math: Plane Geometry Practice CirCles

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math: Plane Geometry Practice

CirCles

2. What is the area of a semicircle with a diameter of 2 ?

F. 2π

G. π

H. π2

J. 1

K. 1

π

3. The area of circle M is 16π, and the radius of circle N is twice the radius of circle M. What is the area of circle N ?

A. 8B. 8πC. 32πD. 64E. 64π

4. Segment AB cuts circle Q, which has an area of 36π, into two equal parts. If A and B are both points on the circle, then what is the length of segment AB ?

F. 2G. 4H. 6J. 8K. 12

5. The circumference of a yo-yo with a diameter of 16 cm is:

A. 4π cmB. 8π cmC. 16π cmD. 36π cmE. 128π cm

14. Two circles have radii in a ratio of 1:2. What is the ratio of their areas?

F. 1:2G. 1:4H. 1:8J. 1:2πK. 1:4π

21. A ferris wheel makes 35 complete revolutions every hour. Through how many degrees does the ferris wheel rotate in 2.5 hours?

A. 40,500B. 31,500C. 26,250D. 18,900E. 15,950


38. In the figure below showing a circle with center O, minor arc HJK has a length of 24π. If the length of OH equals the length of HK, what is the radius of the circle with center O ?

F. 4G. 12H. 24J. 72K. 144

40. The circle below with center O has a radius of 6. If ∠OPQ has a measure of 30 degrees, then what is the area of the section of the circle bordered by OP, OQ, and minor arc PQ ?

O

Q

P

F. 12πG. 24πH. 36πJ. 144πK. Cannot be determined from the given information

43. The area of the circle with center D, shown below, is 32π. If point B is on the circle, what is the area of square ABCD ?

A. 4 2

B. 4πC. 16

D. 16 2

E. 32

O

H

KJ

B C

A D•

24. Points A and C lie on the circumference of the circle with center

O. If ∠AOC is 45° and the radius of the circle is 8, what is the

length of minor arc AC ?

F. πG. 2πH. 3πJ. 4πK. 6π

28. In the figure below, AC is a diameter of the circle and is 4 units long. What is the area, in square units, of the circle?

F. 2πG. 4πH. 6πJ. 8πK. 16π

34. The ratio of the radius of circle A to the radius of circle B is 3:5. The ratio of the radius of circle A to the radius of circle C is 3:4. What is the ratio of the area of circle B to the area of circle C ?

F. 5:4G. 9:16H. 9:25J. 15:11K. 25:16

36. In circle O below, if ∠AOB is 90º and AO is 8 units long, how many units long is minor arc AB ?

F. 2πG. 4πH. 8πJ. 16πK. 32π

CA

O

A B

8


49. Given an ellipse and a circle, what is the maximum number of points of intersection the 2 figures can have?

A. 0 onlyB. 1 onlyC. 0, 1, or 2 onlyD. 0, 1, 2, or 4 onlyE. Infinitely many

45. Amelia and Megan have choreographed a ballet piece to be performed on a stage. If each dancer executes a pirouette that covers a circular area of 28.3 square feet, and there is one foot of space in between the circular areas their pirouettes create, what is the minimum length of the stage that will allow the dancers to perform side by side, without collision?

A. 20B. 13C. 12D. 7E. 4


7. In the figure below, a = 4, b = 5, and c = 8. Given that the triangles are similar and that the side of length b corresponds to the side of length y, what is the value of z if x = 10 ?

x

zy

a

bc

A. 8.0B. 10.0C. 12.5D. 15.0E. 20.0

8. What is the area of the figure shown below?

F. 60G. 64H. 70J. 80K. 84

anGles and shaPes

3. In ∆XYZ below, if XY = YZ , then a = ?

X Z

Y

a°

25°

A. 155ºB. 145ºC. 130ºD. 125ºE. 115º

5. In the figure below, ∆ABC is an isosceles triangle where AB is

equal to BC . If the measure of ∠C is 35 degrees and the measure

of ∠ABD is 4

5 the measure of ∠ABC, then what is the measure,

in degrees, of ∠BDA ?

A

B

CD

A. 110B. 88C. 57D. 35E. 22


16. If one side of a rectangle is 2n – 3 units in length and the side perpendicular is 3n + 5 units in length, which of the following gives the area of the rectangle?

F. 5n + 2G. 6n – 15H. 6n2 – 15J. 6n2 + n – 15K. 6n2 + 8n – 15

17. What is the length of the diagonal of a rectangle with a side of 4 and an area of 12 ?

A. 6B. 5C. 4D. 3E. 2

18. In the figure below, the area of rectangle ACDE is 16 and B is the

midpoint of AC . What is the area of ∆ABE ?

A

E D

B C

F. 2G. 4H. 6J. 8K. 16

19. A regular pentagon, P1, has a perimeter of 40 inches. The length

of each side in another regular pentagon, P2, exceeds the length

of each side of P1 by 3 inches. What is the perimeter, in inches,

of P2 ?

