2018–RegionalMathematicsContest
GeometryTest
Ineachofthefollowing,choosetheBESTanswerandrecordyourchoiceontheanswersheetprovided.Toinsurecorrectscoring,besuretomakeallerasurescompletely.Thetie-breakerquestionsattheendwillonlybeusedtoresolvetiesinfirst,second,and/orthirdplace.Theywillbeusedintheordergiven.Completethefirst25questionsbeforeattemptingthetie-breakerquestions.Figuresarenotnecessarilydrawntoscale.
1.Inordertoconstructtheincenterofatriangle,whichofthesestepsisnecessary.
A.Constructtheperpendicularbisectorofthesidesofthetriangle.
B.ConstructtheAngleBisectorsfortheanglesofthetriangle.
C.ConstructtheMediansofthetriangle.
D.ConstructtheAltitudesofthetriangle.
E.Noneofthese.
2.GivenΔABC.DisthemidpointofAB,EisthemidpointofAC,andFisthemidpointofBC.Whichofthefollowingstatementscannotbeproven.
A. ABC EFCD @ D
B.DBFEisaparallelogram
C. DAE BFEÐ @ Ð
D. ECFÐ and DBFÐ aresupplementaryangles
E.Noneofthese.
3.Giventhestatement:“Ifafigureisarectangle,thenitisaparallelogram.”Ifthestatementismadethat“Ifthefigureisnotaparallelogram,thenitisarectangle”wouldbewhichofthefollowing?
A.Inverse
B.Converse
C.Contrapositive
D.Syllogism
E.Noneofthese
4.Thesidesofaparallelogramare12cmand20cm.Onediagonalmakesanangleof30degreeswiththe20cmside.Findthelengthofthediagonal.
A.23.3cm
B.24.0cm
C.11.3cm
D.16.7cm
E.Noneofthese.
5.GiventheVennDiagram,whichstatement(s)aretrue?
A.AllGimbelsareSprockets
B.AllWidgetsareSprockets
C.AllSprocketsareGimbels
D.AllBrokinsareGimbels
E.Noneofthese
6.Theareaofthesectorofacircleisknowntobe 224 inp .Ifthediameterofthecircleis16in,thenfindthemeasureofthecentralangleofthesector.
A.135°
B.33.75°
C.67.5°
D.90°
E.Noneofthese.
7.Whichanglehasthelargestmeasureinthefollowingpicture.
A.BAE
B.DAE
C.CGF
D.BAD
E.Notenoughinformation.
8.Given:segmentCDisaperpendicularbisectorofsegmentACandthem BAD m ABCÐ = Ð .WhattypeoffigureisADBC?
A.Chevron
B.Rectangle
C.Rhombus
D.Square
E.Noneofthese
9.GiventhatthesegmentABistangenttothecircleandBCisadiameterofthecircle.SegmentBDis5unitslong.Theradiusofthecircleis6.5units.FindthelengthofsegmentAC.
A.14.1units
B.12units
C.17units
D.33.8units
E.Notenoughinformation.
10.Theverticesofaparallelogramlieonthecoordinateplaneat(0,4),(1,1),(6,5)and(7,2).Findtheperimeteroftheparallelogram.
A. ( )2 53 41+
B. ( )73 26+
C.9.24
D. ( )2 10 37+
E.Noneofthese.
11.Thepoints(1,1),(1,3)and(4,1)encloseatriangularregion.Whichofthefollowingsetsofpointsdoesnotencloseacongruenttriangularregioniftheoriginaltriangularregionundergoesasinglerigidtransformationoraseriesofrigidtransformations?
A.(–2,–5),(–5,–3),and(–5,–5)
B.(–4,–2),(–7,–2),and(–7,–3)
C.(3.5,–1.2),(6.5,–1.2),and(6.5,–3.2)
D.(–2,3),(–5,3),and(–5,1)
E.Noneofthese.
12.Youaregivenacirclewithradiusof5m.Findthelengthofsidebinthefollowingfigure.
A.8.7m
B.5.7m
C.3.4m
D.8.2m
E.Noneofthese.
13.Giventhat1sideofarectangleliesontheline2 23
y x= + .Whichlinecannotcontainasideofthe
rectangle?
A.2 43
y x= -
B.3 62
y x= - +
C.6 54
y x= - -
D.2 83
y x= +
E.Noneofthese.
14.Givenacirclewithcenterat(3,2)andradiusof3units.Whichofthefollowingpointsdoesnotlieonthecircle?
A. ( )3,5
B. ( )1, 2 5+
C. ( )5, 2 3+
D. ( )4, 2 2 2-
E.Noneofthese.
15.GiventheinscribedquadrilateralABCD.SegmentBDbisectsangleABC.Ifthe 80m BADÐ = ° andthe 47m BDCÐ = ° ,thenfindthem ADBÐ .
A.33°
B.47°
C.67°
D.53°
E.Noneofthese.
16.Giventhatthecentralangleinacirclehasameasureof2.5radians.Findtheapproximatelengthofthearcinterceptedbythatanglewhentheradiusofthecircleis8cm.
A.20cm
B.0.35cm
C.6.4cm
D.50.2cm
E.Notenoughinformation.
17.Given ABCD withED CB! .ThelengthofABis9inches,thelengthofADis3inches,thelengthofDEis5.2inches,andthelengthofBCis15.6inches.FindthelengthofsideAC.
A.5.6inches
B.8inches
C.8.4inches
D.12.7inches
E.Noneofthese.
