2.1 Segment Bisectors
Page 1 of 72.1GoalBisect a segment. Findthe coordinates of themidpoint of a segment.Segment BisectorsGeo-ActivityFolding a Segment Bisector1 On a piece of paper, draw AB&*. 2 Fold the paper so that point B is on top of point A.Key Words midpoint segment bisectorA bisectBB3 Label the point where the fold intersects AB&* as point M.A4 Use a ruler to measure the lengths of AM&**, MB&*, and AB&*.What do you notice?AMBMB&*. The midpoint of aIn the Geo-Activity, M is called the midpoint of ABsegment is the point on the segment that divides it into two congruentsegments.Student HelpVOCABULARY TIPBi- means “two,” and-sect means “to cut.”So, bisect means “tocut in two.”A segment bisector is a segment, ray, line, or plane that intersects asegment at its midpoint. To bisect a segment means to divide thesegment into two congruent segments.CMAMM is the midpoint of AB&*.BADB &*( is a bisector of ABCD&*.2.1Segment Bisectors53
Page 2 of 7Student HelpEXAMPLE&*. Find AM and MB.M is the midpoint of ABSTUDY TIPThe midpoint of asegment divides thesegment in half.A26AMBSolutionMBIf you know the length12of AB&*, multiply AB by to find AM and MB.Find Segment Lengths1&*, so AM and MB are each half the length of AB&*.M is the midpoint of AB1212AM MB p AB p 26 13ANSWER AM 13 and MB 13.EXAMPLEFind Segment Lengths2&*. Find PS and RS.P is the midpoint of RSR7PSSolution&*, so PS RP. Therefore, PS 7.P is the midpoint of RS&*.You know that RS is twice the length of RPRS 2 p RP 2 p 7 14ANSWER PS 7 and RS 14.Find Segment Lengths1. Find DE and EF.2. Find NP and MP.18DEEXAMPLE3FMN11Use Algebra with Segment Lengths&*.Line l is a segment bisector of ABFind the value of x.lA5xMSolutionAM MBLine l bisects AB&* at point M.5x 35Substitute 5x for AM and 35 for MB.5x35 55Divide each side by 5.x 7CHECKSimplify. Check your solution by substituting 7 for x.5x 5(7) 3554Chapter 2Segments and AnglesP35B
Page 3 of 7Midpoints If you know the coordinates of the endpoints of a linesegment in a coordinate plane, you can find the coordinates of themidpoint of the segment using the Midpoint Formula.THE MIDPOINT FORMULAWordsStudent HelpREADING TIPThe numbers 1 and 2 inx 1 and y2 are calledsubscripts. You read x 1as “x sub 1” and y2 as”y sub 2.”The coordinates of themidpoint of a segment are theaverages of the x-coordinatesand the y-coordinates of theendpoints.Symbolsyy2y 1 y22My1A(x1, y1)The midpoint of thesegment joiningA(x1, y1) and B(x2, y2) x x2y y2B(x2, y2) 11 2 , 2 .is M 4EXAMPLE x1 x22x1x 1 x2 y 1 y2, 22x2 xUse the Midpoint Formula&*.Find the coordinates of the midpoint of ABb. A( 2, 3), B(5, 1)a. A(1, 2), B(7, 4)SolutionStudent HelpSKILLS REVIEWFor help plotting pointsin a coordinate plane,see p. 664.First make a sketch. Then use the Midpoint Formula.a.b.yyA( 2, 3)B(7, 4)1M1M1A(1, 2)xB(5, 1)1xLet (x1, y1) (1, 2)and (x2, y2) (7, 4).x xy y 22 1 7 2 4 , 2211 2 , 2M Let (x1, y1) ( 2, 3)and (x2, y2) (5, 1).x xy y 22 2 5 3 ( 1) , 223 , 1 211 2 , 2M (4, 3)Use the Midpoint Formula&*. Then find the coordinates of its midpoint.Sketch PQ3. P(2, 5), Q(4, 3)4. P(0, 2), Q(4, 0)5. P( 1, 2), Q( 4, 1)2.1Segment Bisectors55
Page 4 of 72.1 ExercisesGuided PracticeVocabulary Checkl1. In the diagram shown at the right,name the midpoint and a segment&*.bisector of ABSkill CheckACBM is the midpoint of the segment. Find the segment lengths.2. Find RM and MS.3. Find FM and MG.6R50MSF4. Find MQ and PQ.PG5. Find YM and YZ.PM4MYM20Z48K&*. Find the value of the variable.M is the midpoint of JK6.7.x 14JM33KJM16r&*.Find the coordinates of the midpoint of PR8.9.yP ( 2, 1)P(2, 3)10.yyR (5, 6)114xR (4, 1)1P ( 3, 0)1x2R (1, 3)Practice and ApplicationsExtra PracticeSee p. 677.Recognizing Midpoints In Exercises 11–14, determine whether M is&*. Explain your reasoning.the midpoint of AB11.12.A13.BAMBMA14.MBAMB15.56Chapter 2&*, that is bisected bySketch a line segment, PQline l at point R.Visualize It!Segments and Anglesx
