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EOY Revision Packet
Mathematics, IM2
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Mathematics
IM2
EOY Revision Packet
IM2: Ch.1 – Ch.5( S1 – S4), Ch.6 – Ch.7(S1 – S3), Ch8, and Ch.10
Highlights:
 Solve problems on square roots, cube roots, and nth roots.
 Recognize rational and irrational numbers and express a repeating decimal as a fraction.
 Add, subtract, multiply, and divide real numbers.
 Round irrational numbers.
 Identify radicals in simplest form. Evaluate nth roots and simplify expressions with nth roots.
 Rationalize the denominator in rational expressions.
 Apply the properties of integer exponents and express fractional exponents in radical form.
 Apply the product and the quotient rule of exponentiation.
 Simplify exponential and radical expressions.
 Solve radical equations.
 Evaluate radical expressions given in contextual situations.
 Solve for missing variables or for one variable in terms of the others in radical expressions.
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Expand, simplify, and evaluate algebraic expressions.
Recognize monomials and polynomials and write a polynomial in standard form.
Identify the degree of a polynomial and the number of terms.
Operate on polynomials. Use special products to simplify binomial expressions.
Rationalize a radical binomial denominator.
Factor the GCF and use identities to factor binomials and polynomials.
Factor polynomials by grouping.
Model a situation with polynomial functions.
Recognize non-linear functions.
Apply the square root principle and the zero product principle.
Solve by completing the square or using the quadratic formula. Work with the sum and product of
the roots of a quadratic equation.
Understand the use of complex numbers and know when two complex numbers are equal.
Apply operations on complex numbers.
Factor and solve a quadratic equation with complex roots over the set of complex numbers.
Graph quadratic functions and know the special properties of a parabola.
Solve quadratic inequalities and applications on quadratic functions.
Graph absolute value functions and apply their special properties.
Graph square root, cube root, reciprocal, and piecewise defined functions.
Solve inequalities graphically.
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Mathematics, IM2
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Define odd and even functions and know their properties and the properties of their graphs.
Calculate the rate of change. Estimate the rate of change from a graph.
Know the sense of variation from the average rate.
Calculate and interpret the average rate from the graph of a function.
Derive the equation of a linear model from a table of values.
Derive the equation of a piecewise linear model from a graph.
Derive the equation of a piecewise cumulative linear function given its description.
Sketch the graph of the absolute value of a function given the graph of the function.
Derive the equation of an absolute value function given the graph.
Work with step functions.
Derive the equation of a quadratic model and an exponential growth from a table of values.
Define percent rate and apply to growth and decay.
Apply arithmetic operations on functions and create new functions.
Define the composite of two functions and use it to build new functions.
Find the inverse of special functions.
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Identify an event, the sample space, and the outcome of an experiment.
Construct a tree diagram for an experiment and a two-way table for a two-stage experiment.
Recognize dependent and independent events.
Apply the fundamental principle of counting.
Use factorials to count the number of arrangements.
Apply the rules for permutations and combinations and identify the difference between ordered
and unordered selections.
Calculate probabilities using the fundamental principle of counting, combinations, permutations,
and factorials.
Understand and calculate experimental probability. Compare it to theoretical probability.
Calculate the theoretical and experimental probability of a complementary event.
Define mutually exclusive events.
Apply the product rule for the probability of independent or dependent events.
Use probability trees in the cases of independent or dependent events.
Calculate the odds of an event occurring.
Work with Venn diagrams.
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Apply the angle addition postulate.
Apply the fact that angles in a linear pair are supplementary and the fact that supplements of
congruent angles are congruent.
Apply the fact that complements of congruent angles are congruent and the fact that vertical
angles are congruent.
Apply the alternate interior angles theorems and the theorems on interior angles on the same side
