sabis mathematics im1 end of year revision packet ... an arithmetic sequence, find its common...

End of Year Revision Packet Mathematics-IM1 of 33

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SABIS® School Network Mathematics

IM1

End of Year Revision Packet

Chapters 1 – 12 S4

Highlights:

Identify the elements of different sets of numbers.

Convert numbers from decimal form to fractional form and vice versa.

Test a number for divisibility by a given prime number.

Give the prime factorization of a given number using knowledge of prime and composite

numbers as well as factors and multiples.

Find the greatest common factor and the lowest common multiple of two numbers.

Write a fraction in simplest form.

Express a fraction as a mixed numeral and vice versa.

Add, subtract, multiply, and divide fractions.

Perform operations on directed numbers.

Order directed numbers.

Simplify numerical expressions.

Express numbers in scientific notation, understand significant figures, and round to a

nearest given value.

Find the range of values of rounded numbers and find the error of given measurements.

Solve a linear equation in one variable and solve word problems leading to linear equations.

Solve a linear equation in more than one variable for one variable in terms of the other.

Solve a linear equation involving absolute values.

Solve a linear inequality in one variable and graph its solution.

Solve word problems leading to linear inequalities.

Solve absolute value inequalities and graph the solution.

Solve word problems leading to absolute value inequalities.

Solve problems involving ratios and proportions.

Solve different types of questions and word problems involving percentages.

Define a relation and determine its range, draw its arrow diagram, list the pairs in it, and

represent it in a coordinate plane.

Define a function and know that it can be represented by a rule, in a coordinate plane, or by

a table.

Analyze graphs of different types of functions.

Recognize the general form of the rule of a linear function and graph it.

Recognize nonlinear functions.

Recognize an arithmetic sequence, find its common difference, apply the rule of its general

term, and graph it.

Use arithmetic sequences to solve problems in context.


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Evaluate simple positive and negative powers, solve exponential equations, and evaluate

simple expressions that involve exponents.

Apply different rules related to exponents.

Recognize numbers written in scientific notation and understand what standard form is.

Apply operations on numbers written in scientific notation and solve problems involving

scientific notation.

Graph simple exponential functions, deduce one graph from the other using shifting, and

use exponential graphs to solve problems in context.

Recognize a geometric sequence, find its common ratio, apply the rule of its general term,

and graph it.

Know the general formula for compound interest and solve related problems.

Solve word problems on half-life of radioactive material.

Work out numerical expressions involving square roots.

Simplify radical expressions and express them using only one radical.

Add, subtract, multiply, and divide radical expressions.

Graph a linear equation in the coordinate plane.

Find the slope of a line.

Use the slope y-intercept form or the point-slope form to graph a line.

Identify parallel, perpendicular, intersecting, and coincident lines.

Apply the midpoint formula and the distance formula.

Check if an ordered pair is a solution of a given system of linear equations.

Solve systems of linear equations by graphing, by substitution, or by linear combination.

Graph the solution set of a linear inequality in two variables.

Graph the solution set of a system of linear inequalities.

Understand undefined terms in geometry like point, line, plane, and space.

Recognize collinear, non-collinear, coplanar, and non-coplanar points.

Define distance, segment, midpoint, and ray.

Define angle and be able to identify the interior and the exterior of an angle.

Recognize adjacent and vertical angles. Know how to measure an angle and apply the arc

addition postulate.

Solve questions about complementary and supplementary angles.

Solve questions about the bisector of an angle.

Identify alternate angles, corresponding angles, and interior angles on the same side and

apply their properties.

Define a regular polygon and know some facts about them.

Define a triangle and know its elements. Define the interior and exterior of a triangle.

Know and apply the triangle inequality.

Classify triangles by sides and angles, and know the relations between the sides and the

angles of a triangle.


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Understand the distance from a point to a line geometrically and analytically.

Know and apply the angle sum theorem and the exterior angle theorem.

Define median, altitude, angle bisector, and perpendicular bisector. Know how to handle

related questions using geometric and analytic approaches.

Define congruent triangles and list corresponding parts. Know the relationships between

special lines in congruent triangles.

Know and apply the SAS postulate and the ASA, SSS, HL, and AAS theorems.

Define similar polygons and identify corresponding parts.

Define similar triangles and apply the AA postulate, SSS and SAS similarity theorems.

Know the relationships between special lines in similar triangles.

Define quadrilateral and special quadrilaterals like parallelograms, rectangles, rhombuses,

squares, and trapezoids.

Use and apply properties of special quadrilaterals using geometric and analytic approaches.

Find the perimeter and the area of a given shape in the coordinate plane.

Define a circle and congruent circles. Know all the parts of a circle. Identify the location of a

point with respect to a circle analytically.

Write the equation of a circle given the center and a point on the circle.

Define central angles and arcs. Find the arc measure and arc length.

