Placement Test Practice Problems. Book II. Geometry, Trigonometry, and
Statistics. Eric Key, University of Wisconsin-Milwaukee. David Ruszkiewicz,
Milwaukee ...
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. EHR0314898. 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).
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
1
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/
2
Areas of polygons and circles AGa1 Find the area of region composed of rectangles and/or right triangles a.
b. 3in
6ft
5in 4ft
c. 12m
9m
4m 18m
d.
4cm 13cm
6in
e.
5cm
5in 4in
3
5in 5in 3in
AGa2 Express shaded area of region composed of rectangles as fractions of total area a.
b.
c.
d.
e.
4
AGa3 Find area of triangle. a.
b.
c.
d.
e.
5
AGa4 Find area of circle or semicircle given radius or diameter a.
b.
c. shaded region only
d.
e.
6
AGa5 Find area of region composed of rectangular and circular or semicircular parts a. shaded area only
8
c. shaded area only
b. shaded area only
12
d. shaded area only
e.
7
AGb1 Find perimeter of a parallelogram given two sides a.
c.
b.
d.
e.
8
AGb2 Find perimeter of rectilinear figure a.
b.
c.
d.
e.
9
AGb3 Find circumference of circle given radius or diameter a.
c.
b.
d.
e.
10
Measuring Solids AGc1 Find surface area of a rectangular box a.
c.
b.
d.
e.
11
AGc2 Find volume of a rectangular box a.
c.
b.
d.
e.
12
AGc3 Find surface area of polycube a.
b.
c.
d.
e.
13
AGc4 Find volume of a polycube a.
c.
b.
d. There is a square hole cut through the center
e. The shape below can be folded to form a cuboid. Find the volume.
14
Angle Sum for a Triangle AGd1 Find the third angle given two angles of a triangle a.
b. 48
X
41
c.
d.
e.
15
AGd2 Find the acute angle in right triangle given the other acute angle a.
b.
c.
d.
e.
16
AGd3 Find exterior angle given two interior angles a.
b. 72 76
45
53
x
x
c.
d. 40
8 x + 15
3 x + 20
4x + 5
x
e.
x
111 39
17
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 ?
18
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 ABCD
c. Identify base and altitude of parallelogram
d. Find base of parallelogram given its area is 105 square inches.
e. Identify base and altitude of ∆CEG
19
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
20
AGe3 Identify right, acute, and obtuse angle
E
D
A a. ABE
B
C b. DBC
c. DBE
d. CBE
21
e. DBA
AGe4 Identify parallel and perpendicular lines
a. j ⊥ _____
b. k ______
d. n ____
e. k ⊥ ____
c. m ⊥ ____ or ____
22
Parallel and Perpendicular Lines BGb1 Recognize and apply definitions of parallel and perpendicular lines Classify the following as intersecting, parallel, or perpendicular a.
d.
b.
c.
e.
23
BGb4 State the relationship between perpendicularity and a right angle Which of the following line segments are perpendicular? a.
b.
c.
90
88
d.
92
e. Two lines intersecting at a right angle are ________________.
24
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. CE = 3, BE = 3, CD = 4, DA = 4 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?
25
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?
mn
mn l
1
4 3
8 7
2
3
9
c.
m
l
6
1 m
4
2
n
n
What is the value of x ?
d. What is the value of x ?
mn
mn 2x + 5
m
x + 60
n
n
80 + x
5x
e. Find the value of all the angles. mn 5 45
8 9
7 6
4 2
m
3 1
m
55 12 10 11
n
26
Pythagorean Relationship BGc1 Given two sides of a right triangle, find the third side a.
b.
c.
d.
e.
27
BGc2 Given hypotenuse and leg of a triangle, find the area a.
b.
c.
d.
e.
28
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?
29
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?
30
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.
31
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.
32
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.
33
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.
34
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?
35
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 ___________.
36
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 ?
37
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?
38
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 ?
39
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
π 5
, 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?
40
DCGc4
Recognize and use the relationship between tangent and radius
a. If OB = 6 and AO = 10, then AB = ?
b. If BOA = 60 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?
41
Congruence and Similarity DBGa1
Recognize and apply definition of congruence and similarity A
a. The two triangles shown are congruent.
B
∆CDO ≅ ______ ∠C ≅ _____
O
CO ≅ ______
DO = ______
C
D
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. 42
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.
d. Find the values of a and b.
c. Find the values of a and b.
e. Name two triangles that are similar to ∆ABC .
43
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
44
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. y
y
7
2
θ
θ
x
6
c.