Page 5 of 7Homework HelpExample 1:Example 2:Example 3:Example 4:Exs. 16–19Exs. 20–23Exs. 26–29Exs. 30–35Finding Segment Lengths M is the midpoint of the segment. Findthe segment lengths.16. Find KM and ML.17. Find DM and ME.38K82ML18. Find YM and MZ.DM19. Find AM and MB.17YE2.7MZAMBFinding Segment Lengths Line l bisects the segment. Find thesegment lengths.20. Find CB and AB.21. Find MP and MN.llA36CB22. Find FE and DE.MP23. Find UT and ST.lDlF29.5N15SE3.6UTBiking The Minuteman Bikeway is a 10.5 mile bike path that runsfrom Arlington to Bedford, Massachusetts.24. Caitlin and Laurie begin at opposite ends of the MinutemanBikeway and meet at the halfway point on the path. How far doeseach rider bike?25. Caitlin starts on the path 4.3 miles from the Arlington end. Lauriestarts on the path 3 miles from the Bedford end. How far will eachrider bike before reaching the halfway point on the path?Using Algebra Find the value of the variable.26.IStudent HelpICLASSZONE.COMHOMEWORK HELPExtra help with problemsolving in Exs. 30–35 isat classzone.com28.A6pM72BAr 3M15B27.29.A19Mq 7BA4M2s 6B&*.Midpoint Formula Find the coordinates of the midpoint of PQ30. P(0, 0), Q(4, 6)31. P(3, 8), Q(7, 6)32. P( 5, 6), Q(9, 7)33. P( 12, 0), Q(6, 1)34. P( 4, 4), Q(4, 0)35. P(3, 2), Q( 7, 4)2.1Segment Bisectors57
Page 6 of 7Student HelpLOOK BACKFor more aboutbaseball, see p. 50.Strike Zone In Exercises 36 and 37, use the information below.In baseball, the strike zone is the region a baseball needs to passthrough for the umpire to declare it a strike if the batter does notswing. The top of the strike zone is a horizontal plane passing throughthe midpoint of the top of the batter’s shoulders and the top of theuniform pants when the player is in a batting stance. Source: Major League Baseball36. Find the coordinate of T.37. Find the coordinate of T.6063TT4542242200Student HelpVOCABULARY TIPLines of latitude runparallel to the Equator.Lines of longitude runnorth-south.EXAMPLELatitude-Longitude CoordinatesFind the coordinates ofthe place halfway betweenSan Francisco (37.8 N, 122.4 W)and Los Angeles (34.1 N, 118.2 W).122 W120 W118 W116 WNEVADA38 NSan Francisco(37.8 N, 122.4 W)Solutionx x2y y2 37.8 34.1 122.4 118.2 , 2211 2 , 2M (35.95 , 120.3 W)ANSWERC ALIFOR NIA36 N34 NLos Angeles(34.1 N, 118.2 W) The place halfway between San Francisco and Los Angeleshas coordinates (35.95 N, 120.3 W).Latitude-Longitude Coordinates Find the coordinates of the placehalfway between the two cities in California.38. Fresno: (36.7 N, 119.8 W)Napa: (38.3 N, 122.3 W)39. Bishop: (37.4 N, 118.4 W)Los Angeles: (34.1 N, 118.2 W)40. San Francisco: (37.8 N, 122.4 W) 41. Santa Barbara: (34.4 N, 119.7 W)Palo Alto: (37.4 N, 122.1 W)58Chapter 2Segments and AnglesOakland: (37.8 N, 122.3 W)