A. 70B. 66C. 55D. 38E. 25

10. In the figure below, parallel lines M and N are crossed by trans-versal E. If the measure of angle 1 is 70 degrees and the measures of angle 2 and angle 3 are equal, then what is the measure of angle 2 ?

M

N

E

3 2

1

F. 25o

G. 35o

H. 40o

J. 50o

K. 55o

12. What is the hypotenuse of a right triangle whose legs measure 15 and 36 ?

F. 5G. 10H. 30J. 39K. 51

15. A triangle with an area of 48 has a base that is 6 times as long as its height. Which of the following equations could be used to solve for the height of the triangle, h ?

A. h(h + 6) = 48

B. h(6h) = 48

C. h h6

248

( )=

D. h h1

648

=

E. h h

1

6

248

=


22. In the figure below showing three straight lines intersecting at a common point, which of the following expressions gives all possible values of y ?

y°

x°

F. xG. 90 – xH. 90 + xJ. 180 – xK. 180 + x

24. In right triangle PQR below, PQ has a length of 6 units, and

PR has a length of 12 units. If QR has a length between 6 and 12, what is the measure of ∠PRQ ?

P

Q R

126

F. 30°G. 45°H. 60°J. 90°K. 120°

20. In the figure below, line A is parallel to line B, and line C inter-sects them both. Which of the following lists 3 equal angles?

F. ∠l, ∠n, ∠kG. ∠l, ∠m, ∠kH. ∠l, ∠m, ∠pJ. ∠m, ∠n, ∠kK. ∠m, ∠n, ∠p

21. In the figure below, BC and DE are parallel. If AB = 4,

AD = 8, and DE = 6, what is the length of BC ?

A

CB

D E

A. 4

3B. 2

C. 3

D. 4

E. 4 5


Use the following information to answer questions 35–36.

In the figure below, D and F are on AB and E and G are on AC .

The measurements given are in centimeters.

175

50

120

180

B C

G

A

ED

F

35. What is the area of triangle AFG, in square centimeters?

A. 21,000B. 36,750C. 42,000D. 73,500E. 87,500

36. What is the length of AC , in centimeters?

F. 130G. 600H. 650J. 780K. 910

37. If ∆ABC is inscribed in equilateral triangle DEF such that the vertices A, B, and C divide each side of DEF into 2 equal seg-ments, then ABC must be:

A. equilateralB. a 45-45-90 triangleC. isosceles (2 sides congruent)D. a 30-60-90 triangleE. obtuse (one obtuse angle)

25. A square with an area of 36 is inscribed in a circle with center O. Which of the following is closest to the area of circle O that does not overlap with the area of the square?

A. 77.0B. 56.5C. 36.5D. 20.5D. 19.0

26. Leo runs from point A to point B and then from point B to point C. Fritz runs directly from A to C. How much longer is Leo’s trip than Fritz’s?

A

C

B

60 ft

80 ft

F. 15 feetG. 20 feetH. 40 feetJ. 60 feetK. 100 feet

27. If an edge of a cube has a length of 4 meters, what is the volume of the cube in cubic meters?

A. 16B. 32C. 64D. 144E. 256

31. A statue in the shape of a right triangle has a height of 7 meters, a base of 24 meters, and a hypotenuse of 25 meters. A sculp-tor is commissioned to make a replica of the statue. If the new hypotenuse is 13 meters long, what is the new base?

A. 12

B. 1212

25

C. 13

D. 30

E. 444

7


49. An isosceles right triangle has a perimeter of 4 4 2+ units. What is the length of one of the legs?

A. 2B. 2

C. 2 2D. 4

E. 4 2

51. One side of a right isosceles triangle is 6 units long. Which of the following could be the length of one of the other two sides?

A. 2

B. 3

C. 3 2

D. 6 3E. 12

53. In the figure below, QLNO is a parallelogram and LMOP is a square. If ∠ Q measures 45° and segment PO measures 3 units, what is the ratio of the length of segment LQ to the length of segment LN ?

L M N

P 345º

Q O

A. 1

3

B. 2

2

C. 1

2

D. 1

E. 2

40. The diagonal of a rectangle is 16. How wide is the rectangle if the length is 6 ?

F. 10

G. 11

H. 4 10

J. 2 55

K. 55

45. The height of an equilateral triangle is 4 3 inches. What is the perimeter of the triangle, in inches?

A. 8

B. 12

C. 8 3

D. 24

E. 16 3

46. In the figure below, ∠T measures 30 degrees in right triangle

TUV. If x and y represent the lengths of the indicated sides, what

is the ratio of y

x?

T

U

V

x

y30°

F. 1

2

G. 1

H. 2 2

2

J. 2 3

3

K. 3


57. Three distinct chords are drawn in a circle with center O. What is the maximum number of distinct, nonoverlapping regions of nonzero area into which these chords divide the interior of the circle?