Define locus and know the locus of points with special characteristics. Find the equation of a

given locus.

Construct geometric figures, given some properties about them. Justify their construction.

Find the image and the pre-image of a point under a translation. Find the image of a segment

and a polygon under a translation. Know that a non-trivial translation has no invariant point.

Know the relation between a point and its image under a reflection.

Find the image of a point under a reflection about a point, a vertical line, or a horizontal line.

Find the image of a polygon under a reflection.

Know the invariant points under a reflection.

Find the image of a point and a polygon after a rotation. Identify the invariant points under a

rotation.

Find the image of a point and a polygon under a given dilation. Identify the invariant points

under a dilation.

Identify the image of a point under the composite of two transformations.

Define glide translation. Identify the image of a point under two specific transformations.

Know the effect of composing a transformation and its inverse. Describe the inverse of a

reflection, rotation, dilation, or translation.

Recognize a figure with a line of symmetry or a rotational symmetry. Find the degree of

rotational symmetry of a figure.


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Classify data as qualitative or quantitative.

Find the mean, mode, median, and quartiles of a set of ungrouped data.

Solve applications on the mean of a set of ungrouped data.

Find the range and the interquartile range of a set of ungrouped data.

Find the mean, median, and mode of a set of data after a linear transformation.

Find the range of a set of data after a linear transformation.

Form a frequency table for a set of ungrouped data.

Form relative and cumulative frequency table given the frequency table.

Solve applications on the cumulative frequency table.

Draw and read a line plot, a bar graph, a box-and-whisker plot, a circle graph, a histogram,

and a line graph.

Estimate using interpolation or extrapolation.


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Vocabulary:

Number sets, Natural numbers, Whole numbers, Set of integers, Set of rational numbers,

Repeating decimals, Terminating decimals, Set of irrational numbers, Set of real numbers,

Divisibility, Factors, Multiples, Prime, Composite, Prime factorization, Greatest common

factor, Lowest common multiple.

Equivalent fractions, Simplest form, Mixed number, Improper fraction, Directed numbers,

Ordering numbers.

Numerical expressions, Scientific notation, Standard notation, Significant figures,

Rounding, Range of values.

Linear equations, Solution, Solve, Word problems, Absolute value, Linear inequality, Solve

and Graph.

Absolute value inequality, Ratio, Proportion, Cross multiplication property, Multi-rate

problems, Percentage, Profit, Discount, Cost, Original price, Percentage profit, Percent

change, Percent discount, Percent error, Simple interest, Capital, Interest rate, Mixture

problems.

Relation, Ordered pair, Abscissa, Domain, Ordinate, Range, Arrow diagram, Function,

Element, Rule of a function, Tables, Graph of a function, Speed.

Function notation, Variable, Independent variable, Dependent variable, Linear functions,

Non-linear functions, Sequence, Arithmetic sequence, Common difference, General term,

nth term, Patterns.

Integer exponents, Exponential expressions, Base, Power, Exponential functions, Horizontal

shift, Vertical shift, Geometric sequence, Common ratio, Compound interest, Half-life of

radioactive material.

Square root, Radical sign, Radicand, Principle square root, Radical expressions, Rationalize,

Unlike terms, Pythagorean theorem.

Linear equation, Slope of a line, Horizontal lines, Vertical lines, Undefined slope, Rate,

Intercepts, Slope y-intercept form, Point-Slope form, Parallel lines, Perpendicular lines,

Intersecting, Coincident, Midpoint formula, Distance formula.

System of linear equations, Solution of a system, System with no solution, System with a

unique solution, System with infinitely many solutions.

Solve by graphing, Graphically, Substitution, Approximate values of the solution.

Adding a linear equation to another, Adding a multiple of a linear equation to another,

Linear combination, Word problems, Algebraic way, Linear inequality in 2 variables,

Shade, Strict and not-strict inequalities, System of inequalities.

Undefined terms, Point, Line, Plane, Space, Collinear points, Non-collinear points, Coplanar

points, Non-coplanar points.

Distance, Coordinate, Betweenness, Segment, Congruent, Midpoint of a segment, Ray,

Opposite rays.


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Angle, Interior angle, Exterior angle, Adjacent angles, Vertical angles.

Measure of an angle, Angle addition postulate, Right angle, Acute angle, Obtuse angle,

Complementary angles, Supplementary angles, Linear pairs, Congruent angles, Bisector of

an angle.

Perpendicular lines, Parallel lines, Transversals, Alternate angles, Corresponding angles,

Interior angles on the same side.

Polygon, Side, Vertices, Diagonal, Convex, Concave, Interior angles, Exterior angles,

Regular polygon.

Triangle, Vertex, Side, Interior, Exterior, Triangle inequality, Equilateral, Isosceles,

Scalene, Acute, Obtuse, Right.