of a transversal.
Find the locus of the points equidistant from the endpoints of a segment.
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Mathematics, IM2
Page 3 of 17
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Apply the angle sum theorem and the exterior angle theorem.
Apply the ASA postulate, SSS, HL, AAS, and SAS theorems to prove that two triangles are
congruent.
Apply the isosceles triangle theorem and its converse.
Know that the median through the vertex of an isosceles triangle is also an altitude and an angle
bisector.
Apply the mid-segment theorem.
Apply the properties of the mid-segment of a trapezoid.
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Apply the fact that opposite angles and sides of a parallelogram are congruent.
Apply the theorem that states that diagonals of a parallelogram bisect each other.
Recognize the cases when a quadrilateral is a parallelogram.
Recognize the cases when a parallelogram is a rectangle.
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Know the sine, the cosine, and the tangent of an acute angle in a right triangle.
Know the cosecant, the secant, and the cotangent of an acute angle in a right triangle.
Find the trigonometric ratios of an acute angle in a right triangle.
Find the leg of a right isosceles triangle given its hypotenuse.
Given the length of one of the sides in a semi-equilateral triangle, find the other two.
Know the trigonometric ratios of a 30° angle, a 45° angle, and a 60° angle.
Know the relations between the six trigonometric ratios.
Given any trigonometric ratios of an angle, find the other ratios.
Know the Pythagorean identity.
Apply the relationships of the trigonometric ratios of complementary angles.
Use trigonometric ratios to find the missing angles of a right triangle.
Use a calculator to find an angle given its trigonometric ratios.
Apply the knowledge of trigonometric ratios in real life situations.
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Mathematics, IM2
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Vocabulary:
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Rational, Integer, Decimal representation, Irrational, Square root, Principal square root, Cube
root, nth root, Simplest form, Index, Exponent, Radicand, Exponential form, Fractional form,
Radical expression, Radical equation
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Term, Algebraic, Constant, Coefficient, Like terms, Algebraic expression, Monomial, Binomial,
Polynomial, Degree, Standard form, Leading coefficient, Radical binomial, Conjugate, GCMF,
Factor
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Quadratic equation, Square root principle, Zero product principle, Quadratic formula,
Discriminant, Root, Sum, Product, Complex number, Real part, Imaginary part, Commutative,
Associative, Distributive, Quadratic inequality, Quadratic function, Domain, Parabola, Lowest
point, Highest point, Vertex, Upward, Downward, Wider, Narrower, Axis of symmetry, Absolute
value function, Corner, Reciprocal, Piecewise
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Even function, Odd function, Symmetrical, Vertical shifts, Average rate, Secant, Increasing,
Decreasing, Linear, Slope, Non-linear, Step function, Exponential function, Percent rate of
increase, Percent rate of decrease, Sum, Difference, Product, Quotient, Composed functions,
Inverse functions
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Outcome, Sample space, Event, Dependent event, Independent event, Fundamental principle of
counting, Factorial, Arrangement, Permutation, Combination, Ordered and unordered selections,
Certain, Impossible, Theoretical probability, Experimental probability, Complementary events,
Mutually exclusive events, Probability trees, Venn diagram, Odds
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Angles in a linear pair, Vertical angles, Supplements, Complements, Parallel lines postulate,
Corresponding angles postulate, Alternate interior angles, Perpendicular bisector
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Angle sum theorem, Exterior angle theorem, Isosceles triangle theorem, Median through the
vertex, Mid-segment theorem
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Parallelogram, Diagonals, Bisect, Rectangle
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Sine, Cosine, Tangent, Secant, Cosecant, Cotangent, Pythagorean identity
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Mathematics, IM2
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Revision Exercises
1. Evaluate each of the following roots, if it exists as a real number.
6
a) 729
5
d)  32
5
c) 32
4
b) 256
2. Without actually computing the value of the expression, decide whether it has a rational value.
Justify your answer in each case.
a) 5  2
b)
4
3
1 3
 64
c) 2
d)
6
12 36
c) 64a b
d)
1

