.::

-,

J

--,..

--,q,

,..,..;•7 ;j:le. Sr ' ",..... y...,:: 7;::%■:,..1.,.: .;;;....

, 4,4 _ - '-'4'

if

t -

1 An overhead view of the intersection of a plane and a right triangular prism is

modeled below.

1111111111 Plane

Which of the following best represents the shape formed by this intersection?

A A B

2 In triangle XYZ, XY ) mzX < 45°. Which o 1

statements is a reaso

A mz_Z > 45°

B Triangle XYZ is an i right triangle.

• C Triangle XYZ is an isosceles triangle.

D Triangle XYZ is an isosceles triangle.

3 Which geometric con necessary for finding circumcenter of a trii

A Angle bisector

B Perpendicular bis

C Median

D Altitude

and

?. following

le conclusion?

les

se

on is

314

4 Four 30 0 -60 0 -90 0 triangles are shown below. The side lengths in units are showi or the first three triangles.

4,[3'N6

2 / Li

213. 6 A 18x

If AC > BC, which expression can be used to represent the length of BC in units?

A 9x

B

C

D 18\ffx

5 Figure JKLM is graphed on the coordinate grid below.

1111111101MM• ,IIMM1115•11.111•11 IMMENIIII••••• . MEM MUM IIIMIMMIIIIIE111112

i

SIELMISILIIMIN111111 111111111111111111 11111411•11111•111••••1111

_ _ • earn 1 MM. ItairwIMBIDERRE

•11111HIMMININII• •1111111=1117111111111 N•111111•111111Nfill •111111•MIMEN•PI _10-9 -8-7-6-5-4 - -2-1 Ommostimai

11 4 --I4 if leggion i MI ' NM ma LE EAVEIEE11 MI Inv mama •• mi. lumminommuna

71MMEIS'E Which of the following best describes figure JKLM?

A Irregular quadrilateral

B Parallelogram

C Kite

D Trapezoid

11315

A )111111116.

rf' 6 Carter's house is 7 miles directly south

of a grocery store and 7 miles directly west of a gas station, as represented in the diagram below.

Grocery Store

7 mi

Carter's 7 mi Gas House Station

7 In Elements, the ancie tjGreek mathematician Euclid s d five statements that are as ined to be true without proof, Lh as "All right angles are congr e t," to prove many theorems. What such statements called? II

A Postulates I II

B Theorems

C Indirect proofs

D Counterexamples

What is the direct distance from the grocery store to the gas station in miles, to the nearest tenth of a mile?

Record your answer and fill in the bubbles on your answer document.

8 A geometry student concluded the following:

If a polyhedron has a pair of congruent parallel faces, then the polyhedron is a prism.

Which figure can be used as a counterexample to the student's conclusio

316

9 A home furnishings store carries rectangular rugs in a range of sizes. Each dimension of the largest rug they sell is 2.5 times the size of the corresponding dimension of the smallest rug they sell. Based on this information, which of the following statements is true?

A The area of the largest rug is 2.5 times the area of the smallest rug.

B The area of the largest rug is 5 times the area of the smallest rug.

C The area of the largest rug is 6.25 times the area of the smallest rug.

D The area of the largest rug is 15.625 times the area of the smallest rug.

10 Greta built a sand pyramid at the beach. Its base was a square with a side length of 6 feet, and its height was 4.5 feet. Which of the following is closest to the volume of sand in the pyramid?

A 54 ft3

B 162 ft3

C 81 ft3

11 A cable from the top of a poi:

attached to a stake in the gro I

6 meters from the base of th; role,

as shown in the diagram.

-•

..•• • ••:. •••:?

•

. .

6 m .

The angle between the cable the ground is 57°. Based on t information, which of the folio

is

closest to the length of the ca

A 9m

B 3m

C 7m

D 11 rn

12 A carpenter cuts a rectangular rd along its diagonal. If the boarc 32 inches long and 24 inches

e,

how long is the diagonal in inc

Record your answer and fill in bubbles on your answer docurr t.

D 36 ft3

317

13 If points A, B, C, and D are coplanar

but are arranged such that no more

than two of the points are collinear, which of the following statements must always be true?