A. 10B. 8C. 7D. 6E. 4

58. In the figure below, lines x and y are parallel and angle measures are as indicated. If it can be determined, what is the value of x?

x°

50°

30°x

y

F. 50G. 80H. 100J. 130K. Cannot be determined from the given information

54. In the figure below, ∆ABC is an equilateral triangle and D is the

midpoint of AC with AD = 3. What is the area of ∆ABC ?

A

B

CD

F. 4

G. 9 3H. 15J. 21

K. 18 3

55. What is the maximum number of points of intersection for a circle and a rectangle which lie in the same plane?

A. 2B. 4C. 6D. 8E. Infinitely many


30. In the right triangle∆ABCshown below, BC is 18 feet long. The

tangent of∠Cis 5

12. About how many feet long is AB ?

A

B C18

F. 6.92G. 7.50H. 12.50J. 16.00K. 19.50

37. In the right triangle below, if tan a = 6

7, then cos b = ?

A. 7

6

B. 7

C. 6 85

85

D. 6

E. 85

57. A straight thirty-foot-long ramp leads from the ground to a level platform. If the ramp tilts at a 35° angle from the ground, how high is the platform?

A. 35 sin 30°

B. 35 sin 35°

C. 30

35cos º

D. 30 sin 35°

E. 30

35cos º

b

a

triGonometry

19. The legs of a right triangle are 5 inches and 12 inches, respec-tively. What is the sine of the smallest angle?

A. 5

13

B. 5

12

C. 12

13

D. 13

12

E. 13

5

26. If 0° < θ < 90°, and cos θ = 4

5, then tan θ = ?

F. 4

5

G. 3

4

H. 1

J. 5

4

K. 4

3

28. In the figure below, ∆SUT is a right triangle. If SU is 5 units long, and UT is 12 units long, then what is the tangent of ∠T ?

F. 5

13

G. 5

12

H. 12

13

J. 12

5

K. 13

5

U

S

T

5

12


answers and exPlanations

CirCles2. H

(F) Incorrect.Thesemicircleshouldbehalftheareaofthecircle,notdouble.

(G) Incorrect.Theproblemaskedforasemicircle;thiswouldbetheanswerforacircle.

(H) Correct.Cutthediameterinhalftofindtheradius,andthenplugitinto πr2.Remembertocuttheresultinhalfbecauseit’sasemicircle.

(J) Incorrect.Youshouldhaveaπinyouranswer.

(K) Incorrect.Noneedtoinvertπhere.Divideitby2.

3. E

(A) Incorrect.Thisisapartialanswer:thevalueoftheradiusofcircleN.

(B) Incorrect.Youneedtosquarethevalueoftheradius.

(C) Incorrect.Trapanswer!Don’tjustdoubletheareaofM.FigureoutM’sradius,doublethat,anddetermineN’sradiusfromthere.

(D) Incorrect.Youshouldhaveaπinyouranswer.

(E) Correct.First,findtheradiusofCircleMworkingbackwardsfromitsarea.Thendoubleitandplugthatvalueintoπr2.

4. K

(F) Incorrect.Youcanballparkthisoneout:theproblemisaskingforthediameter,whichmustbelongerthantheradius.

(G) Incorrect.Youcanballparkthisoneout:theproblemisaskingforthediameter,whichmustbelongerthantheradius.

(H) Incorrect.Thisisthevalueoftheradius,whichisapartialanswer—theproblem(indirectly)askedforthediameter.

(J) Incorrect.Sincetheareais36π,theradiusofthecirclewillbe6.Doublethat.

(K) Correct.DeterminetheradiusofcircleQworkingbackwardsfromitsarea.Thendoubleittofindthediameter.

5. C

(A) Incorrect.Thecircumferenceformulacallsforπtimesthediameter.

(B) Incorrect.Thisisπtimestheradius—thecircumferenceformulacallsforπtimesthediameter.

(C) Correct.Thecircumferenceisfoundbymultiplyingthediameterbyπ.

(D) Incorrect.Thecircumferenceformulacallsforπtimesthediameter.

(E) Incorrect.Thecircumferenceformulacallsforπtimesthediameter.

14. G

(F) Incorrect.Trapanswer!Actuallydeterminetheareasofthetwocircles;don’tassumeanything.

(G) Correct.Plugthevaluesofthetworadiiintoπr2.

(H) Incorrect.Plugthevaluesofthetworadiiintoπr2.

(J) Incorrect.Plugthevaluesofthetworadiiintoπr2.

(K) Incorrect.πispartofbothareas,soitcancelsoutwhenyoucomparethetwo.

21. B

(A) Incorrect.Useyourcalculatortomultiplythenumberofrevolutionsby360by2.5.

(B) Correct.Onerevolutionofaferriswheelmeansonefulltriparoundacircle.Thatmeansthat35timesanhour,theferriswheeltravels360degrees,anditdoesthis2.5times.Multiply35×360×2.5.

(C) Incorrect.Useyourcalculatortomultiplythenumberofrevolutionsby360by2.5.

(D) Incorrect.Useyourcalculatortomultiplythenumberofrevolutionsby360by2.5.

(E) Incorrect.Useyourcalculatortomultiplythenumberofrevolutionsby360by2.5.