4
d. y
4
θ 4
e. If sin θ =
x
2 , for an acute angle θ , state two ways to find cos θ and tan θ . 3
45
x
DCTa2 Use knowledge of sides and angles in 30-60-90 and 45-45-90 triangles, e.g., solve equations of the form sin θ =b or cos θ =b. a. Find the value of θ given cos θ =
1 , Quadrant I. 2
b. Find the value of θ given sin θ =
3 , Quadrant II. 2
c. Find the value of θ given cos θ =
− 2 , Quadrant III. 2
d. Find the value of θ given sin θ =
−1 , Quadrant IV. 2
e. Find the values of θ given sin θ =
1 . 2
46
DCTa3
Convert between degree and radian measure
a. Express 45 in radians.
b. Express 270 in radians.
c. Express
4π in degrees. 3
d. Express
5π in degrees. 6
e. Express
7π in degrees. 3
47
DCTa4
Evaluate expressions involving inverse trigonometric functions.
a. Find the value of Sin −1
3 . 2
b. Find the value of Cos −1
2π . 3
3 c. Find the value of Cos −1 − . 2
5π d. Find the value of Sin −1 sin . 6
π e. Find the value of Cos −1 cos − . 3
48
DCTw1
General story/word problem involving subscore 1 objectives (DCTa1-a4).
a. Solve the right triangle ABC. c
∠A = 36 c = 68
B
A
a C
b
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?
49
Identities DCTb1
Apply identities involving Pythagorean formulas
1 − cos 2 θ , sin θ ≠ 0 a. Simplify sin θ
b. Simplify sec 2 θ + tan 2 θ
sin θ cos θ + c. Simplify cos θ sin θ , sin θ ≠ 0, cos θ ≠ 0 1 sin θ cos θ
d. Prove the identity: csc θ =
e. Prove the identity:
sin θ 1 − cos 2 θ
cot θ 1 = , sin θ ≠ 0, cos θ ≠ 0, tan θ ≠ 0 2 sin θ + cos θ tan θ 2
50
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 cos θ =
3 and θ is in Quadrant IV, find cos 2θ , sin 2θ , & tan 2θ . 5
51
DCTb3
Apply identities involving Quotient and Reciprocal Formulas.
In a-c, simplify, leaving answers in terms of sines and cosines. a. sin θ + cot θ
b. sec θ + tan θ
c.
tan θ + cot θ sec θ csc θ
d. Prove the identity: tan θ = sin θ sec θ
e. Prove the identity: cot θ = cos θ csc θ
52
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. sin (π − θ ) =
1 what’s sin θ ? 7
b. cos (π + 30 ) =
c. sin(π + 45 ) =
In d-e, write each trigonometric function in terms of its complement.
d. sin 29
e. cos 41
53
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
54
Applications DCTc1 function.
Solve equations which are reducible to a linear equation in one trigonometric
a. Solve 2 sin θ + 6 = 4sin θ for 0 ≤ x ≤ 2π
b. Solve 2 cos θ = −1 for 0 ≤ x ≤ 2π
c. Solve 2 sin 2 x + sin x − 1 = 0 for 0 ≤ x ≤ 2π
(
)
d. Solve 2 cos x + 2 ( 2 cos x − 1) = 0 for 0 ≤ x ≤ 2π
e. Solve 9 tan 2 x − 3 = 0 (hint: difference of squares) for 0 ≤ x ≤ 2π
55
Solving Triangles DCTc2
C
Solve right triangles. b
a
Solve the following right triangles: A a. a = 80, β = 60
b. a = 30, c = 70
c. a = 49, α = 45
d. b = 90, β = 13
e. c = 28.6, α = 67.7
56
α
β c
B
DCTc3
Solve triangles using Law of Sines or Law of Cosines. C
γ b
A
a
α
β c
a. a = 7, b = 8, c = 2 . Find γ .
b. a = 10, b = 4, c = 8 . Find β .
c. a = 3, b = 2, α = 100 . Find c
d. a = 50, b = 28.3, γ = 28 . Solve the triangle.
e. a = 14.3, b = 16.5, β = 115 . Solve the triangle.
57
B
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?
58
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.
59
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.
π a. Graph one period of y = sin x + 2
π b. Graph one period of y = cos 2 x − 6
π c. What is the period & amplitude of y − 2 = sin x − ? 2
3π d. What is the period & amplitude of y = 3cos x + 2
?
π e. What is the period & amplitude of y + 2 = tan x + ? 4
60
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
π π y = 2500 cos t + 4 60 where t is the number of minutes that have elapsed lift-off. a. Graph the equation for 0 ≤ t ≤ 120
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
y = a sin ( 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.
61
Milwaukee Temperature Data Time Temperature (months) (F) 1 2 3 4 5 6 7 8 9 10 11 12
31 30 39 44 58 67 76 79 63 51 40 33
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. 14 12
$ in Millions
10 8
6 4 2 0
2000
2001
2002
2003
2004
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?
62
c. Here is a nationwide survey of people’s ice cream preference in 2001 and 2005 by flavor.
Favorite Ice Cream 2001
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?
Other 10% Vanilla 28%
Strawberry 15%
Mint 18% Chocolate 29%
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?
Favorite Ice Cream 2005
Other 6%
Vanilla 22%
Strawberry 28%
Mint 10%
63
Chocolate 34%
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
64
