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× × = =