7
3. Simplify each of the following radicals.
3
a)
3
e)
27x6 y12
x 4 y11
216
4
b)
256x 4 y16
18 x 7 y 4
64
f)
3
g)
x3 y 6
8
5
a)
1
1
b)
5
2
x ,x≠0
c)
6
5
x ,x≠0
32x10 y35
4
48 x5 y 6
4
h)
4. Rationalize the denominator in each of the following expressions.
2
5
3
a)
b)
10 3 5  2 3 625
1
6. Convert
to exponential form.
, y0
1
5
x
a)
, x0
 y
b)
4
7. Give the
value of each expression.
1/2
3/2
4
 
a)  9 
 16 
 
b)  25 
c)
4 3 16  33 54
3
d) 12
3
d)
1 3
 4
2
2
 1
3  , y0
y
c)  
 16 
 
c)  25 
3/2
d)
x
, y0
y
 1,000 


d)  27 
8. Express each of the following in the simplest radical form.
3
a)
3
6
x x
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b)
x4
 x
4
,x≠0
3
5. Simplify each of the following radical expressions.
43 5  3 5
16 x8
2
c)
x
3 3
x
d)
a3
1/3
 a
3
4
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9. Whenever
defined, express
3 each of the following expressions in the simplest radical form.
4
x
a)
x
 3
6
3
b)
3
c)
3
25
5
d)
6
x
x
10. Simplify each of the following.
 3 
a)
4
1
4
 3 
b)
2
3 3
3/2
1/2
d) y  y 
1/ 2
1/ 2
c) x  x 
3
1/3
11. Simplify.
1 Assume all expressions are defined.

3 y


x
a)  
 4x x 
3
b)
1
32 x 4 y 2
3
1
3
c)
2 x2 y6
d)
 2 2
3 
 x
12. Solve.
3
a) 5x  7  2  5
3
b) 3x  8  5
c)
3
x 5  4
t /t
13. The radioactive element Carbon-14 decreases at a rate that is proportional to the amount
1  h present.
This is an example of exponential decay which is given by the formula:
P(t )  Po  
2
where th is
the half-life, i.e. the time it takes an amount of C-14 to decay to half its size, and t is the time
required for the amount to decrease from Po to P(t).
Carbon14 is measured in units called dpm (disintegrations per minute). If the half-life of
C-14 is 6500 years, what will 15.3dpm of Carbon14 become after 15,000 years?
14. Simplify.
a) (2b2  13b)  (b3  3b2 +8b)
b) (cd2 + 2c3 + 12c2d  d3)  (12dc 2  2 d3  8d2c)
c) (6ab  2 b2 + 3a2) + (3a2 + 4 b2  9ab)
d) (2a2b2 + 13a2  7b2) + (4a2 + 12 b2 + 8a2b2)
15. Multiply and write your answer in standard form.
a) (x  3)(2x + 5)
1 
3