A A tetrahedron can be drawn with points A, B, C, and D as vertices.

B A circle can be drawn such that points A, B, C, and D lie on

...... . . the circle.

C A quadrilateral can be drawn with points A, B, C, and D as vertices.

D A triangle can be drawn such that points A, B, C, and D lie on its sides.

14 A park is in the shape shown below, composed of a rectangle and an isosceles triangle.

15 A net of a geometric figur is

shown below.

Which geometric figure is formed by this net?

A Rectangular prism

B Sphere

C Cylinder

D Cone

12 m

30 m

Which of the following represents the area of the park?

A 480 m 2

B 600m 2

C 450 m 2

D 615m 2

318

16 In parallelogram ABCD below, line segments EF and GH have been drawn con pairs of midpoints of consecutive sides.

A

The proof that triangle AEF triangle CHG is shown below.

Statements Reasons _

1. ABCD is a parallelogram; E is the midpoint of AB; F is the midpoint of AD; G is the midpoint of BC; H is the midpoint of CD.

1. Given

. 2. Angle A --==: angle C 2. Opposite angles of a parallelogram

are congruent. _ . 1 1 1 3. AE = —2AB' CH = —2 CD, AF = fAD, and

CG = 1-BC 2

3. Definition of midpoint

_ .

4. AB = CD and AD = BC 4. Opposite sides of a parallelogram are congruent.

1 1 5. AE = -2CD and AF = —28C 5. Substitution

6. AS = CH and AF = CG 6. Substitution

7. AS CH and AF -=--- CG 7. Definition of congruence

8. Triangle AEF :=-- triangle CHG 8.

Which of the following is the missing reason for step 8?

A Side-Side-Side congruence postulate

B Side-Angle-Side congruence postulate

C Angle-Side-Angle congruence theorem

D Angle-Angle-Side congruence theorem

319

17 Paul is wrapping a gift in a box that has the shape of a hexagonal prism. The base is a regular hexagon with a side length of 8 inches. The height of the box is 10 inches.

8 in.

10 in.

F-8 in. H Base

Which of the following is closest to the surface area of the box?

A 333 in. 2

B 646 in. 2

C 1145 in. 2

D 813 in. 2

18 Pedro measured the side le !ths of six quadrilaterals. The sets of •ij- lengths, in units, are listed below.

4, 7, 4, 7 5, 5, 10, v

1 18, 3, 12, 6 3, 14, 6,

2, 1, 9, 8 10, 8, 8, 21'

If Pedro uses inductive reasi ll ing, which conjecture could he 1 e based on these data?

1 A The side lengths of a q all rilateral always include both odd . d even numbers of units.

B The longest side of a quzirilateral is always shorter than t : sum of the other three sides.

C The shortest side of a q rilateral is always less than half length of the longest side.

D The difference between lengths of the two longest sides i a quadrilateral is always g ;ter than the length of the shorte•t ide.

320

19 Square iKLM was dilated to create J'K'L'M', as shown on the coordinate grid below.

_ ---i--, ,--- • I i ! , i

. 7 .

• 1 • . L_k_

Fjr-r— 11 I in II M IF- 1 j Almmommor -II 1.

111 li . i -3411,11111,1 1-1--- 2- ' : ---4---,--- --I ' 1-1 1--1 1(7 1--

- I-1 I 1 r , --L- 1 - - ''11V

I . -10-9 --e -7 -6 -5 ii- 1_0 1 2 3 4 9 6 7 8 910

- 1 ' flr. '-2 1

=11 i

•

-1-1- rm __ tompi• ifil

-;,:_34 . - • ••• mum

:ups irdriml- - .. lima 1........m ........ .. i . ...... -mum Which statement appears to be true?

A Square JKLM and square J'K'L'M' are congruent.

B Square JKLM was enlarged to create square YK'L'M'.

C The scale factor used to create

square J'K'L'M' was

D The center of dilation used to create square J'K'L'M' was (-7, 0).

20 Points D and F lie on circle A, AD 1 AF, and line DE is a tangent to circle A. Which of the following statements is a reasonable conclusion?

