Placement Test Practice Problems

Book II

Geometry, Trigonometry, and Statistics

Eric Key, University of Wisconsin-Milwaukee David Ruszkiewicz, Milwaukee Area Technical College

This material is based upon work supported by the National Science Foundation under Grant No. EHR-0314898. Any opinions, findings and conclusions or recommendations expressed in this material are those

of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).

1

Contents Chapter 7. Geometry

1. Perimeters and Areas of Polygons and Circles 3 2. Measuring Solids 11 3. Angle Sum for a Triangle 15 4. Basic Vocabulary 19 5. Parallel and Perpendicular Lines 23 6. Pythagorean Relationship 27 7. Incidence Properties 30 8. Qualitative Objectives 33 9. Isosceles Triangles 34 10. Circle Properties 37 11. Congruence and Similarity 42

Chapter 8. Trigonometry

1. Basic Definitions 45 2. Applications 49 3. Identities 50 4. Solving Triangles 56 5. Graphing 59

Chapter 9. Statistics 1. Interpreting Bar Graphs and Pie Charts 62 2. Calculating the Mean and Median of a data set 64

2

Introduction What follows are warm-up exercises designed to illustrate the objectives of the

UW-System Mathematics Placement Test and the AccuPlacer. The designers of those

test make no warranty of the reliability of these examples. However, the authors believe

that the exercises you find here well-illustrate the mathematics concepts covered by

these examinations.

There are two volumes of exercises. This volume contains exercises on

geometry, trigonometry, and statistics. A companion volume contains exercises on

arithmetic and algebra. It is our intention that these exercises be used as warm-up

exercises throughout the school year rather than as a way to try to quickly “cram” for

the exam. It is our hope that each and every student performs on these test at a level

that his/her high school grades warrant. Placing into course beyond one’s level of

preparation is just as detrimental to one’s learning of mathematics as placing below

one’s level of preparation.

Please feel free to disseminate these problems as widely as you can, so long as

you acknowledge the source:

Eric Key, University of Wisconsin-Milwaukee

David Ruszkiewicz, Milwaukee Area Technical College

The Milwaukee Mathematics Partnership, supported by the National Science

Foundation under Award Number HER-0314898.

The Milwaukee Mathematics Partnership (MMP) is a collaborative effort among the University of

Wisconsin-Milwaukee, Milwaukee Public Schools, and the Milwaukee Area Technical College to

improve mathematics teaching and learning at all levels, from pre-kindergarten through college. Visit

our website at http://www.uwm.edu/org/MMP/

3

Areas of polygons and circles

AGa1 Find the area of region composed of rectangles and/or right triangles

a. b. c. d. e.

5in

4in

6in

3in

5in

5in 4cm

5cm

13cm

4ft

6ft 3in

5in

9m 12m

18m

4m

4

AGa2 Express shaded area of region composed of rectangles as fractions of total area

a. b. c.

d. e.

5

AGa3 Find area of triangle.

a. b. c. d. e.

6

AGa4 Find area of circle or semicircle given radius or diameter

a. b. c. shaded region only d. e.

7

AGa5 Find area of region composed of rectangular and circular or semicircular parts

a. shaded area only b. shaded area only

c. shaded area only d. shaded area only e.

8 12

8

AGb1 Find perimeter of a parallelogram given two sides

a. b. c. d. e.

9

AGb2 Find perimeter of rectilinear figure

a. b. c. d.

e.

10

AGb3 Find circumference of circle given radius or diameter

a. b.

c. d.

e.

11

Measuring Solids AGc1 Find surface area of a rectangular box

a. b. c. d. e.

12

AGc2 Find volume of a rectangular box

a. b. c. d. e.

13

AGc3 Find surface area of polycube

a. b. c. d. e.

14

AGc4 Find volume of a polycube

a. b. c. d. There is a square hole cut through the center e. The shape below can be folded to form a cuboid. Find the volume.

15

Angle Sum for a Triangle

AGd1 Find the third angle given two angles of a triangle

a. b. c. d. e.

X 41

48

16

AGd2 Find the acute angle in right triangle given the other acute angle

a. b. c. d. e.