 4 x   x  
3
4

b) 
 x 1  x 4 
    
c)  5 2  2 5 
b) (x2 + y2)2
c) (2e + 3s)(2e + 3s)
16. Expand and simplify.
a) (6s  6t)2
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17. Rationalize the denominator of each expression.
54
1
a)
2 4
b)
54
2 5
2 5
c)
18. Factor each of the following.
a) a3 + 27
b) a6  64
c) (a + b)3 + (a  b)3
d) 3x3 + 6x2 + 3x
e) 20y2  20y3 + 5y4
f) 27x2y  36xy + 12y
a) x 2  59x  60
b) x 2  28x  60
c) x 2  17x  60
d) 9f 2  23f + 10
e) 64g 2  48g + 9
f) 4a 2 + 26a + 30
19. Factor each polynomial.
20. While playing catch with his son, Tim throws the ball as hard as possible into the air. The height
h, in feet, of the ball is given by h  16t2  64t  8, where t is in seconds.
a) Describe the height function.
b) Express the height function as the product of its factors.
c) Find the height at which the son catches the ball, given that it took 3 seconds for the ball to
reach the son’s glove.
21. Solve for x.
a)
216   24 x 2
5
d) x2 + 4x  21 = 0
1 (5 x  2)2  3
b) (3x + 2)2 = 4
c) 3
e) x2 + 71 = 18x  9
f) 5x2 + 15x + 90 = 0
22. Solve by completing the square.
a) x2  2x  8 = 0
b) 3x2 + 7x 4 = 0
c) 2x2 + 7x + 5 = 0
23. The sum S and the product P of the roots of a quadratic equation are given. Find the equation and
determine the roots.
a) S = 4, P = 5
b)
S 
1
3
P
2,
2
S
c)
3
5
P
8,
8
24. Express the given function in the form y = a(x  h)2 + k. Identify h and k.
a) y = 8x2  11x  2
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b) y = 6x2 + 11x  12
c) y = 6x2 + 8x 3
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25. Identify the vertex, the axis of symmetry, the extreme value of the given function, then
graph it.
a) y = 3x2  2x  5
b) y = 5x2 + 10x + 11
c) y = 12x2  9x  7
26. Write an equation of the parabola that has the following characteristics and graph it.
a) Minimum at (1, –3), congruent to the graph y = (x + 1)2.
2
b) Axis of symmetry x = 5, congruent to the graph y = 3x , passes through
1
(2, )
3
27. Solve the given quadratic inequality.
a) 2x2 11x  6 > 0
b) 6x2 + 5x + 11 < 0
c) 4x2 + 3x  7 < 0
28. The height h(t) in feet of a ball at time t, is determined by the equation h(t) = 16t2 + 96t + 4.
Over what interval of time is the height of the ball more than 52 feet?
29. A ball is thrown into the air from the top of a 250 feet tall building with an initial upward velocity
of 96 ft/sec.
a) What is the maximum height the ball reaches?
b) How long does it take the ball to reach the maximum height?
c) When will the ball return to the ground?
d) If the ball lands in a 300 feet deep pit, when will it hit the bottom of the pit?
30. A computer store sells 15 printers per week when it is priced at $300. Using a survey, they find
that for every $8 decrease in the price, two additional sales will be made. What price would
produce the highest revenue for the company? What is the maximum revenue?
31. It is known that rainstorms begin with light rain, develop to heavy rain, and then drop back to
light rain. The rate r at which it rains is given by the function r = 2|t  3| + 6 where t is the time
(in hours), and r is measured in inches per hour.
a) Sketch the graph.
b) When does the rain get the heaviest?
c) How long does it take for the rain to stop completely?
32. On the same set of axes, draw the graphs of each given pair of functions.
a) y  5  x , y  x  5
y
c)
1
x6
y
,
3
x6
33. Sketch the graph of the function
g(5), and g(7).
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3
3
b) y  x  4  2 , y  x  2  4
6
g ( x)  
8  x
if x  2
if 2  x  6
. Give the value of each of g(1), g(2),
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Mathematics, IM2
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34. A long-distance calling plan charges 80 cents for any call up to 15 minutes in length, and 10 cents
a minute for each additional minute. A fraction of a minute is considered 1 minute. Write a
formula for the cost C(t) of a call in cents as a function of its length in minutes, and sketch the
graph C(t).
35. Graph the two functions y  2x2  12x  14, and y  2x2  8x + 7.
Deduce the solution of 3x2  7x  12  x2  13x + 9.
36. If R(– 1, – 2) and S (1, b) lie on the graph of an odd function, find b.
37. Given that f(– 4) = 5, find f(4) knowing that f is an even function.
38. The following table shows the global unit sales of a company producing games’ home consoles.
1997 2000 2003 2006 2009 2012
Year
Number of units
5.5
6.5
5.5
2.5
26
10
(millions)
a) Calculate the average rate of change of the number of units sold per year over each of
the 3-year periods given in the table.
b) Which period witnessed the highest average rate of increase in the sales?
c) Interpret the variation in the average rates of change over the given periods.
d) What are the expectations for the average rate of change of the number of units sold per year in
the future?
39. A car rental company A charges a flat-fee of $30 per day to rent a new economy car, plus a
mileage charge of $0.20 per mile..
a) Construct a table of values for the total cost of renting a car for one day and driving 10, 20, 30,
40, and 50 miles.
b) Sketch the graph of the cost C1 against of the number of miles driven x.
c) What do the y-intercept and the gradient of the graph represent?
d) Find a symbolic representation for this function.
Another car rental company B charges a flat fee of $20 per day to rent an economy car plus a
mileage charge of $0.30 per mile.
e) Find a symbolic representation for the cost function C2 for renting from company B in terms of
the number of miles driven x. .
f) Use the graph to estimate the number of miles that produce the same cost for renting a car from
either companies.
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40. A motorist drives his car at an average speed of 40 miles per hour (mph) for 2 hours, then at 60
mph for the following 3 hours and finally at an average speed of 50 mph for a whole hour to reach
his destination.
a) Draw a graph for the distance traveled after t hours.
b) Derive a symbolic representation for the distance function.
c) Use the proper equation to find the distance traveled after 2.5 hours.
d) How long did it take the motorist to travel a distance of 100 miles?
41. The scatterplot in the following diagram is used to track the performance of a company share
during a previous year. The prices shown are the share prices at the beginning of the
corresponding month.
Share Price ($)
40
p
35
30
25
20
t
Jan
Apr
Jul
Month
Oct
Jan
a) Find the monthly average rate of change of the price of the share over each quarter.
b) Assuming that the average rate of change of the price of the share is constant over each quarter,
express the price of the share as a piecewise linear function of the time in months.
c) Use the function defined to calculate the prices of the share at the beginnings of March, June, and
November.
42. A parking garage charges $2 for the first hour and $1 for each hour thereafter up to a maximum of
6 hours.
a) Represent this information using a step function.
b) Sketch the graph of the function defined.
43. The water level in a lake in a certain month of next year can be estimated from the equation
y = 2t2 – 25t + 100, where t is the month of the year, and y is the depth of the water in feet.
a) What is the y-intercept and what does it represent?
b) What are the coordinates of the vertex and what do they represent?
c) During which month is the water level expected to reach its minimum value?
d) A state of emergency is to be announced if the water level drops below 50 feet. With the help
of the calculator, identify the months expected to be listed under this state of emergency.
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
x