Distance from a point to a line, Angle sum theorem, Exterior angle theorem, Median,

Altitude, Angle bisector, Perpendicular bisector, Concurrent.

Congruent triangles, Correspondence, Included angle, Included side, SAS, ASA, SSS, HL,

AAS.

Similar polygons, Similar triangles, AA postulate, SAS similarity theorem, SSS similarity

theorem.

Quadrilateral, Parallelogram, Rectangle, Rhombus, Square, Trapezoid, Perimeter, Area.

Circle, Interior, Exterior, Center, Radius, Diameter, Chord, Location of a point, Secant,

Tangent.

Central angles, Central arcs, Arc measure, Arc length, Minor arc, Major arc, Semi-circle.

Locus, Equidistant, Compound locus.

Construction, Drawing a figure to scale.

Translation, Pre-image, Image, Non-trivial translation, Invariant point.

Reflection, Benchmark reflection, Rotation, Benchmark angle, Dilation, Reduction,

Enlargement, Similarity transformation, Scale factor, Composite of two transformations,

Glide translation, Inverse of a transformation.

Line of symmetry, Point of symmetry, Rotational symmetry, Degree of rotational symmetry.

Qualitative data, Quantitative data, Mean, Mode, Median, Quartiles, Range, Interquartile

range, Upper quartile, Lower quartile, Five-number summary, Linear transformation

Frequancy table, Frequancy, Relative frequency, Cumulative frequency, Data, Line plot, Bar

graph, Box plot, Modified box plot, Circle graph, Histogram, Line graph, Linear

interpolation, Linear extrapolation


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Revision Exercises:

Ch. 1 Basics Section 1 Number Sets

1. For each of the numbers listed below, identify the smallest set from among the sets , W, ,

, that contains it. 22 9

2.202002..., , 6, , 7.2, , 8.00, 4.757 3

Section 2 Factors and Multiples

1. Find the greatest common factor of each pair of numbers.

a) 2940 and 2520 b) 4840 and 528 c) 1401 and 4056

2. Find the least common multiple of each set of numbers.

a) 108, 90, and 35 b) 7, 100, and 200 c) 48, 32, and 64

3. Jimmy changes the password on his laptop on a regular basis. He always chooses the password

to be the sum of all the prime numbers which are less than or equal to the number of the day in

the month.

a) List all days of the month on which Jimmy cannot change his password.

b) What will his password be if he changes it on the 19th of October?

Section 3 Fractions

1. Locate the fractions

3

5 ,

1

2 ,

1

5 , and

3

10 on the number line then write them in increasing order.

0 1

2. Multiply. Express the answer as a fraction in simplest form or as a mixed number.

a)

3 5

25 2

b)

1 10

5 33

c)

5 9

27 25

d)

5 7

4 4

e)

2 15

15 2

f)

32 96

48 64

g)

642

7

h)

156

54

i)

1 3

2 5


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3. Add. Express the answer as a mixed number.

a)

1 35 2

3 4

b)

2 75 2

25 10

c)

1 3 51 2 3

4 5 7

d)

1 12 7

5 2

e)

3 37 8

8 7

f)

3 1 101 2 5

7 14 21

4. Subtract. Express the answer in simplest form.

a)

6 35 2

7 7

b)

3 47 2

11 11

c)

7 56 4

8 12

d)

1 16 2

4 3

e)

4 73 2

7 8

f)

2 35 3

7 8

5. Divide. Express the answer in simplest form as a fraction or as a mixed number.

a)

1 11 2

2 3

b)

4 33 5

7 7

c)

2 35 4

3 4

d)

58 1

4

e)

26 1

5

f)

5 32

7 4

Section 4 Directed Numbers

1. Refer to the number line below to state the coordinate of the specified point.

8 7 6 5 4 3 2 1 O 1 2 3 4 5 6 7 8

A B C D E F G H I J K L M N P Q R

a) The point between C and P that is twice as far from C as it is from P.

b) The point between C and P that is five times as far from P as it is from C.

c) The point to the left of L that is twice as far from R as it is from L.

Section 5 Simplifying Numerical Expressions

1. Simplify.

a) 8 + 10 5 b) 90 10 2 c) 100 + 20 5

d) 8 10 5 e) 90 10 2 f) 100 20 5

2. Evaluate each expression for

2

3x

,

3

5y

, and

1

4z

.

a)

x y

y z

b) 1

xz

y

c)

2 3

2 5

x y

y z

d) 1

x z

y


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Section 6 Estimation

1. Round 3,409,981.348 meters to the nearest.

a) meter. b) 10 meters. c) kilometer (1,000 meters).

2. The dimensions of a rectangular sheet of paper are measured to the nearest inch as 11 in. by

17 in.

a) What is the least possible value of the area of this sheet of paper?

b) The cost of printing colored images is $0.05 per square inch. What is the range of the cost

for printing a colored image on this sheet of paper?