36. G

(F) Incorrect.Setupaproportion.Thisishalfthevalueofthearc.

(G) Correct.Thecentralangle’sratioto360°is

proportionaltothearc’sratiotothecircumference:90360 16

= xπ

.

(H) Incorrect.Ballpark:thisishalfthecircumference.

(J) Incorrect.Thisistheentirecircumference.

(K) Incorrect.Ballpark:thisisgreaterthantheentirecircumference.

38. J

(F) Incorrect.Ballpark,andthisistoosmall.Didyousetuptheproportioncorrectly?

(G) Incorrect.Ballpark,andthisistoosmall.Didyousetuptheproportioncorrectly?

(H) Incorrect.Ballpark,andthisistoosmall.Didyousetuptheproportioncorrectly?

(J) Correct.SincebothOHandOKareradii,HOKis

anequilateraltriangle,makingthemeasureofeach

angle60°.∠HOK’sratioto360°isproportionaltothe

arc’sratiotothecircumference:60360

24= πx

.Findthe

radiusbyworkingbackwardsfromthere.

(K) Incorrect.Thisisdoublethevalue.Didyousetuptheproportioncorrectly?

24. G

(F) Incorrect.Sketchandballpark:πisonly1

16ofthe

circumference.Thearcshouldbelargerthanthat.

(G) Correct.Thecentralangle’sratioto360°is

proportionaltothearc’sratiotothecircumference:45

360 16= x

π.

(H) Incorrect.Setuptheproportion:thecentralangletothe360°,proportionaltothearctothecircumference.Thisshouldnotmatchyourcalculation.

(J) Incorrect.Sketchandballpark:4πisaquarterofthecircumference.Thearcshouldbesmallerthanthat.

(K) Incorrect.Sketchandballpark:6πisnearlyhalfofthecircumference.Thearcshouldbesmallerthanthat.

28. G

(F) Incorrect.Remembertosquaretheradius.

(G) Correct.Cutthediameterinhalftofindtheradius,thenplugitintoπr2.

(H) Incorrect.

(J) Incorrect.

(K) Incorrect.Remembertocutthediameterinhalf;don’tplugthediameterintoπr2.

34. K

(F) Incorrect.Trapanswer!Don’tassumeanything.Theratiooftheradiiisnotthesameastheratiooftheareas.

(G) Incorrect.Trapanswer!ThiswouldbetheratioofcircleAtocircleC.

(H) Incorrect.Trapanswer!ThiswouldbetheratioofcircleAtocircleB.

(J) Incorrect.Pluginfortheradiianddetermineeachcircle’sarea.MakesureyouarecomparingBtoC.

(K) Correct.Plugintheeasiestnumbersavailable:3fortheradiusofA,4fortheradiusofB,and5fortheradiusofC.Thendetermineeachcircle’sareausingπr2.


45. B

(A) Incorrect.Thisstageisbigenough,buttheproblemaskedfortheminimumpossiblelength.Eventhoughitworks,it’snotthebestanswer.

(B) Correct.Theradiusofthecircleeachdancerrequirescanbefoundbydividing28.3byπandtakingthesquarerootoftheresult.Doublethisradiustodeterminethediameter.Combinebothdiameters(roughly6)andaddthefootbetweenthedancers.

(C) Incorrect.Don’tforgetthefootofspacebetweenthem.

(D) Incorrect.Youmayhaveforgottentofactorinbothcircles,oryoumayhaveaddedtogethertheirradiiinsteadoftheirdiameters.

(E) Incorrect.Ballpark:thisisn’tbigenoughtocoverevenonedancer’spirouette.

49. E

(A) Incorrect.Thetwofigurescanintersect.

(B) Incorrect.Thetwofigurescanintersectmorethanonce.

(C) Incorrect.Thetwofigurescanintersectmorethantwice.

(D) Incorrect.Ifthetwofiguresarenotidentical,thisansweriscorrect,butacircleisaformofellipse,sotheycanbeidentical—nothingintheproblemrulesthisout.

(E) Correct.Iftheellipsehappenstobeacircle,thetwofigurescanbeidentical.

anGles and shaPes3. C

(A) Incorrect.Subtractthevalueofboth∠’sXandZ,notjustoneofthem.

(B) Incorrect.∠’sXandZareequal.Subtracttheirtotalfrom180°.

(C) Correct.SinceXY=YZ,∠X=∠Z.Addtheseandsubtractfrom180°.

(D) Incorrect.∠’sXandZareequal.Subtracttheirtotalfrom180°.

(E) Incorrect.∠’sXandZareequal.Subtracttheirtotalfrom180°.

40. F

(F) Correct.Determinethecircumference.Sinceboth

OPandOQareradii,anglesPandQareequal.Angle

POQ’sratioto360°isproportionaltothesectorarea’s

ratiotothecircle’sarea:120360 36

= xπ

.

(G) Incorrect.Noneedtodoublethesectionarea.

(H) Incorrect.Thisistheareaoftheentirecircle.