17

AGd3 Find exterior angle given two interior angles

a. b.

c. d.

e.

x

111�

39�

40� 8 15x +

4 5x + 3 20x +

x

45�

72�

x

76�

53�

x

18

AGd4 Recognize that the angle sum is 180 degrees.

a. Given the diagram below,

1. What is the measure of d∠ ?

2. What is the measure of e∠ ?

3. What is the measure of a∠ ?

4. What is the measure of b∠ ?

5. What is the measure of c∠ ?

6. What is the measure of f∠ ?

19

Basic Vocabulary

AGe1 Identify the base and altitude of a triangle or parallelogram

a. For the triangle below, identify the base and the altitude b. Identify base and altitude of ABD∆ c. Identify base and altitude of parallelogram ABCD

d. Find base of parallelogram given its

area is 105 square inches. e. Identify base and altitude of CEG∆

20

AGe2 Identify parallelogram, rectangle, square, trapezoid, rhombus

a. A quadrilateral with four right angles and four congruent sides b. A quadrilateral with exactly one pair of parallel sides c. A quadrilateral with both pairs of opposite sides parallel

d. A quadrilateral with four right angles e. A quadrilateral with four congruent sides

21

AGe3 Identify right, acute, and obtuse angle

a. ABE� b. DBC� c. DBE� d. CBE� e. DBA�

E D

C B A

22

AGe4 Identify parallel and perpendicular lines

a. _____j⊥ b. ______k � c. ____ ____m or⊥

d. ____n � e. ____k ⊥

23

Parallel and Perpendicular Lines

BGb1 Recognize and apply definitions of parallel and perpendicular lines

Classify the following as intersecting, parallel, or perpendicular a. b. c.

d. e.

24

BGb4 State the relationship between perpendicularity and a right angle

Which of the following line segments are perpendicular? a. b. c. d. e. Two lines intersecting at a right angle are ________________.

92� 88�

90�

25

DBGb2 Recognize and apply the fact that a line connecting the midpoints of two sides of a

triangle is parallel to the third side

a. 3, 3, 4, 4CE BE CD DA= = = = are &DE AB parallel? b. What do you know about &AC DE ?

c. ,AE EC BD CD≅ ≅ , are &AB ED parallel? d. ,AE EC BD CD≅ ≅ , are &AB EC parallel?

e. , ,AF FB AD DC BE EC≅ ≅ ≅ . Which line segments are parallel?

26

DBGb3 Recognize and apply relationships among the angles formed when parallel lines

are cut by a transversal

a. Which angles are congruent? Why? b. Which angles are congruent? Why? m n� m n�

c. What is the value of x ? d. What is the value of x ? m n� m n�

e. Find the value of all the angles.

m n�

9 8

7 6

4

3 2

1

l

4 3

2 1

12

11 10 9

8 7

6

5 4

3

2 1

55�

80 x+

2 5x + m

n 5x

60x + m

n

45� m

n

m

n

m n

l

27

Pythagorean Relationship

BGc1 Given two sides of a right triangle, find the third side

a. b. c. d. e.

28

BGc2 Given hypotenuse and leg of a triangle, find the area

a. b. c.

d. e.

29

DBGc3 Solve verbal problems involving the Pythagorean relationship

a. A rope 17 m long is attached to the top of a flagpole. The rope is able to reach a point on the ground 8 m from the base of the pole. Find the height of the flagpole. b. The base of an isosceles triangle is 14 cm long. The equal sides are each 22 cm long. Find the altitude of the triangle. c. A flagpole has cracked 10 feet from the ground and fallen as if hinged. The top of the flagpole hit the ground 12 feet from the base. How tall was the flagpole before it fell? d. In the Old West, settlers often fashioned tents out of a piece of cloth thrown over tent poles and then secured to the ground with stakes forming an isosceles triangle. How long would the cloth have to be so that the opening of the tent was 4 meters high and 3 meters wide?

e. A ramp was constructed to load a truck. If the ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp?

30

Incidence Properties

BGg1 Recognize that two points determine a line

Use the figure above for questions a-d. a. True/False. A, O, and B are collinear b. Must B, O, and Y be collinear? c. What other point forms a line with X ? d. What other point forms a line with Y ? e. Can a given point be in two lines? In twenty lines?