Chapter Summary

TB pages 33 – 35 Chapter Test TB pages 36 – 38

Ch. 2 Linear Equations and Linear Inequalities Section 1 Linear Equations

1. The greatest of four consecutive integers is 8 less than twice the least. Find the integers.

2. Jose is saving to buy a coat that costs $85. He has $20 already saved. Every week he plans to

add $5 more to his savings. Based on this plan, how many weeks will it take Jose to save

enough money to buy the coat?

Section 2 Linear Equations Involving Absolute Value

1. Solve. If the equation has no solution, state so.

a)

1 2| |

3 3a

b)

7 4| | 1

9 9b

c)

3 13 | | 2

4 4c

Section 3 Linear Inequalities

1. Solve each inequality and graph its solution set on a number line.

a)

3 4

3

x

1 2

x

b) 13 4(x + 2) + 3x 2 0

2. An employee is paid $20 for the first working hour and $12 for every additional hour. How

many hours should the employee work to earn more than $80?

Section 4 Absolute Value Inequalities

1. Solve and graph.

a)

2 5

3 6

x

b) 8 | a 8 | 4


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Section 5 Ratios and Proportions

1. A cable is 144 cm long. It is cut into two pieces whose lengths are in the ratio of 3 to 4. How

long is each piece?

2. A plane travels a distance of 1,000 km in 2.5 hours while flying against the wind and in 2

hours while flying with the wind. Find the speed of the wind.

Section 6 Percentages

1. May paid a 5% sales tax on her new car. What was the total price of the car if the sales tax was

$425?

2. Roberto invested $2,000 for two years at a simple interest rate of 8.5% per year. How much

interest did he earn?

Chapter Summary


Ch. 3 Functions Section 1 Relations

1. The range of a relation is {0.5, 1, 2, 3.5, 5.5}. Consider the relation that pairs each number in

the domain with its half in the range.

a) Give the domain of this relation.

b) Draw an arrow diagram that joins each pair in this relation.

c) List the set of ordered pairs of the relation.

d) Graph the set of ordered pairs in the relation in a coordinate plane.


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Section 2 Functions

1. The table below represents a function.

x y

1 4

2 8

3 12

4 16

5 20

a) Give the domain and range of this function.

b) List the set of ordered pairs in the function.

c) Graph the function in a coordinate plane.

d) Give a rule that can be used to define this function.

e) Use the rule to find the value of b if the ordered pair (8, b) were also an element of the

function.

2. Determine whether the graph is the graph of a function. Explain.

a) y

x O

b) y

x O

c) y

x O

d) y

x O

e) y

x O

f) y

x O


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Section 3 Linear Functions

1. Consider the function f given by f(x) = 2x 2.

a) Find f(0) and f(1).

b) Draw the graph of f.

c) What is the value of a for which f(a) = 4?

d) Is there a value for x in the domain of f for which f (x) = x?

2. a) Find the range of g(x) = x – 5 defined over the domain D = {x: –2 x 5}.

b) Find the range of h(x) = 2x defined over the set of whole numbers.

c) Find the range of f (x) = 10x if the domain of f is the set of real numbers between 0 and 2

inclusive .

Section 4 Arithmetic Sequences

1. The 2nd and the 7th terms of an arithmetic sequence are –2 and –17, respectively.

a) Is the sequence increasing or decreasing?

b) Find the values of the 4th and the 40th terms.

2. The first five terms of an arithmetic sequence are

plotted on the coordinate grid to the right.

a) What are the coordinates of the point that

corresponds to the 6th term of this sequence?

b) What is the rule for this sequence?

1 2 3 4 5 6

1

2

3

4

5

6

7

8

9

x

y

0

Chapter Summary

TB page 99 Chapter Test TB pages 100 – 102


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Ch. 4 Exponential Functions Section 1 Exponential Expressions

1. Express

3

81

x

as a power of 3 and solve

33

81

x

.

2. Simplify using only positive exponents.

a)

12 4

2 3

4

16

a b

a b

b)

22 5

6 7

2

10

a b

a b

c)

04 8

2

6

15

a b

a b

d)

32 4

2

3

2

a b

a b

e)

33 2

1 0

2

3

a b

a b

f)

22 3

23 2 1

5

25

x y z

x y z

Section 2 Operations With Scientific Notation

1. Simplify. Express your answer in decimal notation.

a) 5101 + 6102 + 4103 b) 110 + 2102 + 4104

2. The half-life of radioactive Iodine-129 is 1.56×107 years and the half-life of radioactive

Nickel-59 is 7.5×104 years. How many times greater is the half-life of Iodine-129 than the

half-life of Nickel-59?

Section 3 Exponential Functions

1. The graph of y = 3x is shown to the right.

By using the given graph, sketch the

graph of y = 32x.

Explain your reasoning.