(J) Incorrect.Noneedtosquarethesectionarea.

(K) Incorrect.Thesolutioncanbedetermined.

43. C

(A) Incorrect.Partialanswer–thisistheradiusofthecircle(andthediagonalofthesquare).

(B) Incorrect.πshouldnotappearintheanswerfortheareaofasquare.

(C) Correct.Fromthevalueofthearea,determinethe

radius,whichcanbedrawnatBD,makingitthe

diagonalofthesquare.Thediagonalis 4 2 ,making

thesidesofthesquareequalto4.

(D) Incorrect.Youcandropthe 2 fromyour

calculationsafterfindingthesidesofthesquare.

(E) Incorrect.Youmayhavesquaredthevalueofthediagonal,notthevalueoftheside.


10. K

(F) Incorrect.Subtractthevalueofangle1from180°anddividetheresultinhalf.

(G) Incorrect.Subtractthevalueofangle1from180°anddividetheresultinhalf.

(H) Incorrect.Subtractthevalueofangle1from180°anddividetheresultinhalf.

(J) Incorrect.Subtractthevalueofangle1from180°anddividetheresultinhalf.

(K) Correct.Subtractthevalueofangle1from180°anddividetheresultinhalf.

12. J

(F) Incorrect.Youcanballparkthisoutbecausethehypotenusemustbelongerthaneitherleg.

(G) Incorrect.Youcanballparkthisoutbecausethehypotenusemustbelongerthaneitherleg.

(H) Incorrect.Youcanballparkthisoutbecausethehypotenusemustbelongerthaneitherleg.

(J) Correct.Thisisa5-12-13triangleinwhicheachsidehasbeentripled.

(K) Incorrect.You’vemadea +b=cinsteadofa2+b2=c2.

15. C

(A) Incorrect.Theproblemsaysthatthebaseissixtimesaslong;thisformulamakesit6unitslonger.Readcarefully!

(B) Incorrect.Remembertodividebhby2inthetriangleareaformula.

(C) Correct.Thisis12

bh .

(D) Incorrect.Thebaseis6timesaslongastheheight,notviceversa.

(E) Incorrect.Thebaseis6timesaslongastheheight,notviceversa.

5. C

(A) Incorrect.Thisisthevalueof∠ABC.

(B) Incorrect.Thisisthevalueof∠ABD.

(C) Correct.SinceAB = BC,∠A=∠C.Addtheseand

subtractfrom180°tofind∠ABC.Multiplythisvalue

by45

tofind∠ABD.Addthistothevalueof∠Aand

subtractfrom180°tofind∠BDA.

(D) Incorrect.Thisisthevalueof∠’sAandC.

(E) Incorrect.Thisisthedifferencebetween∠’sABDandABC.

7. E

(A) Incorrect.Thisisthevalueofc.

(B) Incorrect.Thisisthevalueofx.

(C) Incorrect.Thisisthevalueofy.

(D) Incorrect.Thisisthevalueofx+b.

(E) Correct.Inthesimilartriangles,acorrespondstox,

andcto z.Setupaproportion:ax

cz

= .

8. G

(F) Incorrect.Didyouassumetheupperpartofthefigureisexactlyhalfthelower?Itisn’t;dothecalculationscarefully.

(G) Correct.Dividethefigureintotwoparts,andyougeta6-8-10triangleontopandan8×5rectangleonbottom.Addtheareasofthesetwofigures.

(H) Incorrect.Dividethefigureintotwoparts:atriangleontopandarectangleonbottom.Addtheirareas.Thisshouldnotmatchyourcalculations.

(J) Incorrect.Dividethefigureintotwoparts:atriangleontopandarectangleonbottom.Addtheirareas.Thisshouldnotmatchyourcalculations.

(K) Incorrect.Dividethefigureintotwoparts:atriangleontopandarectangleonbottom.Addtheirareas.Thisshouldnotmatchyourcalculations.


19. C

(A) Incorrect.EachsideofP1is8,whichmeanseachsideofP2is11.Multiply11×5.

(B) Incorrect.Bothfiguresarepentagons.IfyoufiguredoutthateachsideofP2is11,youmayhavemultipliedbysixinsteadoffive.

(C) Correct.EachsideofP1is8,whichmeanseachsideofP2is11.

(D) Incorrect.EachsideofP1is8,whichmeanseachsideofP2is11.Multiply11×5.

(E) Incorrect.Didyousubtract3fromthelengthofthesidesofP1insteadofadd?

20. F

(F) Correct.Rememberthatwhenalineintersectstwoparallellines,allsmallanglesareequal.Youcaneliminatealltheotheranswerchoicesbecausetheyincludeoneofthelargerangles.

(G) Incorrect.misnotoneofthesmallangles.

(H) Incorrect.mandparenotamongthesmallangles.

(J) Incorrect.misnotoneofthesmallangles.

(K) Incorrect.mandparenotamongthesmallangles.

21. C

(A) Incorrect.Thisisapartialanswer,onehalfoftheproportionyoushouldsetup.