31

BGg2 Recognize the different intersection sets between circles, or two lines, or a circle and a

line, or a plane and a line, or two planes

Consider the figure above for the following questions. Which of the following are true/false? a. D is on line h. b. h is in S. c. h is in T.

d. AB����

is in plane T. e. S contains AB����

f. T and S contain D. g. A, B, and C are collinear. h. A, B, C, and D are coplanar.

i. . Plane T intersects plane S in AB����

. j. Point C is in T and S.

32

The figure above is a rectangular solid. a. Name a fourth point that is in the same plane as A, B, and C. b. Name a fourth point that is in the same plane as B, E, and F. c. Name a fourth point that is in the same plane as H, G, and C.

d. Are there any points in CG����

besides C and G? e. Are there more than four points in plane ABCD? f. Name the intersection of ABFE and BCGF. g. Name two planes that do not intersect.

33

Qualitative Objectives

CGq2 Geometric proportionality and scaling

a. The two polygons below are similar. Find the missing side x.

b. STV PQR∆ ∆∼ . Find the value of x, y, and the missing angle P.

c. B D∠ ≅ ∠ . Is ABE CDE∆ ∆∼ ? d. Are the triangles shown below similar?

e. Find PT and PR.

34

Isosceles Triangles

DBGe1 Recognize and apply the definition of an isosceles triangle

a. How many sides are congruent on an isosceles triangle? b. What side of an isosceles triangle is not congruent to the rest? c. Besides two sides, what else is congruent in an isosceles triangle? d. What is the perimeter of the triangle below? e. Construct an isosceles triangle. Label the vertices A, B, and C.

35

DBGe2 Recognize that congruent angles are opposite congruent sides

a. What two angles are congruent? b. What is the length of AB? c. AB= BC. What is the measure of angle X ? d. What is the length of BC?

e. Can you determine the length of AB? If not, why not?

36

DBGe3 State or recognize that the altitude of an isosceles triangle bisects the base

Let ABC be an isosceles triangle with AB = AC. Suppose AM BC⊥

For questions a-g, which of the following are true?

a. MB MC≅ b. ABC ACB≅� � c. AMC AMB≅� �

d. BAM CAM∆ ≅ ∆ e. AM is the altitude of ABC∆

f. ABM ACM≅� � g. AM is the angle bisector of angle BAC. h. ___________ is the perpendicular bisector of ___________.

37

Circle Properties

DBGf1 Recognize that a tangent to a circle is perpendicular to the radius drawn to the

point of tangency

Line m is tangent to the circle with center O.

1. What is the measure of OPM∠ ? 2. AB is tangent to circle O. Find CB.

3. AC is tangent to circle O. What is the area of ACO∆ ?

38

DBGf2 Recognize that an angle inscribed in a semicircle is a right angle

a. Determine the measure of angle x. Justify your assertions.

b. The diameter of the circle below is 13. What is the length of CB?

c. What is the area of ABC∆ ? d. What is the diameter of the circle below?

39

DBGf3 Recognize that 2 tangents from an external point are the same length

a. The two lines from point P are tangent to the circle. What is the length of segment PA? b. If PA and PB are tangent to the circle from point P, what must be true about PA and PB?

3. Is PAB∆ isosceles?

d. PC & BN are tangent to both circles. Is PN BN= ?

40

DBGf4 Recognize and use the relationship, in a sector of a circle, between area, arc length,

radius and central angle

a. What is the radian measure of an angle x that intercepts an arc of 8 cm in a circle of radius 10cm?

b. At the same central angle x, what is the arc length if the radius is 5 cm?

c. At a central angle of 5

π, approximately what ratio has the arc to the radius?

d. What is the area of the sector?

50o e. An 16-inch pizza (ie, radius is 8 in) is divided into four equal sectors. What is the surface area of each slice?

41

DCGc4 Recognize and use the relationship between tangent and radius

a. If OB = 6 and AO = 10, then AB = ?

b. If 60BOA = �� and OB = 6, then AO = ?

c. If AB = 9 and BO = 8, then AO = ? d. AB is tangent to circle O and circle N. AB = ? e. What kind of quadrilateral is ABON?

