Name:___________________________________________________Date:___________Period:_____
HospitalProblem
Boise,ID;Helena,MT;andSaltLakeCity,UTarethreelargecitiesinthenorthwesternpartof
theUnitedStates.Althougheachcityhasalocalhospitalforminorneedsandemergencies,an
advancedmedicalfacilityisneededfortransplants,research,andsoforth.Imaginethe
potentialofahigh-powered,high-tech,extremelymodernmedicalcenterthatcouldbeshared
bythethreecitiesandtheirsurroundingcommunities!
Youhavebeenhiredtodeterminethebestlocationforthisfacility.
Placetheadvancedmedicalfacilityonthemapandjustifywhyyouplaceditwhereyoudid.
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ConstructingPerpendicularBisectors:TheHospitalProblem
1
Inthehospitalproblem,youwerelookingforapointthatwasanequaldistancefromthethreecities.
Thispointiscalledthecircumcenter.Thecircumcenterofatriangleisthepointwherethe
perpendicularbisectors(mid-points)ofthesidesintersect.
1. Findthecircumcenteroftheacute,rightandobtusetrianglesusingconstructions.Then
constructthecirclethatpassesthroughtheverticesofeachtriangle.
Triangle1:
Triangle2
ConstructingPerpendicularBisectors:TheHospitalProblem
2
Triangle3
2. Findthedistanceincentimetersfromthecircumcenterandtheverticesofeachtriangle.
a. Triangle1:____________________________
b. Triangle2:____________________________
c. Triangle3:____________________________
3. Wherewasthecircumcenterlocatedforeachofthefollowingtriangles?
a. AcuteTriangle_________________________
b. RightTriangle_________________________
c. ObtuseTriangle_______________________
4. Doyouhavetoconstructallthreeperpendicularbisectorstofindthecircumcenterofthe
triangle?Whyorwhynot.
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5. Doyouthinkthecircumcenteristhebestplacetobuildasharedfacility,likethehospital,for
threecities?Explain.
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ConstructingPerpendicularBisectors:TheHospitalProblem
3
TeacherDirections
Adaptedfrom:NCTMIlluminations:HospitalProblem
Objective:Useconstructionstofindthepointequidistantfromtheverticesofatriangle.
Materials:
Compass:1perstudent
Ruler:1perstudent
Computer:withaccesstogeogebra
DigitalProjector:
Geogebrafile:http://tube.geogebra.org/student/m168553
Directions:
Note:Thecircumcenterisaninterestingandusefulpointthatgivesusameaningfulreasonto
constructperpendicularbisectors.Afullunderstandingofcircumcenterisnotexpectedthis
unit,butafacilitywithconstructingperpendicularbisectorsis.
Havestudentsreadthedirectionsandletthemfindthebestlocationforthehospital.Students
mustexplainwhytheyplacedthehospitalinthelocationtheydid.Havestudentssharetheir
answerswiththeclass.AnswerswillvaryfrompointsontheanglebisectorofHelena,Boiseand
SaltLakeCity.Makesuretoendtheconversationwiththatthehospitalshouldbeequidistant
fromallthreecities.Reasoncanbetobefairtoeachhospitaloranothercleverreasonwhythe
hospitalcannotbeclosertoonecity.
Oncestudentsunderstandthatwearetryingtofindthepointequidistanttothethreecities
opentheGeogebraapplet.HaveastudentcometothecomputerandmovetheHospitaluntilit
isreasonablyclosetoequidistantfromeachcity.Distanceswillchangedynamically.Onceitis
complete,changethelocationsofthecitiestorepresentanequilateraltriangleandhave
anotherstudentfindwherethehospitalshouldbeplaced.Oncetheyhavefoundthelocation,
askstudentsiftheythinkthereisaneasierwaytofindthislocation.
ClicktheButtonBoisetoHelenaPB.AskstudentswhatdoesPBstandfor.(PB=Perpendicular
Bisector)
ClicktheButtonBoisetoSaltLakeCityPB.Askstudentswhatdotheynotice.
AskstudentswhatdotheythinkwillhappenwhenweclickHelenatoSaltLakeCityPB.
Finally,askstudentsforthedefinitionofacircle.ThenclicktheDrawCirclebutton.Resizethe
circlesothatthecircumferenceofthecirclepassesthrougheachvertex.
Hidethecircleandperpendicularbisectorsbyclickingthebuttonsonce.Changethetriangle
onemoretime.Thistimeusetheperpendicularbisectorstofindthelocationofthehospital.
Verifythatacirclewillpassthroughtheverticesagain.
Havestudentsreadthedirectionsandcompletetheremainderoftheactivity.Studentsshould
befamiliarwithconstructingtheperpendicularbisectorofasegment.Ifnotmodelthestepsto
constructingaperpendicularbisector.
ConstructingPerpendicularBisectors:TheHospitalProblem
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