A Point E lies on circle A.

B DE ==-: AD

C AF DE

D AF II DE

21 What is the midpoint of FM witl-endpoints F(-10, 5) and M(-2, - 13)?

A (-6, -4)

B (-4, -4)

C (-4, -9)

D (-6, -9)

22 A sprinkler rotates in a compll and reaches a radius of 6.2 m

What is the area that the sprint covers in square meters, to the square meter?

Record your answer and fill in t bubbles on your answer docurni

rcle

rest

321

23 Veronica is painting an arrow on the side of a building. The arrow has the shape of a concave heptagon composed of a rectangle and an isosceles right triangle, as represented in the diagram below.

25 Bert wants to determine th

section formed by the inte a sphere and a plane passi its center. Which of the foll best help Bert solve the pr

4 3 A

ross

tion of through ing would em?

2 ft

7 ft

Which of the following measurements best represents the perimeter of the arrow in feet?

A (22 + n) ft

B (22 + 3n) ft

C (20 + 3n) ft

D (20 + n) ft

-radius C ,-

C A sphere is the set of r three-dimensional spac given distance from its

s in at are a ter.

-

24 One end of a board rests against the top of a platform, and the other end is on the floor.

Platform Board

If the board is twice as long as the platform is high, what angle does the board make with the floor?

A 20°

B 45°

C 60°

D 30°

322

lar 26 A pet supply store offers aquariums in two sizes. Each is in the shape of a recta prism. Each dimension of the larger aquarium is twice the size of the correspon( dimension of the smaller aquarium.

Ad11111V 2h h

2w

2/

If the volume of the smaller aquarium is 2000 cubic inches, which of the followi represents the volume of the larger aquarium?

A 16,000 in. 3

B 8000 in. 3

C 12,000 in. 3

D 4000 in. 3

27 is graphed on the coordinate grid below.

, lumnimi . mmusum Ihmo••111 --r . 111•11•••=111• 1•0•1111••■••••0111•11

111011W11111 INIIIIIIIIIMMI

MEM ' ••••••••111111

III 1111111E1M - •111011•111111• ••11111111•1111•• L .+-1 11111g1211• I -10-9 -8 -7 -6 -5 -4 3 -2 -1 0 1 2 3 4 5 6 7 8 9 11

1 • 11 morinninism - 1 sr -2 simumissimm

-.. - rim% - IMININII i ffi. ,..MILIMMIIIIMINIIII -H ma

I -- MI -7.MMIIIIIIIIIIIII iEl _-8 Min

I -10.111111 II-

Which of the following coordinates for vertex K would produce an isosceles right triangle .1 KL?

A (7, —2)

C (-1, 4)

B (7,3)

D (1,2)

best

323

324

28 Mack examined a set of quadrilaterals and made the following conjecture based on his observations:

If a quadrilateral has three acute interior angles, then its fourth interior angle must be obtuse.

Which of the following best explains the validity of this statement?

A The statement is valid because the sum of the interior angles in a quadrilateral is 360 0 , and the sum of three acute angles is less than 270°.

B The statement is valid because all quadrilaterals have at least one - obtuse angle.

C The statement is invalid because a quadrilateral may have four acute angles.

D The statement is invalid because a rectangle has no obtuse angles.

30 Shouze wants to determine h width of a highway. He knows that t e distance between two street Ii hts is 50 meters. From a point d r ctly opposite one street light, th gle between the sight lines to th wo street lights is 63°.

50m

;63f,- -

What is the width of the high y in meters, to the nearest ten of a meter?

Record your answer and fill ir ie bubbles on your answer docu nt.

29 What is the slope of the line passing through points (4, 7) and (-2, 9)?

A —3

B —1

C

31 The three-dimensional figure shown is composed of eight identical cubes.

Which of the following could represent a top, front, or side view of the figure?

A

32 Which of the following represents the length in units of AK with endpoints A(-2, —1 and K(8, 4)?

A 3 \

B q-s- C 461

D 4109

325

33 The side lengths, in units, are shown for each of the triangles below.

cu

r .,,, A 1— 6 4 ... co

I— 8 6 10 11 .2? — c.) cz Which of the following is a valid conclusion based on these examples? 6. a

A Only isosceles triangles are acute triangles.