42

Congruence and Similarity

DBGa1 Recognize and apply definition of congruence and similarity

a. The two triangles shown are congruent.

______CDO∆ ≅

_____C∠ ≅

______CO ≅

DO = ______

Explain how you can deduce that AB CD�

b. The pentagons shown are congruent. C corresponds to _________

_______CANDY ≅

_________ I= ∠

CY = ___________

If CA NA⊥ , name two right angles in the figures.

c. Suppose you know that ABC DEF∆ ≅ ∆ . Name three pairs of corresponding sides. d. Two similar polygons are shown. Find the values of x, y, and z. e. Draw two equilateral hexagons that are clearly not similar.

O

C D

A B

43

DBGa2 Given similar triangles, recognize proportionality of sides

a. Complete.

_______ABC∆ ∼

? ? ?

AC AB BC= =

15 18

? ?=

15 12

? ?= a = ______ and b = ________

b. Find the values of a and b. c. Find the values of a and b.

d. Find the values of a and b. e. Name two triangles that are similar to ABC∆ .

44

DBGa3 Recognize that congruency implies similarity but not conversely

a. Tell whether the triangles are congruent, similar, neither, or no conclusion is possible. b. Tell whether the triangles are congruent, similar, neither, or no conclusion is possible. c. Tell whether the triangles are congruent, similar, neither, or no conclusion is possible. d. Draw two parallelograms that are clearly similar but not congruent. e. Explain how you can tell at once that quadrilateral RSWX is not similar to quadrilateral RSYZ

45

TRIGONOMETRY

Basic Definitions

DCTa1 Find the value (numerical or literal of a trigonometric function using a right

triangle, coordinate, or circular definition).

For problems a-d, give the values of the six trigonometric functions of θ . a. b. c. d.

e. If 2

sin3

θ = , for an acute angle θ , state two ways to find cosθ and tanθ .

6

2

y

θ x

4

7

y

θ x

4

4

y

θ x

46

DCTa2 Use knowledge of sides and angles in 30-60-90 and 45-45-90 triangles, e.g., solve

equations of the form sinθ =b or cosθ =b.

a. Find the value of θ given 1

cos2

θ = , Quadrant I.

b. Find the value of θ given 3

sin2

θ = , Quadrant II.

c. Find the value of θ given 2

cos2

θ−

= , Quadrant III.

d. Find the value of θ given 1

sin2

θ−

= , Quadrant IV.

e. Find the values of θ given 1

sin2

θ = .

47

DCTa3 Convert between degree and radian measure

a. Express 45� in radians.

b. Express 270� in radians.

c. Express 4

3

πin degrees.

d. Express 5

6

πin degrees.

e. Express 7

3

πin degrees.

48

DCTa4 Evaluate expressions involving inverse trigonometric functions.

a. Find the value of 1 3

2Sin− .

b. Find the value of 1 2

3Cos

π− .

c. Find the value of 1 3

2Cos−

−

.

d. Find the value of 1 5sin

6Sin

π−

.

e. Find the value of 1 cos3

Cosπ− −

.

49

DCTw1 General story/word problem involving subscore 1 objectives (DCTa1-a4).

a. Solve the right triangle ABC.

36A∠ = �

68c = b. What is the angle of elevation of the sun when a tree 6 m tall casts a shadow of 10 m long?

c. The approach pattern at a local airport requires pilots to set an 11�angle of descent toward the runway. If a plane is flying at an altitude of 9000 m, at what distance measured along the ground from the airport must the pilot began the descent?

d. How far from the base of a building is the bottom of a 30 ft ladder that makes a 75� angle with the ground? e. The distance directly below a hot air balloon to where the balloon is staked to the ground is 275 ft.