(B) Incorrect.Didyousetuptheproportioncorrectly?

(C) Correct.BecauseBCandDEareparallel,ABCand

ADEaresimilar.Setupaproportion:ABAD

BCDE

= .

(D) Incorrect.Thisisthevalueofseveralotherlinesegmentsinthefigure.

(E) Incorrect.Didyousetuptheproportioncorrectly?

16. J

(F) Incorrect.Thisshouldnotmatchyourtargetanswer.

(G) Incorrect.Thisshouldnotmatchyourtargetanswer.

(H) Incorrect.Thisshouldnotmatchyourtargetanswer.

(J) Correct.Pluginfornandmultiplythevaluesofthesidesbyeachother.

(K) Incorrect.Thisshouldnotmatchyourtargetanswer.

17. B

(A) Incorrect.Recognizethe3-4-5triangle,orplugthesidesintothePythagoreanTheorem.Thisshouldnotmatchyourcalculations.

(B) Correct.Dividetheareaby4togettheotherside.Sincethesidesoftherectangleare3and4,thediagonalwillbe5;ACThashiddena3-4-5triangleinsidearectangle.

(C) Incorrect.Thisisthevalueofonesideoftherectangle,notthediagonal.

(D) Incorrect.Thisisthevalueoftheothersideoftherectangle,notthediagonal.

(E) Incorrect.Thediagonalofarectangleisalwayslongerthaneitherside.

18. G

(F) Incorrect.Ballpark:IfABishalfthelengthofAC,

thenABEmustbemorethan18

oftherectangle.

(G) Correct.PluginforthevaluesofthesidesofABCD

suchthattheymultiplytobecome16,suchas8and2.

ThismakesAB=4andAE=2.Plugthesevalues

into12

bh.

(H) Incorrect.Pluginvaluesforthesidesoftherectangleandfigureoutthesizeofeachofthethreetriangles.Thisshouldnotmatchyourcalculations.

(J) Incorrect.ThisistheareaofEBD.

(K) Incorrect.ThisistheareaofABCD.


26. H

(F) Incorrect.Ballpark:thedifferencemustbegreaterthanthis.

(G) Incorrect.Trapanswer:thisisthedifferencebetweenACandAB.

(H) Correct.Thistriangleisproportionaltoa6-8-10,soACis100.Subtractthisfrom(60+80)tofindthedifference.

(J) Incorrect.ThisisthevalueofBC.

(H) Incorrect.ThisisthevalueofAC,notthedifference.

27. C

(A) Incorrect.Cubetheedge,don’tjustsquareit.

(B) Incorrect.Thisishalfthevolume.

(C) Correct.43.

(D) Incorrect.Thisis123.Noneedtotripletheedgebeforecubingit.

(E) Incorrect.Thisis44.

31. B

(A) Incorrect.Didyousetuptheproportioncorrectly?

(B) Correct.Areplicawillbeasimilartriangle,

proportionaltotheoriginal.2513

24=x

.

(C) Incorrect.Thisisthehypotenuseofthenewtriangle.

(D) Incorrect.Didyousetuptheproportioncorrectly?


35. B

(A) Incorrect.Thisisapartialanswer.You’llfindthisvaluemidwaythroughyourproportioncalculations.

(B) Correct.SetupaproportiontofindAF:DEFG

ADAF

= .

ThenplugthevaluesofAFandFGinto12

bh .

(C) Incorrect.Didyousetuptheproportioncorrectly?

(D) Incorrect.Remembertodividebhby2.


22. G

(F) Incorrect.Trapanswer!xandylookequal,butwedon’tknowthis.Plugin,andremembernottousethesamevalueforbothvariables.

(G) Correct.Whetherxandyareequalornot,thisanswerwillwork.

(H) Incorrect.Thisshouldnotmatchyourtargetanswer.

(J) Incorrect.Thisshouldnotmatchyourtargetanswer.

(K) Incorrect.Thisshouldnotmatchyourtargetanswer.

24. F

(F) Correct.BecausePQishalfthelengthofPR,thisisa30-60-90triangle.

(G) Incorrect.Reviewtherulesfor30-60-90vs.45-45-90.Youmayhaveconfusedthem.

(H) Incorrect.Ina30-60-90triangle,thesideoppositethe30°angleistheonethat’shalfthehypotenuse.

(J) Incorrect.Youcan’thavetwo90°anglesinthesametriangle.

(K) Incorrect.Thismakesthetrianglegreaterthan180°.

25. D

(A) Incorrect.Ballpark:thisisbiggerthantheentirecircle.

(B) Incorrect.Thisisroughlytheareaofthecircle.

(C) Incorrect.Thisisclosetotheareaofthesquare.

(D) Correct.Findthesideofthesquarebytakingtherootofthearea.Thenfindthediagonalbymultiplyingthesideby 2 .Halfthatdiagonalwillbetheradiusofthecircle,sofinditsarea.Thensubtracttheareaofthesquarefromtheareaofthecircle.