B Triangle side lengths always increase in a counterclockwise direction.

C The longest side in a triangle is always opposite the largest angle.

D The shortest side in a triangle is always more than half the length of the lc side

34 A playground, when viewed from above, has the shape of a rectangle that is long and 20 meters wide. The water fountain and the entrance to the playgn opposite vertices of the rectangle.

Water Fountain ft

/

/

/

/

/

/

' _./

/

/

/

/

/ ' /

/

/

e

Entrance 21 m

Which of the following best represents the direct distance from the entrance of playground to the water fountain?

A 21 m

B 26 m

C 28m

D 29 m

20 m

meters are at

326

35 Triangle ABC is graphed on the coordinate grid below.

I =10 -9 -8 -7 -6 -5 -4 =3"-.91.....1.- 91 1 5 3 45 6 / 9 15 1

A midsegment of a triangle connects the midpoints of two of its sides. Which oil le

following lines does not contain a midsegment of triangle ABC?

A y = 0

C y = — 1

B y = 2x

D y = —4x + 2

36 A medallion is in the shape of a regular octagon, as shown below. The side le

is 2 centimeters.

2 cm

Which of the following is closest to the area of the medallion?

A 16.0 cm 2

B 32.0 cm 2

C 19.3 cm 2

D 38.6 cm 2

327

37 Below is a proof that an altitude of an isosceles triangle divides the triangle ir congruent triangles.

Given: Triangle ABC is isosceles; BL is an altitude of triangle ABC.

Prove: Triangle ABL triangle CBL

A

:WO

Statements Reasons ,

1. Triangle ABC is isosceles; BL is an altitude of triangle ABC.

1. Given

2. AB ----- BC 2. Definition of isosceles trian pi La.

3. BL I AC 3. Definition of altitude

4. Angle BLA and angle BLC are right angles. 4. Definition of perpendicular _

5. 5. Definition of right triangle _

6. BL -----= BL 6. Reflexive property . 7. Triangle ABL' .-- triangle CBL 7. Hypotenuse-Leg theorem

Which of the following best represents the missing statement for step 5?

A AL CL

B LBLA LBLC

C Triangle ABC is a right triangle.

D Triangle ABL and triangle CBL are right triangles.

38 A circular swimming pool is 24 feet in diameter. If 1 foot is approximately equa 0.305 meter, which of the following is closest to the area of the surface of the swimming pool?

A 42m 2

B 168m 2

C 452m2

D 138m 2

328

li I 61‘ / III

11

/ 1 I 1114 1 1 1

1 11 1 1 1 3 1111

). --- -. 1 _-;-.- -I ' ___;___1

1_

1 1 1 1 1 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

1 -1--. 1 -3 1--1-1 -4 ; 1

11 11 1v111 1

_ '11All 1 I

/ i

-

1

-r---'-5 4

1

1 2 • 4 6 6 -1

MIMI 1 1 -2

-3

1 5 1

. 1 111 1 i

1 1 ,

MOM MIME MU

i ._

I 1 _1_4 I 1 I i 32

• c 6 -5 -4 -3 -2 -1 -

1 5

1 2 3 4 5 i•

6

I

A

39 Which of the following represents a line segment with a length of 10 units?

40 In the figure below, lines d and e intersect lines f and g.

g I e and Li L2. Which of the following statements must be true?

A mL2 = 90°

B c/ f

C f g

D due

329

41 Which set of statements represents a valid deductive argument?

A All equilateral triangles are similar. All dilations produce similar figures. All produce equilateral triangles.

B All rhombuses have diagonals that bisect their interior angles. All squares diagonals that bisect their interior angles. All rhombuses are squares.

C The cross section of a prism made by a plane parallel to its bases is congn the bases. The bases of a cube are squares. The cross section of a cube m plane parallel to its base is a square.

