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EOY Revision Packet
Mathematics, IM2X
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Mathematics
IM2X
EOY Revision Packet
IM2: Ch.6 – Ch.12
Highlights:
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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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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.
Know the properties of the centroid of a triangle.
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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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Find the image of a point, a segment, and a line in the coordinate plane under a dilation.
Know and apply the AA postulate and the similarity theorems to prove that two triangles are
similar. Use the ratio of similarity in different instances.
Identify the image of the midpoint of a segment under a dilation.
Know the theorems of medians, altitudes, and angle bisectors in similar triangles.
Apply the proportionality theorem as well as its converse.
Apply the theorem of the angle bisector in a triangle.
Apply the Pythagorean Theorem and its converse and the metric relations in a right triangle.
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Mathematics, IM2X
Page 2 of 17
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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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Apply the fact that the bisector to a chord through the center of the circle bisects the chord.
Apply the fact that congruent chords in a circle are equidistant from the center.
Apply the fact that a line tangent to a circle is perpendicular to the radius containing the point of
tangency.
Apply the fact that tangent segments to a circle from an external point are congruent.
Apply the fact that the line through an external point to the center of a circle bisects the angle
formed by the tangents to the circle through this point.
Define an arc. Apply the arc addition postulate.
Apply the different theorems about measures of arcs, chords, and angles (central, inscribed).
Apply the different theorems related to angles formed by chords intersecting inside or secants
intersecting outside the circle, and the angles formed by chords and tangents.
Apply the relation between the measures of the segments formed by two intersecting chords, by
two secant segments intersecting at an exterior point, or by a secant and a tangent intersecting at
an exterior point.
Know the rules related to the sum of the measures of the interior angles or the exterior angles of
convex and regular polygons.
Recognize inscribed and circumscribed polygons and circles.
Recognize the circumcenter and the incenter of a triangle.
Know the relation between the interior angles of a cyclic quadrilateral.
Find the radii of the inscribed and circumscribed circles of a regular polygon.
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Define the radian measure of an angle and know how to convert between radians and degrees.
Find the length of a circular arc, given the angle and the radius.
Find the area of a circular sector.
Apply the rules for finding the volume of a cylinder, a prism, a pyramid, a cone and a sphere.
Use Cavalieri’s principle to explain why two volumes are equal.
Find the volume of a part of a cone and a cylinder with a spherical lid.
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Mathematics, IM2X
Page 3 of 17
Vocabulary:
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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, Centroid
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Parallelogram, Diagonals, Bisect, Rectangle
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Dilation, Image, Similar, Bisector of an angle, Proportionality theorem, Metric relations,
Pythagorean Theorem
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Sine, Cosine, Tangent, Secant, Cosecant, Cotangent, Pythagorean identity
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Line through the center of a circle, Chord, Equidistant from the center, Tangent, Concentric,
Common tangent, Tangent internally, Tangent externally
Arc, Central angle, Minor arc, Major arc, Semicircle, Arc addition postulate, Inscribed angle
Convex polygon, Regular polygon, Inscribed polygon, Circumscribed polygon, Inscribed triangle,
Circumscribed triangle, Cyclic quadrilateral, Inscribed circle, Circumscribed circle
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Radian measure, Circular sector, Circular segment
Cavalieri’s principle, Cone, Pyramid, Cylinder, Sphere, Prism
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Page 4 of 17
Revision Exercises
1. Given m1 = (2x  15) and m2 = (4x  9). Find x if
a) 1 and 2 are supplementary