(E) Incorrect.It’sclose,butseetheproblemthroughtotheend.Don’tguess.


45. D

(A) Incorrect.Thisisthevalueofanyonesideofthetriangle.

(B) Incorrect.Trapanswer!Eachsideis8,not4.

(C) Incorrect.Reviewthe30-60-90rules.Theheightis

thelongerofthetwolegs;tofindtheshorter,divideby

3 ,sothereshouldbeno 3 intheanswer.

(D) Correct.Theheightofanequilateraltrianglecutsitintotwo30-60-90triangles,wheretheheightisthelongerleg.Use30-60-90rulestodeterminethesides.Addthethreesides.

(E) Incorrect.Reviewthe30-60-90rules.Theheightis

thelongerofthetwolegs;tofindtheshorter,divideby

3 .Thereshouldbeno 3 intheanswer.

46. K

(F) Incorrect.Thisistheratioofxtothehypotenuse.

(G) Incorrect.Reviewthe30-60-90rules.Noneofthesidesareequalinlength,sotheirratiocan’tbe1.

(H) Incorrect.Reviewthe30-60-90rules.Iftheshortest

legisx,thenthehypotenuseis2xandthelongerlegis

x 3 .Thereisno 2 forthistypeoftriangle.

(J) Incorrect.Thisistheratiobetweenthehypotenuseandy.

(K) Correct.Iftheshortestlegisx,andthelongerlegis

x 3 ,thentheratioofthelongerlegtotheshorteris

x 3 /x.Thex’scanceltoleave 3 .

36. H

(F) Incorrect.ThisisthelengthofAE.

(G) Incorrect.ThisisthelengthofAB.

(H) Correct.CalculateABbasedontheinfoyou

determinedduringthepreviousproblem.Setupa

proportion:ADAB

AEAC

= .

(J) Incorrect.Didyousetuptheproportioncorrectly?

(K) Incorrect.Didyousetuptheproportioncorrectly?

37. A

(A) Correct.Thisone’s“deceptivelyeasy”:inthiscase,theobviousansweriscorrect.

(B) Incorrect.ThereshouldbenorightanglesinABC.

(C) Incorrect.All3sidesarecongruent,notjust2.

(D) Incorrect.ThereshouldbenorightanglesinABC.

(E) Incorrect.ThereshouldbenoobtuseanglesinABC.

40. J

(F) Incorrect.Don’tsimplysubtract6from16.

(G) Incorrect.PlugtheknownvaluesintothePythagoreanTheorem,wherethediagonalisthehypotenuse.

(H) Incorrect.PlugtheknownvaluesintothePythagoreanTheorem,wherethediagonalisthehypotenuse.

(J) Correct.PlugtheknownvaluesintothePythagoreanTheorem,wherethediagonalisthehypotenuse.

(K) Incorrect.PlugtheknownvaluesintothePythagoreanTheorem,wherethediagonalisthehypotenuse.


51. C

(A) Incorrect.Ballpark:ifonesideofanisoscelesis6,2cannotbethelengthofanyside.

(B) Incorrect.Ballpark:ifonesideofanisoscelesis6,

3 cannotbethelengthofanyside.

(C) Correct.Thetrickisthatthegivenvalueof6unitscan

bethevalueofthehypotenuse,sotryitout.Thevalue

ofeachlegisthen6

2,whichisequivalentto 3 2 .

(D) Incorrect.Careful: 2 shouldappearinanisosceles

righttriangle,not 3 .

(E) Incorrect.Iftwosidesare6,thisistheirsum,buttheproblemdidn’taskforthat.

53. B

(A) Incorrect.LQPandMNOareboth45-45-90triangles,

makingLQ= 3 2 andMN =6.

(B) Correct.LQPandMNOareboth45-45-90triangles,

makingLQ= 3 2 ,MN=6,andLQLN

= 3 26

.

(C) Incorrect.LQisnothalfofLN.

(D) Incorrect.Thetwoarenotequal.

(E) Incorrect.YoumayhaveusedLMinsteadofLN.

54. G

(F) Incorrect.Ballpark:thisistoosmallfortheareaofatrianglewithsidesof6.

(G) Correct.ADishalfthesideoftheequilateraltriangle,soeachsideis6.Theheight,BD,canbedeterminedusing30-60-90rules.

(H) Incorrect.Theheightcanbedeterminedusing30-60-

90rules,andmustinclude 3 .

(J) Incorrect.Theheightcanbedeterminedusing30-60-

90rules,andmustinclude 3 .

(K) Incorrect.Don’tforgettodividebhby2.

49. C.

(A) Incorrect.Reviewtherulesforisoscelesrighttriangles.

Ifalegis 2 ,thentheperimeteristhesumofthe

twoequallegsandthehypotenuse,whichwouldgive

2 2 +2.

(B) Incorrect.Reviewtherulesforisoscelesright

triangles.Ifalegis2,thentheperimeteristhesumof

thetwoequallegsandthehypotenuse,whichwould

give4+2 2 .