The angle of elevation up the rope to the balloon is 58� . How high is the balloon?

c

b

a

C A

B

50

Identities

DCTb1 Apply identities involving Pythagorean formulas

a. Simplify 21 cos

sin

θθ

−, sin 0θ ≠

b. Simplify 2 2sec tanθ θ+

c. Simplify

sin cos

cos sin1

sin cos

θ θθ θ

θ θ

+, sin 0,cos 0θ θ≠ ≠

d. Prove the identity: 2

sincsc

1 cos

θθ

θ=

−

e. Prove the identity: 2 2

cot 1

sin cos tan

θθ θ θ

=+

, sin 0,cos 0, tan 0θ θ θ≠ ≠ ≠

51

DCTb2 Apply identities involving Angle Sum Formulas or Double Angle Formulas.

a. Find the exact value of ( )cos 75� . Note: ( )cos(75 ) cos 30 45= +� � �

b. Find the exact value of ( )sin 75� .

c. Find the exact value of ( )tan 75� .

d. Find the exact value of ( )tan 165� .

e. If 3

cos5

θ = and θ is in Quadrant IV, find cos 2 , sin 2 , & tan 2θ θ θ .

52

DCTb3 Apply identities involving Quotient and Reciprocal Formulas.

In a-c, simplify, leaving answers in terms of sines and cosines.

a. sin cotθ θ+

b. sec tanθ θ+

c. tan cot

sec csc

θ θθ θ+

d. Prove the identity: tan sin secθ θ θ=

e. Prove the identity: cot cos cscθ θ θ=

53

DCTb4 Use Reduction Formulas or Complementary Angle Formulas.

In a-c, change each expression by using the reduction identity to write it as a sine function.

a. ( ) 1sin

7π θ− = what’s sinθ ?

b. ( )cos 30π + =�

c. sin( 45 )π + =�

In d-e, write each trigonometric function in terms of its complement.

d. sin 29�

e. cos 41�

54

DCTb5 Use even and odd properties of trigonometric functions.

a. ( )cos 18− =�

b. ( )sin 18− =�

c. ( )tan 18− =�

In d-e, write each function in terms of its complement of a positive angle.

d. cos120�

e. sin100�

55

Applications

DCTc1 Solve equations which are reducible to a linear equation in one trigonometric

function.

a. Solve 2sin 6 4sinθ θ+ = for 0 2x π≤ ≤

b. Solve 2cos 1θ = − for 0 2x π≤ ≤

c. Solve 22sin sin 1 0x x+ − = for 0 2x π≤ ≤

d. Solve ( )( )2cos 2 2cos 1 0x x+ − = for 0 2x π≤ ≤

e. Solve 29 tan 3 0x − = (hint: difference of squares) for 0 2x π≤ ≤

56

Solving Triangles

DCTc2 Solve right triangles. Solve the following right triangles:

a. 80, 60a β= = �

b. 30, 70a c= =

c. 49, 45a α= = �

d. 90, 13b β= = �

e. 28.6, 67.7c α= = �

c

b a

C

B A β α

57

DCTc3 Solve triangles using Law of Sines or Law of Cosines.

a. 7, 8, 2a b c= = = . Find γ .

b. 10, 4, 8a b c= = = . Find β .

c. 3, 2, 100a b α= = = � . Find c

d. 50, 28.3, 28a b γ= = = � . Solve the triangle.

e. 14.3, 16.5, 115a b β= = = � . Solve the triangle.

γ

c

b a

C

B A β α

58

DCTc4 General story/word/application problem involving subscore 3 objectives (DCTc1-c3)

a. In Milwaukee, a hill makes an angle of 22�with the horizontal and has a tall building at the top. At a point 100 ft down the hill from the base of the building, the angle of elevation to the top of the

building is 70� . What is the height of the building? b. The most powerful lighthouse on Lake Michigan is 50 m tall. Suppose you are in a boat just off the coast. Determine your distance from the base of the lighthouse if the angle from the boat to the top of

the lighthouse is 14� . c. A fighter jet must hit a small target by flying a horizontal distance to reach the target. When the

target is sighted, the onboard computer calculates the angle of depression to be 27� . If after 150 km

the target has an angle of depression of 42� , how far is the target from the fighter jet at that instant?

d. A vertical tower is located on a hill whose inclination is 5� . From a point 100 ft down the hill to the

base of the tower, the angle of elevation to the top of the tower is 25� . What is the height of the tower? e. A buyer is interested in purchasing a triangular lot with vertices ABC, but unfortunately, the marker at point A has been lost. The deed indicates that CB is 450 feet and AB is 110 feet, and the angle at A

is 80� . What is the distance from A to C?