5 5

5

10

x

y

0


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Section 4 Geometric Sequences

1. The nth term of a pattern is given by the formula below: 1

12

n

na

a) Find a1, a2, and a3.

b) Find the sum of the first four terms of the pattern.

c) Show that 1

11

2n na a

.

2. The half-life of radioactive Carbon-11 is 20 minutes. Suppose there were 200 grams of

Carbon-11 present.

a) How many grams were present 40 minutes earlier ?

b) How many grams were present 1 hour later?

Chapter Summary


Ch. 5 Radical Expressions Section 1 Square Roots

1. Find the square root of each of the following numbers.

a) 1,521 b) 1,225 c) 2,025 d) 3,025

Section 2 Multiplying and Dividing Radical Expressions

1. Simplify.

a)

10 1

3 5

b) 2 4 25 4x x y

2. Simplify.

a)

4 25

5 64

b)

2

4

49

25

x

y

Section 3 Adding and Subtracting Radical Expressions

In 1 – 4, simplify the expression.

1.

73

9

2.

2 3

3 4

3. 4 28 2 7 21 4. 2 200 3 250 4 300


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Section 4 Applications

1. Which of the following is an irrational number?

A. 4.2356 B. 80 C. 144 D.

5

7

2. A vertical pole broke at a height 5 feet off the ground. Its tip

landed on the ground 12 feet away from the bottom of the

pole. What was the original height of the pole?

5 ft

12 ft

3. Find the value of x in the figure below.

3

2

2

3x

Chapter Summary


Ch. 6 Linear Equations in Two Variables Section 1 The Graph of a Linear Equation in Two Variables

1. Graph each equation on a separate coordinate plane.

a) 2

4

yx

b) y 6x = 1

2. Find the value of m for which the point with the given coordinates lies on the graph of the

given equation.

a) 2mx + 7y = 27; C(2, 1) b) 2

2

mx y

; D( 6, 2 )

3. Graph the two equations on the same coordinate plane then determine from the graph the point

of intersection of the two graphs.

a) 4x + y = 7; 2x y = 1 b) 4x y = 6; y 3x = 1


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4. Supply the missing values and then graph the given equation. 1

12

x x

x

y 2 2

Section 2 The Slope of a Line

1. Determine the value of p so that the two given points lie on the line whose slope is given.

a) (0, 3) and (5, 8); slope is 2p + 3 b) (2, p) and (9, 7); slope is 1

2. Determine whether the line passing through the given points is vertical, horizontal, slants

upward to the right, or downward to the right.

a) R(7, 7) and S(9, 9) b) E(1, 1) and F(0, 1)

Section 3 The Slope y-Intercept Form of a Linear Equation

1. Express each equation in the slope y-intercept form and deduce the value of its slope and y-

intercept.

a) 0.6x + 0.4y = 1.2 b) x = 4

2. The equation below gives the total amount y Tania gets paid when she works x hours of

overtime.

y = 20x + 600

a) What is the y-intercept of the line represented by this equation? What does it represent

relative to Tania’s salary?

b) What is the slope of the line represented by this equation? What does it represent relative to

Tania’s salary?

c) How much will Tania get paid at the end of the month, if she works 9 hours of overtime?

Section 4 The Point-Slope Form of the Equation of a Line

1. Write the equation of the line with an undefined slope and passing through the point E(5, –3)

in the slope y-intercept form and draw its graph.

2. Write the equation of the line with undefined slope and that has the same xintercept as the

line whose equation is given by 4x 3y = 12.


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Section 5 Parallel and Perpendicular Lines

1. Write the equation of the line parallel to n: y = 5x + 2 and having a yintercept of (0, 7).

2. Write the equation of the line passing through A(1, 2) and perpendicular to m: 3x + 6y = 4.

Section 6 The Midpoint and Distance Formulas

1. Find the coordinates of B given that M is the midpoint of AB .

a)

5 2 5 2, , ,

7 9 7 9A M

b) A(n+3, 1), M(5n, 1)

2. ABCD is a parallelogram with A(3, 5), B(7, 5), C(5, 2), and D(1, 2). What is the perimeter

of ABCD?

Chapter Summary


Ch. 7 Systems of Linear Equations and Linear Inequalities Section 1 Graphical Solutions

1. Solve by graphing.

a)

27

3

11

3

y x

y x

b)

3 1

3 9 8

x y

x y

2. Find the point on the graph y 2x = 3 where the xcoordinate is twice the ycoordinate.

Section 2 The Substitution Method

1. Solve using the substitution method.

a)

8

8

x y

y x

b)

73

2 3

x y

x y

2. Determine whether the lines 2x + 3y = 13, 3x 2y = 0, and x + y = 5 are concurrent. (Three

lines are concurrent if they meet at the same point.)