(C) Correct.Ifalegis2 2 ,thentheperimeteristhe

sumofthetwoequallegsandthehypotenuse,which

wouldgive4 2 +4.

(D) Incorrect.Reviewtherulesforisoscelesrighttriangles.

Ifalegis4,thentheperimeteristhesumofthetwo

equallegsandthehypotenuse,whichwouldgive

8+4 2 .

(E) Incorrect.Reviewtherulesforisoscelesrighttriangles.

Ifalegis4 2 ,thentheperimeteristhesumofthe

twoequallegsandthehypotenuse,whichwouldgive

8 2 +8.


triGonometry19. A

(A) Correct.Thisisa5-12-13triangle,andthesmallestanglewillbeoppositethesideoflength5.

(B) Incorrect.Thisisthetangentoftheangle.

(C) Incorrect.Thisisthecosineoftheangle.

(D) Incorrect.Thisisthesecantoftheangle.

(E) Incorrect.Thisisthecosecantoftheangle.

26. G

(F) Incorrect.Thisisthecosineoftheangle.

(G) Correct.Theproblemtellsyoucosθ,soyoucanconsider4tobetheadjacentand5thehypotenuse.Thatmakesthisa3-4-5triangle,where3istheopposite.

(H) Incorrect.θ isnot45°.

(J) Incorrect.Thisisthesecantoftheangle.

(K) Incorrect.Thisisthecosecantoftheangle.

28. G

(F) Incorrect.Thisisthesineoftheangle.

(G) Correct.Tangentisdefinedasoppositeoveradjacent,whicharegivenvalues.

(H) Incorrect.Thisisthecosineoftheangle.

(J) Incorrect.Thisisthecotangentoftheangle.

(K) Incorrect.Thisisthecosecantoftheangle.

30. G

(F) Incorrect.

(G) Correct.Sinceyou’regiventhetangent,youcansetup

aproportion:ABBC

= 512

.

(H) Incorrect.Didyousetuptheproportioncorrectly?

(J) Incorrect.Didyousetuptheproportioncorrectly?

(K) Incorrect.ThisisthevalueofAC.

55. D

(A) Incorrect.Thetwofigurescanintersectmorethantwice.

(B) Incorrect.Thetwofigurescanintersectmorethanfourtimes.

(C) Incorrect.Thetwofigurescanintersectmorethansixtimes.

(D) Correct.Drawarectangleinscribedinthecircle,thenmakeitjustalittlelargerbyextendingthesidesbeyondthecircle.Thecircleshouldintersecttwiceneareachofthefourcornersoftherectangle.

(E) Incorrect.Thecircleandrectanglecannotbeidentical.

57. C

(A) Incorrect.Thehighestnumberisrarelytherightanswerin“maximum”questions.

(B) Incorrect.Youcan’tgeteightregions.

(C) Correct.Makesureallthreechordsintersectoneanother,andnevermorethantwoatthesamepoint.

(D) Incorrect.Makesureallthreechordsintersectoneanother,andnevermorethantwoatthesamepoint.

(E) Incorrect.Makesureallthreechordsintersectoneanother,andnevermorethantwoatthesamepoint.

58. G

(F) Incorrect.Thisisthevalueofoneofthegivenanglesintheproblem.

(G) Correct.Theanglesupplementalto50°is130°,andtheanglesupplementalto30°is150°.Drawalineparalleltotheonethatformsthe50°anglesuchthatitformsaquadrilateralwithx,the130°angle,andpartofthe150°angle.Recognizethata50°anglehasnowbeencreatedatthebottomrightofthequadrilateral.Thealternateinteriorangletothismakesup50°ofthe150°angleabove,leaving100°insidethequadrilateral.Add50,100,and130,andsubtractfrom360°tofindx.

(H) Incorrect.Toolarge:130+150+100is380,whichistoomuchforanyquadrilateralyoumightdrawinsidethefigure.

(J) Incorrect.130isthevalueofanotherangleinthefigure.

(K) Incorrect.Youcandeterminetheanswer.


57. D

(A) Incorrect.The35and30arereversed.

(B) Incorrect.Therampis30feetlong,not35.

(C) Incorrect.Youdon’tneedcosineinthisproblem.

(D) Correct.Drawthefigure.Relativetothe35°angle,theheightoftheplatformwillbetheopposite,andtherampwillbethehypotenuse.Eliminateanswersthatdealwithanythingbutsine.

(E) Incorrect.Youdon’tneedcosineinthisproblem.

37. C

(A) Incorrect.Thisisthetangentoftheangle.

(B) Incorrect.Thisisjustthevalueofoneofthesides.

(C) Correct.Theproblemtoldyouthetangent,whichgivesyouthevalueofbothlegs.UsethemtofindthehypotenusewiththePythagoreanTheoremandplugtheadjacentandoppositeinasthecosine.

(D) Incorrect.Thisisjustthevalueofoneofthesides.

(E) Incorrect.Thisisthevalueofthehypotenuse.

math: Plane Geometry Practice CirCles

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