59

Graphing

DCTd1 Recognize graphs from rule or vice versa for y = sinx, y = cosx, y = tanx, etc.

Match each graph with its trigonometric function. a. b. c. d. e.

60

DCTd2 Know and use relationships between any of the following: graph; rule; values of

amplitude/period/phase/intercepts/extrema; qualitative properties (boundaries, periodicity,

range, invertability, even/odd) for y = A sin (Bx+C)+D or similar cos or tan function.

a. Graph one period of sin2

y xπ = +

b. Graph one period of cos 26

y xπ = −

c. What is the period & amplitude of 2 sin2

y xπ − = −

?

d. What is the period & amplitude of 3

3cos2

y xπ = +

?

e. What is the period & amplitude of 2 tan4

y xπ + = +

?

61

DCTw5 General story/word/application problem involving subscore 5 objectives (DCTd1-2)

The distance a certain satellite is north or south of the equator is given by

2500cos60 4

y tπ π = +

where t is the number of minutes that have elapsed lift-off.

a. Graph the equation for 0 120t≤ ≤ b. What is the greatest distance that the satellite ever reaches north of the equator? c. How long does it take to complete one period? The table below gives the average monthly temperatures for Milwaukee for a 12-month period starting with January. Model the monthly temperature with an equation of the form

( )( )siny a b t h k= − +

y in degrees Fahrenheit, t in months, as follows: d. Find the value of b assuming that the period is 12 months

e. How is the amplitude a related to the difference 79 30−� � f. Use the information in part (e) to find k.

Milwaukee Temperature Data

Time (months)

Temperature

( )F�

1 31

2 30

3 39

4 44

5 58

6 67

7 76

8 79

9 63

10 51

11 40

12 33

62

Statistics

BSa1 Interpret pie charts and bar graphs.

a. Below is a bar graph for the sales of Key & Co. for the past 5 years.

0

2

4

6

8

10

12

14

$ in Millions

Which of the following statements are true?

1. Key & Co. had greater sales in 2001 than in the next two years combined. 2. Key & Co. had greater sales in 2004 than in 2000 and 2001 combined.

3. Key & Co. had five times as many sales in 2004 compared to 2002.

4. Key & Co. had greater sales in 2004 than the previous three years combined.

What year were Key & Co. sales the greatest? The Least? What is the average yearly sales for Key & Co. from 2000-2004? What is the median yearly sales for Key & Co. from 2000-2004? b. Which of the following pie charts shows the theoretical outcomes of tossing a die 500 times?

2000 2001 2002 2003 2004

63

c. Here is a nationwide survey of people’s ice cream preference in 2001 and 2005 by flavor.

Favorite Ice Cream 2001

Vanilla

28%

Chocolate

29%

Mint

18%

Strawberry

15%

Other

10%

Favorite Ice Cream 2005

Vanilla

22%

Chocolate

34%Mint

10%

Strawberry

28%

Other

6%

1. What percentage increase did chocolate occurred from 2001-2005?

2. What is the ratio of categories

that increased in popularity to those that decreased?

3. If a percentage point shift

results in annual additional sales of $50,000, how much, in dollars, did combined annual strawberry and chocolate sales increase from 2001 to 2005?

4. Which flavor decreased the

most in popularity from 2001 to 2005?

5. In 2001 the Mint category had

10% popularity, and 5,850 people surveyed preferred Mint, then how many people

were surveyed all together?

64

BSa2 Calculate the mean and median of a data set.

a. Find the mean and median: 5, 7, 8, 14, 6 b. Find the mean and median: 4, 6, 8, 13, 17, 10, 8, 21 c. Find the mean and median: 1.3, 7.1, 7.7, 2.2, 5.9, 3.1, 4.8 d. Find the mean and median: 200, 700, 300, 500, 800, 400, 400, 900 e. Find the mean and median: 48, 49, 49, 49, 50, 51, 51, 52, 52, 52, 52, 55, 57