitute 4x for SA and 6x - 10 for BC.6X-10Subtract 6x from each side.Divide each side by-2.Now find AB.AB 4xAB 4(5) 20Substitute 5 for x.Got It? 1. What is the length of QRlV13n - lVCPowerGeometry.comJ '-7I Lesson 5-2 Perpendicular and Angle Bisectors293

Problem 2IHow do you findpoints that areUsing a Perpendicular BisectorA park director wants to build a T-shirt stand equidistant from the Rollin' Coasterand the Spaceship Shoot.What are the possible locations of the stand? Explain.equidistant from twogiven points?Paddle boatsBy the Converse of thePerpendicular BisectorTheorem, pointsSpaceship Shootequidistant from twogiven points are on theperpendicular bisector ofthe segment that joins, \ Rollin Coasterthe two points.Merry-go-roundTo be equidistant from the two rides, the standshould be on the perpendicular bisector of thesegment connecting the rides. Find the midpoint Aof and draw line i through A perpendicular toRS. The possible locations of the stand are all thepoints on line LGot It? 2. a. Suppose the director wants the T-shirt stand to be equidistant from thepaddle boats and the Spaceship Shoot. What are the possible locations?b. Reasoning Can you place the T-shirt stand so that it is equidistant fromthe paddle boats, the Spaceship Shoot, and the Rollin' Coaster? Explain.Essential Understanding There is a special relationship between the pointson the bisector of an angle and the sides ofthe angle.The distance from a point to a line is the length oftheperpendicular segment from the point to the line. This distanceis also the length ofthe shortest segment from the point to theline. You will prove this in Lesson 5-6. In the figure at the right,JIT?the distances from A to and from Bto are represented by thered segments.In the diagram, AD is the bisector of Z.CAB.If you measure thelengths of the perpendicular segments from D to the two sides ofthe angle, you will find that the lengths are equal. Point D is equidistantfrom the sides of the angle.y294Chapter 5 Relationships Within Triangles

Theorem 5-4 Angle Bisector TheoremTheoremIf.Then.If a point is on the bisectorofan angle, then thepoint is equidistant fromthe sides ofthe angle.QS bisects PQR, 1 QP,SP SRand SR 1 QRP,SYou will prove Theorem 5-4 in Exercise 34.Theorem 5-5 Converse of the Angle Bisector TheoremTheoremIf.Then .If a point in the interiorof an angle is equidistantSP J. QP, XQS bisects yLPQRand SP SRfrom the sides of theangle, then the point is onthe angle bisector.You will prove Theorem 5-5 in Exercise 35.Problem 3\Using the Angle Bisector TheoremAlgebra What is the length ofPM?NR bisects /.LNQ. 1 M andPP 1 WQ.RM RP bI«aThe lengthUse the Angle BisectorofPMTheorem to write an equationyou can solve for x.Angle Bisector Theorem7x: 2x; 25Substitute.How can you use the5a; 25Subtract 2x from each side.expression givenforffPto check youra: 5iJ answer?2X 25Divide each side by 5.Now find RM.Substitute 5 for X in theRM 7xexpression 2x 25and verify that the result 7(5) 35is 35.Substitute 5 forx.(6x 3,Got It? 3. What is the length ofPS?4x 9CPowerGeometry.comLesson 5-2 Perpendicular and Angle Bisectors295

Lesson CheckMATHEMATICALDo you UNDERSTAND?Do you know HOW?4. Vocabulary Draw a line and a point not on the line.Use the figure at the right for Exercises 1-3.Draw the segment that represents the distance from1. What is the relationshipbetween AC and BDlPRACTICESDthe point to the line.15 r2. What is the length of AB?f5. Writing PointP is in the interior of LLOX.Describehow you can determine whether P is on the bisector3. What is the length of DC?of /LLOX without drawing the angle bisector.MATHEMATICALPractice and Problem-Solving ExercisesPracticePRACTICES See Problem 1.MUse the figure at the right for Exercises 6-8.6. What is the relationship between MB and }K1- 18/7. What is value of x?/. r\3x1 \8. Find/M.Reading Maps For Exercises 9 and 10, use the map ofapart of Manhattan.9. Which school is equidistant from the subway See Problem 2.I Subway Statlori stations at Union Square and I4th Street? How do nyou know?10. Is St. Vincent's Hospital equidistant from VillageKids Nursery School and Legacy School? How doNYC MuseumSchoolCDVillage KidsNursery Schoolyou know?)Coleman School11. Writing On a piece of paper, mark a point H forhome and a point S for school. Describe how to findthe set of points equidistant from Hand S.::-v.I Xavier School.v.Sf.Viiicent A Hospl i,\12. According to the diagram, how far is I fromHKl From HFl13. How is HL related to AKHF1 Explain.14. Find the value ofy.(4y 18)296Chapter 5 Relationships Within TrianglesUnion SquareSee Problem 3.Use the figure at the right for Exercises 12-15.15. Find m/LKHL and mLFHt. . Legacy School