b) 1 and 2are complementary

2. In ABC below, DE is parallel to BC . Find x.
A
(100  3x)
B
D
E
(70  2x)
C
3. 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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Mathematics, IM2X
Page 5 of 17
4. 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
5. 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.
6. 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
7. MNP is an isosceles triangle with mM = 110. Find mP.
8. Given a triangle SOU such that mS = 5(x + 1), mO = (5x + 6) and mU = (8x + 1). Find
mS.
9. Find x in the adjacent diagram
(x + 23)o
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10. 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
11. 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
12. Find x in each case.
a)
b)
A
x
A
8
M
6
B
12
x
N
M
6
3
C
13. 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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Mathematics, IM2X
14. Consider the adjacent figure, find x.
Page 7 of 17
A
B
x
12
48
O
16
D
C
15. 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
C
16. Consider the adjacent figure.
Given that G is the centroid of the ACB. Find x.
N
A
2+x
G
x
M
P
B
1
x4
17. In ABC, M is the midpoint of AC , and G is the centroid. Find x if AM = 2
and GM = 8.
18. Given ABC is scalene. MBC
is the midpoint of BC . BN and CP are two altitudes of the triangle.
a) Show that MN = MP = 2 .
b) Deduce that MNP is isosceles.
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19. Consider the adjacent figure.
Page 8 of 17
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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Reasons
F
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Mathematics, IM2X
Page 9 of 17
H
20. Consider the adjacent figure.
G
Given: EFGH is a parallelogram
2
Prove: m1 > m2
1
F
E
21. Sketch the image of each figure under the given dilation
a)
y
y
b)
B(1, 3)
x
O
M(2, 4)
N(2, 4)
Q(2, )
O
P(2, )
C(6, 3)
D(5, )
1
Center of dilation O, scale factor = 2
x
Center
of dilation O, scale factor =
1

2
22. In the statements below, answer as true or false. In the event when the claim is false, state why.
a) If two polygons are congruent, then the polygons are similar.
b) All regular pentagons are similar.
23. In the adjacent figure, the two quadrilaterals are
similar.
Find the values of x and y.
6
x
12
8
9
y
24. The sides of a parallelogram have lengths 12 cm and 15 cm. Find the lengths of the sides of a
similar parallelogram with perimeter 90 cm.
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25. In the adjacent diagram, ABC is similar to
DEF.
Find the scale factor and the length of BC and
DF.
26. Given ABC with vertices A(2, 2), B(1, 0), and
C(5, 2).
If the coordinates of each vertex are multiplied
by 2 to create DEF, show that DEF is similar
to ABC.
Page 10 of 17
y
D
B(1, 3)
F(8, 2)
C(3, 1)
x
A(1, 0)
O
E(5, 0)
27. Given ABC such that AB  AC . D and E belong to AB and AC , respectively, such that
DAE  ABC. Show that ABC is equiangular.
MN // BC
28. Given a triangle ABC. M and N belong to AB and AC , respectively,
, and MN is
AN AM such that
the angle bisector of ANB. Show that C  NBC and NB

MB
.
29. Given an isosceles triangle ABC with vertex A. AM , BN , and CP are the altitudes of the triangle
and H is the orthocenter. Show that:
a) BP  BA = BM  BC
b) MHB  MCA.
30. Refer to the adjacent figure to answer the question.
1
PM  CN
2
Explain why
.
A
31. Given a triangle ABC with right angle at A. AP is an
M
2
altitude, and PM and PN are angle bisectors in
N
triangles ABP and APC respectively. Show that:
a)
BP PA

PA PC
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b)
BM AN

MA NC
4
B
2
6
P
6
C
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Mathematics, IM2X
32. Find x in each of the following cases.
a)
A
x+9
Page 11 of 17
b)
M
2
4x  3
H
8
B
5
C
y
7
N
3
Q
x
C
F
E
33. Refer to the diagram below to answer the question.
A
D
E
I
Show that
IF
IE

1
BC AD
.
B
F
C
34. Given a triangle ABC with right angle at B. H belongs to AC such that BH  AC and AB  CH .
Show that:
a) HC2 = AH  AC
b) BC2 = AB  AC.
35. The altitude to the hypotenuse divides the hypotenuse in two segments whose lengths are in the
ratio 1 to 2. The length of the altitude is 32. Find the length of the hypotenuse.
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36. Find the values of x and y in the figure below.
A
45
10
C
60 B
60
a
x
D
E
37. 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.
38. Given x is an acute angle, simplify the expression cos (90  x) tan (90  x).
39. Show that
tan 
sec   1