D All rectangular pyramids have a rectangular base. All rectangular pyramids faces. All polyhedrons with five faces have a rectangular base.

ions

t to by a

five

42 School district A, shaded in the diagram below, is in the shape of a trapezoid county that is rectangular: 25 miles long and 16 miles wide.

25 mi

16 mi A

1-8 mi

Assuming that the county is evenly populated with students, if a student in thE chosen at random, what is the probability that he or she lives in school district

A 20%

B 26%

C 52%

D 30%

in a

nty is

330

43 A reservoir is in the shape of a circle 45 Joe's eye level is 6 feet abc ' with a diameter of 400 yards. A fence ground. When standing 15

runs along the edge of the reservoir from a tree, he measures t • for the length of a 220 0 arc. I elevation to a bird's nest in

the : away 3ngle of tree to

22

be 60°.

1-15 ft—i

Which of the following meas best represents the height o nest above ground?

A 5-\ff ft

B (6 + 5 .Nff) ft

C 15 -\/-3- ft

D (6 + 15 -a) ft

46 Two sails on a sailboat are si triangles. Each dimension of sail is 1.5 times the correspc dimension of the smaller sail area of the smaller sail is 8 5

meters, what is the area of t sail in square meters?

Which of the following is closest to the length of the fence?

A 768 yd

B 1256 yd

C 244 yd

D 1535 yd

44 Triangles JKL and MNO are shown below.

0

ments e bird's

ar larger

ng the

3 re larger

Angle K angle M, and KO ML. Record your answer and fill Which additional condition is sufficient

bubbles on your answer doc nt.

0 to prove that JK NM?

cn . A JL KM 0 0

LM MN B (.7) . -o

C KM MN

D JL II ON

331

47 Laura took a quick series of photos of a falling leaf, as shown below.

1 2 3 4

_

Which of the following best describes the movement of the leaf from picture t1 icture?

A Rotation, reflection, rotation

B Rotation, translation, reflection

C Rotation, translation, rotation

D Rotation, reflection, reflection

48 For which of the following statements is the converse true?

A If a quadrilateral is a rectangle, then it has congruent diagonals.

B If two triangles are congruent, then they have congruent corresponding a g -s.

C If two line segments are perpendicular, then they intersect at exactly one nt.

D If a polygon is a quadrilateral, then the sum of its interior angle measures i 360 0 .

332

49 The Halversons' attic space is in the shape of an isosceles triangular prism.

Mr. Halverson is 6 feet 2 inches tall, and when he stands 3 feet from the edge of the attic, his head touches the slanted ceiling, as shown in the diagram.

50 In triangle DEF below, GH II

6ft2in.

1-3 ft-A 15 ft

The total width of the attic is 15 feet. Based on this information, which of the following represents the height of the ceiling at its highest point, in the center of the attic?

A 30 ft 10 in.

B 10 ft 8 in.

C 15 ft 5 in.

D 18 ft 6 in.

Which of the following state would best be used as a rea proof that triangle GEH — tri

nts in a

gle DEF?

A Corresponding angles fo ed by

parallel lines and a tran are congruent.

rsal

B Congruent triangles are i ilar.

C The midsegment of a tri le is half the length of the si parallel to.

e t is

D An isosceles triangle ha congruent angles.

o

333

51 The apothem of a regular polygon is the perpendicular distance from the cen to a side. The apothem, perimeter, and area are given for the first three regL polygons below.

.41 C

Lsi,n CO

P = 36 in. P = 40 in. P = 40 in. A C

A = 5413- in. 2 A = 100 in. 2 A = 110 in. 2

Triangle ABC is an equilateral triangle. Which expression can be used to repre nt the area of triangle ABC in square inches?

A [(3x + 6) + 6(4)] in. 2 C [Y8)(x + 2)] in. 2

B [ -r (4)(3x + 6)] in. 2 D [1(4)(x + 2)] in. 2

52 Triangle TUV is graphed on the coordinate grid below.

- -2

-t

-I -7

-9

-10

Triangle TUV is rotated 180 0 around the origin to create triangle T'U'V'. Which c` the following best describes the shape formed by combining the preimage and the iTiage?

A Rhombus C Parallelogram

B Isosceles triangle D Rectangle

334