17. Algebra Findy, ST, and TU.16. Algebra Findx,fK, and/M.x 53y 6 ApplyAlgebra Use the figure at the right for Exercises 18-22.18. Find the value of x.T19. Find TW.20. Find WZ.21. What kind of triangle is AJWZ? Explain.3x - 522. If i? is on the perpendicular bisector of TZ,theni?is ? from rand Z, or ? ? .23. Think About a Plan In the diagram atthe right, the soccer goalie will preparefor a shot from the player at pointP bymoving out to a point on XY.To havethe best chance of stopping the ball,should the goalie stand at the pointon XY that lies on the perpendicularbisector of GL or at the point on XYthat lies on the bisector of Z.GPL1IfBStiiiiLExplain your reasoning. How can you draw a diagramto help? Would the goalie want to be thesame distance from G and L orfrom PG and PLl24. a. Constructions Draw Z.CDE. Construct the angle bisector ofthe angle,b. Reasoning Use the converse ofthe angle bisector theorem to justify yourconstruction.25. a. Constructions Draw Qfi. Construct the perpendicular bisector of Qi? toconstruct APQR.b. Reasoning Use the perpendicular bisector theorem to justify that yourconstruction is an isosceles triangle.26. Write Theorems 5-2 and 5-3 as a single biconditional statement.27. Write Theorems 5-4 and 5-5 as a single biconditional statement.PowerGeometry.comLesson 5-2 Perpendicular and Angle Bisectors297

28. Error Analysis To prove that APQR is isosceles, a student began by statingthat since Q is on the segment perpendicular to PR,Q is equidistant from theendpoints of PR. What is the error in the student's reasoning?Writing Determine whether A must be on the bisector of LTXR.Explain.29.30.32. Prove the PerpendicularBisector Theorem.31.33. Prove the Converse of theProof Perpendicular Bisector Theorem.Given: PM AB, PM bisects ABGiven: PA PB with PM 1 AB at M.Prove: AP BPProve: P is on the perpendicular bisectorof .1P/ \.134. Prove the Angle35. Prove the Converse of theProof Bisector Theorem.Given: QS bisects APQR, 1 h Prove: SP SRProof Angle Bisector Theorem.Given; h 1SP SRProve: OS bisects APQi?.36. Coordinate Geometry Use points A(6,8), 0(0,0), and B(10,0).a. Write equations of lines and m such that -L OA at A and m 1 OB at B.b. Find the intersection C of lines and m.c. Show that CA CB.d. Explain why C is on the bisector of AAOB.298Chapter 5 Relationships Within Triangles

challenge37. A,B, and C are three noncollinear points. Describe and sketch aline in plane ABCsuch that points A,B, and C are equidistant fromthe line. Justify your response.38. Reasoning Mis the intersection ofthe perpendicular bisectors oftwo sides of hABC.Line (. is perpendicular to plane ABC at M.Explainwhy a point on is equidistant from A,B, and C.{Hint:See page 48,Exercise 33. Explain why AEAM AEBM AECM.)yStandardized Test Prep39. Fori4(l,3)and B(l, 9), which point lies on the perpendicular bisector of AB? (3,3)CD (1,5) (6,6)CD (3, 12)40. What is the converse ofthe following conditional statement?If a triangle is isosceles, then it has two congruent angles.CD If a triangle is isosceles, then it has two congruent sides.CD If a triangle has congruent sides, then it is equilateral.CH If a triangle has two congruent angles, then it is isosceles.CD If a triangle is not isosceles, then it does not have two congruent angles.41. Which figure represents the statement BD bisects AABClCD4 CDDB4CDBCshort e ponse42. The line y 7 is the perpendicular bisector ofthe segment with endpoints A{2,10)and B{2, k). What is the value of kl Explain your reasoning.Mixed Review43. Find the value ofx in the figure at the right. See Lesson 5-1.2x 1044. Z.1 and A2 are complementary and Z.1 See Lesson 1-5.and Z.3 are supplementary. If mA2 30,what is mZ.3?Get Ready! To prepare for Lesson 5-3, do Exercises 45-47.45. What is the slope of a line that is perpendicular to the line y —3x 4? See Lesson 3-8.46. Line f is a horizontal line. What is the slope ofa line perpendicular to ?47. Describe the line x 5.cPowerGeometry.comI Lesson 5-2 Perpendicular and Angle Bisectorr299

Nov 19, 2018 · Theorem 5-4 Angle Bisector Theorem Theorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. If. . . QS bisects PQR, 1 QP, and SR 1 QR P, S Then. SP SR You will prove Theorem 5-4 in Exercise 34. Theorem 5-5 Converse of the Angle Bisector Theorem Theorem

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