sec   1
tan 
is true for all values of  for which it is defined.
40. 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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41. Supply the missing reasons in the proof below.
Page 13 of 17
A
R
P
In the adjacent figure, AM // BN and O is the midpoint of
RS . Prove that AM  BN .
O
B
Q
Proof
Statements
a)
b)
AM // BN , O is the midpoint of RS
Reasons
a)
b)
c)
Draw the perpendicular through O to
AM intersecting AM in P and BN in
Q
PQ
is perpendicular to BN
d)
OR  OS
d)
e)
QSO  PRO
e)
f)
QSO  PRO
f)
g)
OP  OQ
g)
h)
AM  BN
h)
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c)
M
S
N
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Mathematics, IM2X
42. Supply the missing reasons for the
proof below.
Consider the adjacent figure.
Given that the two circles are tangent
externally at M. AM , AN , and AP
are tangent segments to the circles as
shown. Prove that
i- AN  AP .
ii- O, O, and M are collinear.
Page 14 of 17
A
P
N
M
O
O
Proof
Statements
i- a)
Reasons
AN and AM are tangents to
a)
O
b)
AN  AM
b)
c)
AM and AP are tangents to
c)
O
d)
AM  AP
d)
e)
AN  AP
e)
OM  AM and OM  AM
f)
O, O, and M are collinear
g)
ii- f)
g)
43. In each case below, O is the center of the circle. Find x.
a)
150
b)
O x
110
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150
c)
x
O
250
O x
2x
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Page 15 of 17
100
44. In the adjacent figure, AB // DC and
AC is a diameter of the circle. PQ is
C
D
P
x
tangent to the circle at A.
a) Explain why AB  DC .
b) Deduce the value of each of x, y,
and z.
O
y
A
B
z
Q




45. Given PA and PB are tangents to O.

A
P

AC is a diameter. Prove that BC // OP .
OO
B
C
46. Find the value of x.
a)
b)
2x
30
x
5
x
3
2
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R
47. Given two concentric circles with center O.
M
X
RS is tangent to the inner circle at M.
S
O
RX = 5 and RS = 20.
Find XY.
Y
T
48. Find the value of x.
a)
x
b)
150
c)
90
92
140
100
115
x
100
35
140
x
x
110
115
49. Find the number of sides of a regular polygon if the measure of one interior angle is 108.
50. The measures of the interior angles of a convex quadrilateral ABCD are (4x + 25), (2x),
(3x  20) and (3x  5)o. Find x.
51. What is the value of x in the diagram to the right?
x
10
12
9
52. In the adjacent diagram, ABCD is an inscribed
quadrilateral with its diagonals perpendicular to each
other and BAD is right.
a) Show that ADB  BAC and deduce that ABC is
isosceles.
B
A
H
b) Show that ADC is isosceles.
D
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53. An arc of a circle subtends an angle of 0.7 rad at the center. Find the radius of the circle if the
length of the arc is 5 in.
54. In a circle with center O and radius 2 cm, a central angle whose measure is 1.2 rad subtends AB .
Find, to two decimal places, the area and the perimeter of sector OAB.
55. A container is in the shape of a triangular prism
whose bases are right triangles with sides 6 in., 8 in.,
and 10 in. Standing upright on the triangular bases
the height of the water level in the container reached
15 in. which is half the height of the container. The
position of the container is changed so that it lays on
a horizontal plane on one of its sides, as shown in
the adjacent figure. Estimate, to the nearest tenth of
an inch, the height of water when the container is in
this position.
h
5m
57. Refer to the adjacent diagram to answer the
question.
A structure has the shape of a cylinder with a
conical roof.
a) Calculate the volume of the structure.
b) The exterior of the structure is to be entirely
painted. The cost of 1 m2 of paint is $1.25.
Find the total cost of painting the structure.
9m
56. The radius of a spherical balloon was reduced to
its third due to deflation. What was the percent
decrease in the volume of the balloon?
8m
58. If the area of the base of a cone is doubled and the height is halved, verify that the volume of the
cone remains the same.
59. The adjacent diagram shows a pyramid with a
rectangular base ABCD. The diagonals of ABCD
intersect at O. VO is the altitude.
AB = 12 cm, BC = 6 cm, VO = 3 5 , X is the
midpoint of AB , and Y is the midpoint of BC .
Find
a) the angle of elevation of V from A,
b) VX and VY,
c) the volume of the pyramid, and
d) the lateral surface area of the pyramid
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V
D
C
O
A
X
Y
B