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Section 3 Linear Combinations

1. Solve using the linear combination method.

a)

3 4 7 0

4 3 8 0

x y

x y

b)

3 21

1 1 5

6

x y

x y

2. The price of a sandwich and a soda is $3.60,

the price of a sandwich and a bag of chips is $3.14, and

the price of a soda and a bag of chips is $1.84

What is the price of the soda?

Section 4 Applications of Systems of Linear Equations

1. The length of a rectangle is 4 times its width. If the perimeter is 80 m, find the length and

width of the rectangle.

2. A 5% acid solution is mixed with a 10% acid solution. The mixture is 8 gallons of a 6% acid

solution. What is the amount of the 10% acid solution that was added?

3. A model airplane club publishes a newsletter. Expenses are $90 for printing and mailing each

copy, plus $600 for research and writing. The price of the newsletter is $1.50 per copy. How

many copies of the newsletter must the club sell to break even?

Section 5 Graphing One Linear Inequality in Two Variables

1. Graph the solution set of the given inequality.

a) x + y 4 b) | y | 3

Section 6 Graphing a System of Linear Inequalities in Two Variables

1. Graph the solution set of the given system of inequalities.

a)

2

6

y

y x

x y

b)

0

6

12

y

x

x y

2. Jennifer works at a shop where she is paid $6 per hour. She also works at a library where she

is paid $4 per hour. She wants to earn at least $60 per week but would like to work no more

than 12 hours per week.

a) Write a system of linear inequalities that describes the situation and graph its solution set.

b) Give three possible solutions of the system.

Chapter Summary



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Ch. 8 Lines and Angles Section 1 Introduction to Geometry

1. Sketch and label the figures described.

a) Points M, N, P, and Q are coplanar.

b) Point A is contained in three lines.

c) Line m contains points A, B, and C, but not point D.

d) Plane K contains points, M, N, and Q, but does not contain point S.

e) MN

and PQ

intersect at point M.

Section 2 Segments and Rays

1. Use the diagram below.

3 2 1 0 1 2 3 4

G F E D O A B C

a) Find AF.

b) Name a segment congruent to EA .

c) What is the coordinate of the midpoint of BF ?


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2. Consider the proof below for the following fact:

If M and N are the midpoints of AB and CD ,

respectively and DN = AM, then DC = AB

A

M

B D

N

C

Proof

Statements Reasons

a) M is the midpoint of AB

and N is the midpoint of CD

a) Given

b) AM =

1

2 AB and DN =

1

2 DC b)

c) DN = AM c)

d)

1

2 DC =

1

2 AB d)

e) DC = AB e)

Use one of the reasons listed for each of the missing reasons in the proof.

Given

Property of midpoint

Simplification

Substitution


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3. Use the adjacent diagram.

a) Give two other names for AM

.

b) Name a ray opposite to KA

.

c) Explain why KM

and KA

are not opposite

rays.

d) Find the intersection of MA

and KA

.

e) Find the intersection of KC

and MA

.

f) Explain why AS

and AN

are not opposite

rays.

g) Find the intersection of BC

and CQ

.

h) Find the intersection of FB

and BC

.

i) Find the intersection of AK and MC .

Section 3 Angles

1. Sketch and label the figures described.

a) The intersection of 1 and 2 is a ray.

b) The intersection of 1 and 2 is a point.

c) The intersection of 1 and 2 is three points.

d) The vertex of 1 is contained in the interior of 2 and the vertex of 2 is contained in the

interior of 1.

e) 1 is contained in the interior of 2.

Section 4 Measure of an Angle

1. Find x.

A

N

S

M

K

C Q

R

F B

x

p p

2p 2p


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2. Given m2 = 135°, and 1 and 2 form a linear pair. If 3 is the complement of 1, find

the measure of 3.

3. BAD and DAC are adjacent angles such that mBAD = 65° and mDAC = 32°. Find the

measure of BAC.

Section 5 Perpendicular and Parallel Lines

1. Justify each of the following.

a) m // n

b) CD

// AB

2. Use the adjacent diagram.

a) Find x, y, z and w.

b) Are any of the lines drawn parallel to one

another? Why?

Chapter Summary


m n

A C

20°

D B 20°

m

n

s t

y z 70° w 130°

130° 110° x


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Ch. 9 Plane Figures Section 1 Polygons

1. Give the name of the polygon and classify it as convex or concave.

a) b) c)

2. Find the measure of the sixth angle of a convex hexagon if the measure of each of the other

five angles is 130°.

Section 2 Triangles

1. ABC is an isosceles triangle.

a) Name the vertex angle.

b) Name the base.

c) Name the base angles.

d) Name the legs.

e) Name the side opposite to A.

B C

A

2. Consider a triangle with vertices O(0, 0), B(–4, 4), and C(4, 4). Verify the triangle inequality

by using the distance formula to find the lengths of the sides of OBC.

3. Consider the adjacent figure.

Given that: m1 = 43 and m3 = 35.