x
44. Use the graphing calculator to sketch the graphs of f ( x)  1.25 and g ( x)  0.8 on the same
screen.
a) Is there any symmetry between the two graphs? How can this be justified?
b) Which function defines an exponential growth and which defines an exponential decay?
c) Compare the end behaviors and the intercepts of the two graphs.
d) Compare between the domains and the ranges of both functions.
45. Due to the inflation rate, food prices increase by an average of 9% each year. Consider an item of
food that sells for $10 per unit.
a) Write an equation to calculate the cost of the item after t years.
b) Use a calculator to sketch the graph of the equation.
c) After how many years will the price of the item exceed $15?
d) What will the price of the item be after 10 years?
46. The following table shows the height of a plant versus its age in weeks.
Age in weeks
Height in inches
2
6
4
12
6
24
8
48
10
96
a) Describe the relationship between the height of the plant and its age.
b) Draw a scatter plot for the data given in the table.
c) Write a function to express the height of the plant in terms of its age.
d) If the growth of the plant continues in the same way, what is the expected height of the plant
after 12 weeks?
47. Show that
2 x are
3 inverses of each other.
x the
3 given functions
f (x) = x  2 , x ≠ 2, g(x) = x  1 , x ≠ 1
48. From a class of 20 students, the teacher chooses 4 students to answer 4 different questions. In how
many ways can the teacher assign the questions?
49. The shelves of a library comprise 5 Math books, 4 Science books, and one History book. Two
books are drawn randomly from the shelves.
a) What is the probability that one of the books is a Math book and the other is a Science book?
b) What is the probability that both books have the same subject?
c) What is the probability that the two books have different subjects?
50. Beth has 3 red shirts and 4 yellow shirts. She selects two shirts randomly to pack for a trip. Find
the probability of each of the following selections.
a) Both shirts are red.
b) Both shirts are yellow.
c) One shirt is red and the other is yellow.
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51. The probability that Janna arrives late to work is 0.9 and the probability that her assistant Kathy
arrives late is 0.95.
a) What is the probability that both arrive late to work one day if they commute separately?
b) What is the probability that Janna arrives late but Kathy is on time?
52.The odds of winning a game are 3:1. What is the probability of winning 3 games in a row?
53. If A and B are mutually exclusive events such that P(A) = 0.6, and P(B) = 0.2. Calculate each of
the following probabilities.
a) P(A or B)
b) P(A and B)
54. If A and B are two events such that P(A) = 0.6 and P(B) = 0.4, find P(A or B) in each of the
following cases.
a) A and B are mutually exclusive.
b) A and B are independent.
c) A and B are dependent and P(A and B) = 0.2.
55. The results of a survey among the residents of a neighborhood showed that 90% of them dim their
lights occasionally to save energy and that 60% dim their lights and buy organic products. A
resident is chosen randomly from this neighborhood.
a) What is the probability that the resident dims his light but does not buy organic food?
b) What is the probability that the resident buys organic products but does not dim the light?
56. Daren has an 80% chance of traveling to Europe next summer. Carol has only 40% chance to
travel to Europe next summer.
a) What is the probability that both Carol and Daren travel to Europe next summer?
b) What is the probability that Carol or Daren travels to Europe next summer?
c) What is the probability that Carol but not Daren travels to Europe next summer?
57. The members of a neighborhood association are 18 males and 12 females. If 10 of the males and 6
of the females ride public transportation , what is the probability that a member selected randomly
from the council is a female or rides public transportation?
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58. Given m1 = (2x  15) and m2 = (4x  9). Find x if
a) 1 and 2 are supplementary
b) 1 and 2are complementary


