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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

5in4cm

5cm

13cm

4ft

6ft3in

5in

9m12m

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.











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 15 x +

4 5 x + 3 20 x +

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 ∠ ?





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

ED

CBA



22

AGe4 Identify parallel and perpendicular lines

a. _____ j ⊥ b. ______ k c. ____ ____ m or ⊥

d. ____ n e. ____ k ⊥







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?





















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?











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 60 BOA = 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?





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





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 given1

cos2

θ = , Quadrant I.

b. Find the value of θ given3

sin2

θ = , Quadrant II.

c. Find the value of θ given 2cos2

θ −= , Quadrant III.

d. Find the value of θ given1

sin2

θ −

= , Quadrant IV.

e. Find the values of θ given1

sin2

θ = .





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 −

−

.





50

Identities

DCTb1 Apply identities involving Pythagorean formulas

a. Simplify21 cos

sin

θ

θ

−, sin 0θ ≠

b. Simplify 2 2sec tanθ θ +

c. Simplify

sin cos

cos sin1

sin cos

θ θ

θ θ

θ θ

+, sin 0,cos 0θ θ ≠ ≠

d. Prove the identity:2

sincsc

1 cos

θ θ

θ =

−

e. Prove the identity: 2 2

cot 1

sin cos tan

θ

θ 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 cotsec csc

θ θ

θ θ +

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

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





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 2 x π ≤ ≤

b. Solve 2cos 1θ = − for 0 2 x π ≤ ≤

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

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

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

BA 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

BA

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. Ata 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 thecoast. 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 thetower?

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 x π + = +

?



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 t π 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 startingwith January. Model the monthly temperature with an equation of the form

( )( )sin y 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 6310 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

$ i n M i l l i o n s

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





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

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