Find m4.

G

B

E

A

4 3

2

C

1

F

D


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4. Given is a median of MPQ, MQ = 4a + 6 and QX = a + 7. Find a.

Section 3 Congruent Triangles

1. Given AC and BD bisect each other.

AB DC .

a) Are the two triangles AMB and CMD

congruent? State the theorem or the postulate

that supports your answer.

b) Explain why 1 2.

A

B M D

C

1 2

2. In the adjacent diagrams ABC and DBC are

isosceles triangles with vertices A and D.

a) What is the relation between DBC and DCB?

b) Explain why 1 and 2 are congruent.

A

C B

D 2 1

Section 4 Similar Triangles


a) Find:

AM

AB and

AN

AC .

b) Is AMN congruent to ABC? Explain.

c) Is AMN similar to ABC? Explain.

B C

N M

8

6 3

4

A

PX

M

P Q

X


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Given ABM DEN,

1 2, and 3 4.

a) Is 1 congruent to 3? Explain.

b) Is BAC congruent to EDF?

Explain.

c) Explain why ABC DEF.

A

1 2

B M C

E N F

D

3 4

Section 5 Quadrilaterals


Find the values of x and y.

A B

C D

(3x y)

y 70

2. Show that quadrilateral ABCD with vertices A(1, 2), B(2, 5), C(5, 7) and D(4, 4) is a

parallelogram.

Section 6 Perimeter and Area

1. Find the nature of quadrilateral ABCD below and then calculate its perimeter and area.

0 4 2 2 4

4

2

2

4

x

y

A

B

C

D

Chapter Summary



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Ch. 10 Circles and Constructions Section 1 Circles

1. Write the equation of the circle with center at (2, 2) and passing through the point (4, 6). Is

K(8, 2) exterior, or interior, or on the circle?

2. Determine the value of x in each case.

a)

66

x

b)

x

35

Section 2 Locus

1. Draw a line s and a point B, 6 units away from s.

a) Draw the set of points that are at a distance of 4 units from s.

b) Draw the set of points that are 12 units away from B.

c) How many points belong to both sets?

2. What is the equation of the locus of points equidistant from the lines x = 4 and x = 10?

Section 3 Constructions

1. Given angle B. Construct an angle congruent to B.

B

2. Describe how you can construct a 30 angle.

3. Use a ruler to draw segments of lengths 4 cm and 6 cm. Construct a triangle with side lengths

4 cm, 4 cm, and 6 cm.

Chapter Summary



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Ch. 11 Transformations Section 1 Translations

1. In each case, pick the description that best describes the translation. If more than one

description applies, pick any one of those that do apply.

a)

b)

c)

d)

e)

f)

- A translation of 2 units down

- A translation of 2 units right

- A translation of 2 units down and 2 units to the right

- A translation of 2 units down and 2 units to the left

- A translation of 2 units up and 2 units to the right

- Not a translation

2. Find the image of P under the translation T: (x, y) (x + 3, y 4).

a) P(3, 4) b) P(3, 4)

3. Find the pre-image of Q under the translation S: (x, y) (x, y 7).

a) Q(2, 1) b) Q(3, 4)

4. T: P(x, y) P(x, y) = (x 3, y) and R: P(x, y) (x, y) = (x, y 9). Find the translation

S that maps P to P and give the image of (2, 1) under this translation.

Section 2 Reflections

1. Find the image of the point M under a reflection in the line y = x.

a) M(2, 3) b) M(–2, –4) c) M(–3, 1) d) M(2, –1)

2. Find the image of the point (–3, 7) under a reflection about

a) the origin b) the point (1, 4)


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3. ABC was reflected to DEF as shown below.

4 6

8

2 + a b 3

2c 1 A B

C

D E

F

Find the value of each of a, b, and c.

Section 3 Rotations

1. Find the image of each point under a rotation of 270 about the origin. Draw each point and its

image in a coordinate plane.

(3, 2) (5, 7) (–4, 2) (2, –5) (–3, –2)

2. Find the pre-image A of A(3, 2) under a rotation of 90 about the origin.

What rotation about the origin would produce A as the image of A?

3. The point (6, 9) is mapped onto the point (9, 6) by a rotation of 90°. Find the possible centers

of the rotation.

Section 4 Dilations

1. In the following diagram, the large rectangle is an image of the smaller rectangle under a

dilation. What is the value of x?

4 cm

9 cm

6 c

m

(3x

1.5

) cm

2. Find the image of each point below under a reduction with center at the origin and a scale

factor 1/2.

(4, 2) (8, 2) (0, 0)

3. Find the pre-image of each point below under an enlargement with center at the origin and a

scale factor 2.

(10, 2) (4, 1) (0, 4)


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4. Draw the image K L M of the right triangle KLM under a dilation with center at the origin

and a scale factor 1.5.