Page 7 of 7&*,42. Using Midpoints In the diagram below, B is the midpoint of ACAB 9, and AD 25. Find CD.25A9BCD&* is M(7, 5). The coordinates of43. Challenge The midpoint of ABpoint A are (4, 1). Find the coordinates of point B. Explain.Standardized TestPractice&*. What is the value of x?44. Multiple Choice T is the midpoint of QRA C B D 1729.522P884x 10TR7845. Multiple Choice What is the midpoint of the segment joining(2, 7) and ( 6, 2)?F Mixed Review 2, 92 G H ( 4, 9)J ( 2, 4) 92 , 2 Evaluating Statements Use the diagram at the right to determinewhether the statement is true or false. (Lessons 1.3, 1.5)46. Point A lies on line m.l47. Point E lies on line l.C48. Points B, E, and C are collinear.EA49. Lines l and m intersect at point E.mBFD50. Point E is between points B and C.51. Point F is between points A and B.Classifying Angles Name the vertex and sides of the angle. Thenstate whether it appears to be acute, right, obtuse, or straight.(Lesson 1.6)52.53.54.PJAKPLRBAlgebra SkillsCEvaluating Expressions Evaluate the expression.(Skills Review, p. 670)55. 2 p 15 401256. 120 35 p 357. p 50 14558. p 16 2059. 6 3 p 5 260. 11 p 4 7 2061. 12 p 2 3 p 462. 5 10 p 6 163. 2 (3 4) p 5542.1Segment Bisectors59
Sep 01, 2011 · segment bisector bisect In the Geo-Activity, M is called the midpoint of AB&*. The of a segment is the point on the segment that divides it into two congruent segments. A is a segment, ray, line, or plane that intersects a segment at its midpoint. To a segment means to divide the segment into two congruent segments. M is the midpoint of AB&*.
5.2 Bisectors of a Triangle 277 22. DEVELOPING PROOF Complete the proof of Theorem 5.6, the Concurrency of Angle Bisectors. GIVEN ABC,the bisectors of A, B,and C, DEÆfi ABÆ, DFÆfi BCÆ, DGÆfi CAÆ PROVE The angle bisectors intersect at a point that is equidistant from ABÆ, BCÆ, and CAÆ. P
1 5.1 Bisectors of Triangles I can 1. Identify and use perpendicular bisectors in triangles. 2. Identify and use angle bisectors in triangles. Definitions Perpendicular Bisector –a line, segment, or ray that passes through
2.1 Segment Bisectors goal: Bisect a segment. . Example 2: Angle Measures and Classification Solution bisects , and 1. Find and 2. Determine whether is acute, right, obtuse, or straight L O N M 46o. chp 2.notebook October 11, 2014 Che
A B is an angle bisector of DAC. DAB CAB m DAB m CAB Just as every triangle has three medians, three altitudes, and three perpendicular bisectors, every triangle has three angle bisectors. An angle bisector of a triangle has all of the characteristics of any angle bisector. In FGH, F J bisects GFH. 1. 1 2, so m 1 m 2. 2. m 1 1 2 (m GFH) or 2(m .
IHS/CSM Segment NHTSA Segment Representative Model Tailpipe CO 2 (g/KM) Fuel Economy (mpg) B-segment Car Compact Car Chevrolet Sonic 78 61.1 D-segment Car Mid-Size Car Buick Regal 88 54.9 E-segment Car Full-Size Car Cadillac XTS 102 48.0 C-segment Truck Small SUV Chevrolet Equinox 102 47.5 D-segment Truck Mid-Size CUV Ca
Chapter 6 Test Review 1 Pre-AP Geometry – Chapter 6 Test Review Standards/Goals: D.2.b./G.CO.12.: o I can solve problems with triangles that involve a mid-segment. o I can identify medians, altitudes, perpendicular bisectors, and angle bisectors of triangles and use their properties to solve problems. G.1.b.: I can
bisectors. So, by the Incenter Theorem, ND NE NF. Step 1 Solve for x. ND NE Incenter Theorem 5x 1 2x 11 Substitute. x 4 Solve for x. Step 2 Find ND (or NE). ND 5x 1 5(4) 1 19 So, because ND NF, NF 19. b. Recall that the shortest distance between a point and
Often academic writing is full of technical jargon-technical jargon is an essential ‘tool of the trade’ -jargon eases communication –speeds up exchange of ideas between other professionals-BUT it can also obscure: creates ‘them’ (ordinary ‘laypeople’ culture and [implied] elite ‘professionals’) Beginners don’t always know enough to see errors. Strategies for ‘Being