18.Findthecenterandradiusofthecirclewhoseequationisgivenby:𝑥" + 𝑦" = 2 3𝑥 − 2𝑦 + 6
A.Center=(3,–1),radius=4units
B.Center=(3,–2),radius=5units
C.Center=(3,–2),radius=2 3
D.Center=(3,2),radius=5
E.Noneofthese.
19.GivenCircleOwithangularmeasuresasshown.Findthem AODÐ .
A.82°
B.90°
C.164°
D.76°
E.Notenoughinformation
20.Awomanstandsontopofabuildingdirectlyoverthebaseofthebuildingclosesttoatree.Shenoticesthattheangleofdepressiontothebaseofthattreeis38°.Shealsonoticesthattheangleofelevationtothetopofthatsametreeis15°.Ifthebaseofthetreeis15mfromthebaseofthebuilding,thenhowtallisthetree?
A.15.7m
B.82.4m
C.26.3m
D.34.6m
E.Noneofthese.
21.Thedistance(apothem)fromthecenterofaregularhexagontothemidpointofasideofthathexagonhasalengthof4units.Findthecircumferenceofthecirclethatcircumscribesthehexagon.
A.8p
B.16p
C.16 3p
D.16 33
p
E.Noneofthese.
22.Thevolumeofasphereis 32563
inp .Findtheareaofacirclethathasthesameradiusasthesphere.
A. 28 inp
B. 216 inp
C.3
216 33
inp
D. 264 inp
E.Noneofthese.
23.Giventhat ABCD and ADCD arebothinscribedincircleO.Whichofthefollowingpiecesofinformationissufficienttoprovethat ABC ADCD @ D withoutanyadditionalinformation?
A. ! !BC DC@
B. ! !AD AB@
C. AC passesthroughpointO
D. AD AB@
E.Noneofthese.
24.Youhaveacylinderwithasurfaceareaof 248 cmp andavolumeof 340 cmp .Ifyoudoubletheheightandthediameterofthebaseofthecylinder,thenwhatisthenewvolume?
A. 380 cmp
B. 3160 cmp
C. 3320 cmp
D. 3384 cmp
E.Notenoughinformation.
25.Twoairplanesleavetheairportatthesametime.Oneplanefliesdueeastat420mph.Theotherplaneflies20°WestofNorthat450mph.In2hours,howfarapartaretheplanes?
A.1426miles
B.1231miles
C.999miles
D.1331miles
E.Noneofthese
TieBreaker#1
Name:_______________________________
IdentifyasequenceoftransformationsthatareneededtocarryABCDontoA’B’C’D’.
A
B
C
D
A’
B’
C’
D’
TieBreaker#2
Name:________________________________
Listthestepsneededtoconstructasquareusingonlyastraightedgeandacompass.
TieBreaker#3
Name:________________________________
Aregularhexagonisinscribedinacircle.Iftheareaofthecircleis32p ,thenfindtheareaofthehexagon.
Solutions:
1.B
2.D
3.E
4.B
5.A
6.A
7.E
8.C
9.A
10.D
11.B
12.D
13.E
14.C
15.C
16.A
17.E
18.B
19.E
20.A
21.D
22.B
23.E
24.C
25.A
TieBreaker1
Thesolutioncanconsistofarotationfollowedbyatranslation.
Thesolutioncanconsistofverticalreflectionandahorizontalreflectionthenatranslation.
Thereareothercombinationsthatcouldalsocreatethistransformation.
TieBreaker2
Firstyoumustdrawalinesegmentofanylength.
Markapointonthelinesegment.
Constructalineperpendiculartothesegmentthroughthatpoint.Thisisdonebyusingthecompasstomarkoffthelinesegmentoneachsideofthechosenpointafixeddistancefromthepoint.Oncethesegmentismarkedinbothoftheselocations,thenyouneedtoextendthecompassbeyondthatopening.Thenusingthenewlycreatedmarksasthefixedlocationforthecompass,placeanarcabovethechosenpointonthesegmentwitharcscomingfromthemarksonbothsidesofthechosenpoint.Nowusingthestraightedge,constructalinebetweenthechosenpointonthelineandtheintersectionofthetwoarcs.
Usingtheintersectionofthetwoperpendicularlines,usethecompasstomarkoffequalsegmentsonbothlines.
Thenusingthecompassopentothesidelengthofthesquare,makeanarcwithacenterattheendpointofeachoftheperpendicularsidesthatarenotthepointofintersection.Thatintersectionpointwillbethefourthvertexofthesquare.
Useastraightedgetoconnecttheendpointsofthefirsttwosidesofthesquaretothisfourthvertex.
Note:Therearemanywaysthatthiscanbedone,sothepersonscoringthissectionmusthavesomeknowledgeofconstructions.
TieBreaker3
Theareaofthecircleis32p .So,
2 32rp p= solvingforrgives
4 2r =
Inthehexagon,theradiusofthecircleisshown.
Thestudentscouldshowthatthereare6equilateral
triangleswithsidelengthof4 2 ortheycoulduse
theperimeterandtheapothem.Ineithercasethey
mustfindthedistancefromthecenterofthehexagontothemidpointofasideofthehexagon(theapothem).Thiscanbefoundbyusinga30-60-90triangle(halfofoneoftheequilateraltriangles.)Thiswouldgivethelengthsofthe30-60-90triangletobe:
Theareaofeachofthe6equilateraltrianglescanbefoundas1 4 2 2 6 4 12 8 32
× = = .
Multiplyingby6,wegetthetotalareaofthehexagontobe 48 3 .
Ifyouchoosetousetheperimeterandtheapothem,thenyouwouldgetaperimeterof
6 4 2 24 2× = andtheapothemis 2 6 .Thatgivesanareaof 1 24 2 2 6 24 12 48 32× × = =