0 8 4 4 8

8

4

4

8

x

y

K

M

L

What is the nature of triangle K L M ?

Section 5 Compositions of Transformations

1. Consider the dilation D with center at the origin and a scale factor 2/3 and the reflection R in

the y-axis. Find the image of W(6, 0) under the composition of D and R by applying:

i- D first. ii- R first.

Is D R = R D?

2. Consider ABC with vertices A(2, 1), B(4, 1), and C(3, 3).

a) Draw ABC and its image XYZ under the translation (x, y) (x + 2, y 2).

b) Draw the image of XYZ under the reflection in the y-axis.

c) On a different coordinate grid, draw ABC and its image UVW under the reflection in the

y-axis.

d) Draw the image of UVW under the translation (x, y) (x + 2, y 2). Does this image

coincide with the image in part (b)?


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Section 6 Elements of Symmetry

1. Consider the set of points (0, 1), (0, 2), (1, 2), (1, 3), (1, 0), and (1, 2). Find the smallest

set of points that contains the given set and has

a) the y-axis as a line of symmetry.

b) the x-axis as a line of symmetry.

c) the origin as a point of symmetry.

2. Each figure below has a rotational symmetry. Give its degree of rotational symmetry.

a)

b)

c)

3. Show that (2, 1) is a point of symmetry for the set of points (4, 5), (0, 7), (2, 0), (3, 1),

(2, 1), (1, 1), (5, 7), (1, 9), and (2, 2).

Chapter Summary



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Ch. 12 Statistics Section 1 Qualitative and Quantitative Data

In questions 1 2, state if the data is quantitative or qualitative.

1. Students at a college were surveyed. Below is some of the information they were asked to

provide.

- Class (Freshmen, Sophomore, …)

- Number of years in college

- Major

- Minor

- Average number of credits per semester

- Favorite professor

2. At the end of the term, students of a college were asked to fill in a questionnaire about the

courses they took during that term. Some of the information requested was:

- Name

- Gender

- Number of courses they completed during the term

- Difficulty of the courses (Choose from 1 to 5: 1 is very easy, 5 is extremely challenging)

- Difficulty of the final exams (Choose from 1 to 5: 1 is very easy, 5 is extremely

challenging)

Section 2 Mean, Mode, Median, and Quartiles

1. In the set {3, 5, 8, 12, 16, 22}, the median is 10 and the mean is 11 while in the set {3, 5, 8,

12, 16, 40}, the median is 10 and the mean is 14.

Write a set consisting of 6 numbers that has

a) a mean of 15 and a median of 14.

b) a median of 8 and a mode of 18.

c) a mean of 12 and a mode of 5.


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2. Ten females and ten males were requested to answer the following question:

“How much are you willing to pay for a meal at the restaurant?”

The answers were as follows:

Sample 1

(males)

$15, $50, $35, $10, $20, $25, $25, $20,

$15, $25

Sample 2

(females)

$15, $40, $35, $40, $50, $45, $15, $20,

$15, $25

a) Find the mean, the median, and the mode for each sample.

b) Compare the two samples.

Section 3 Frequency Tables

1. Below are the playoffs statistics for the 2008 season for a basketball player.

Month Jan Feb Mar Apr October Nov Dec

Total number of minutes

played 659 281 583 118 75 517 602

a) What is the total number of minutes played during the season?

b) What is the relative frequency of the time played in March?

c) What is the cumulative frequency of the minutes played up until March?

2. The ages of 50 teenagers are listed below.

14 13 14 13 16 17 15 15 13 18

17 14 17 14 14 13 18 15 14 13

15 14 14 15 17 14 14 18 14 13

16 15 16 14 15 15 16 13 15 17

15 14 14 15 18 16 16 14 16 16

a) Arrange the data in a frequency table. Include the cumulative and relative frequencies.

b) Find the mean, median, range and interquartile range of the data.


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Section 4 Visual Representation of Data

1. A survey was conducted on 40 people about the number of cups of coffee they consume daily?

The result of the survey follows.

Number

of cups

Cumulativ

e

frequency

Frequency

0 7

1 13

2 23

3 30

4 35

5 38

6 40

a) Fill the last column in the table.

b) Find the five number summary of the number of cups consumed for the given sample.

c) Represent the data using a box plot.

d) Give one advantage and one disadvantage of representing the survey using a circle graph.

2. The temperature, in degrees Celsius, was recorded for 50 successive days. The result was

summarized in the table below.

Temperature Number of

days

[9 15) 11

[15 18) 12

[18 21) 8

[21 24) 7

[24 27) 7

[27 36] 5

a) Draw a histogram to represent the data.

b) What percentage of the days was the temperature at least 18C?

c) The data, as presented in the table, is not suitable to draw a line graph. Explain.

sabis mathematics im1 end of year revision packet ... an arithmetic sequence, find its common...

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