59. In ABC below, DE is parallel to BC . Find x.
A
(100  3x)
B
D
E
(70  2x)
C
60. Refer to the diagram below to answer the question.
A
B 30
85
C
M
E


Prove that CD // EF .
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D
115
N 
G
F
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61. Find the length AM in each case. Justify your answer.
a)
b)
A
A
2x  2
x+4
25
25
B
6
M
6
C
B
5x  1
M
4x + 4
C
62. Consider the two points in the coordinate plane P(1, 2) and Q(3, 0).
a) Find the coordinates of the midpoint of PQ .
b) Find the slope of PQ .
c) Find the slope of the line perpendicular to PQ .
d) Find the equation of the perpendicular bisector of PQ .
e) Find a point on the perpendicular bisector of PQ .
f) Show that the point found in e) is equidistant from P and Q.
63. Find the value of x in each figure below.
a)
b)
x
B
A
E
128
110
160
C D
x
(x + 8)
F
G
64. MNP is an isosceles triangle with mM = 110. Find mP.
65. Given a triangle SOU such that mS = 5(x + 1), mO = (5x + 6) and mU = (8x + 1). Find
mS.
66. Find x in the adjacent diagram
(x + 23)o
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67. For each of the figures below, state if the indicated triangles are congruent. If so, state why.
a)
b)
A
M
30
A
N
P
N
M
C
B
30
P
B
ABC and PMN
ANP and MNB
A
68. Given: ABC is an isosceles triangle with base BC ,
MB = NC, and AS is a median.
M
Prove: AS  MN
N
O
B
C
S
69. Find x in each case.
a)
b)
A
x
A
8
M
6
B
12
x
N
M
6
3
C
70. Consider the adjacent figure.
N
9
B
B
C
C
4x + 2
Given that MN is the mid-segment of the
trapezoid ADCB. Find MN.
3x + 6
M
N
18
A
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A
71. Consider the adjacent figure, find x.
B
x
12
48
O
16
D
C
72. Given a parallelogram ABCD. M, N, and P are the midpoints of AD, AC and BC respectively.
Show that:
a) M, N, and P are collinear
b) N is the midpoint of MP
73. Consider the adjacent figure.
D
C
Given: ABCD is a parallelogram, DE  AB ,
CF  AF
Prove: DE  CF
A
E
B
Supply the missing reasons.
a)
Statements
ABCD is a parallelogram
a)
b)
AD  BC
b)
c)
DC // AB
c)
d)
DAE  CBF
d)
e)
DE  AB and CF  AF
e)
f)
DEA and CFB are right angles
f)
g)
DEA  CFB
g)
h)
DEA  CFB
h)
i)
DE  CF
i)
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H
74. Consider the adjacent figure.
G
Given: EFGH is a parallelogram
2
Prove: m1 > m2
1
F
E
75. Given x is an acute angle such that tan x =
16
63
and sec x =
65
16 .
Find the other four trigonometric
ratios of x.
76. Given x is an acute angle, simplify the expression cos (90  x) tan (90  x).
77. Show that
tan 
sec   1

sec   1
tan 
is true for all values of  for which it is defined.
78. A salvage ship’s sonar locates wreckage at a 15° angle of depression. A diver is lowered 50
meters to the ocean floor. How far does the diver need to walk along the ocean floor to the
wreckage?
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