Geometry 1 FINAL REVIEW 2011
1) Always, Sometimes, or Never.
If you answer sometimes, give an example for when it is true and an example for when it is not true.
a) A rhombus is a square.
b) A square is a parallelogram.
c) Both pairs of opposite sides of a trapezoid are parallel.
d) The diagonals bisect the angles of a parallelogram.
e) A rhombus is a rectangle.
f) The diagonals of a kite bisect each other.
g) The legs of a trapezoid are congruent.
h) If FGIJ is an isosceles trapezoid, then < F is supplementary to < I.
i) If the diagonals of a parallelogram are congruent, it is a square.
j) The diagonals of a parallelogram are perpendicular.
k) If one angle of a parallelogram is right, the parallelogram is a rectangle.
l) The diagonals of a trapezoid bisect each other.
2) ABCD is a parallelogram. AD =
x2
6
+
1
8
and BC =
x
3
. Solve for x and the length of BC
3) In parallelogram MATH, if <M = 145°, then <A=
4) Quadrilateral ABCD has coordinates A (1,4), B (4,9), C (–1,12), D (–4,7). What is the most descriptive name
for ABCD? Justify your response with words and math!!!
5) In parallelogram ABCD, <A is three times as large as <D. Find <C.
6) If two of the angles of a parallelogram are in the ratio 2:1, what are the four angles?
page 1
7) Given: Parallelogram ABCD
A
B
AB = 10x − y
BC = 8x + 2y + 14
DC = 7x + 4y + 23
E
AD = 3x − 2y + 3
∠ADE = (10x − 12y − 10)o
D
C
a) Solve for x and y.
b) What are the lengths of AB, BC, CD, AD ?
c) What is the measure of ∠EDC ?
d) What is the most descriptive name for parallelogram ABCD?
8) Give the most descriptive name that you can for each quadrilateral:
9) a. Graph the vertices of quadrilateral ABCD: A(0,–6), B(–4,2), C(4,6) and D(8,–2).
b. Find the slopes of AB, BC, CD, and DA.
c. What is the most descriptive name for quad ABCD?
d. Find the slopes of AC and BD.
e. Now, what is the most descriptive name for quad ABCD?
f. (At this point, you should have enough information to know that quad ABCD is a square.) Verify that all
sides of square ABCD are equal by finding the lengths of AB, BC, CD, and DA.
page 2
10)
ABCD is a parallelogram
o
< BAD = 3x + 5
D
A
( )
o
< DCB = (2x + 35)
E
B
Find the measure of < ADC.
C
11)
ABCD is a parallelogram
D
A
AE = 4x − y
DE = 2x + y
E
BE = 6
EC = 6
Solve for x and y.
B
C
12) QUAD is a parallelogram. Give the most descriptive name for QUAD.
UF = 2x–y, FD = 18, QF = x+y, FA = 6+2y, ∠QFD = (10x)o
U
A
F
(10x)°
Q
D
13)
ABCD is a parallelogram
AE = 4
D
A
EC = x + y
BE = 2y
E
ED = 3x − 7
Solve for x and y.
B
C
14) Solve for x:
6
16
5
7
8
15)
x
23
14
Given: ΔGHI ~ ΔXYZ
m∠G = 3x + 4,
m∠X = 2y + 3,
m∠Y = 4y + 2x + 2,
m∠I = 2y + x – 9
Find:
page 3
m∠Y
16) Name the most specific quadrilateral that you can based on the marks on the diagrams.
91°
91°
D
A
A
D
Z
B
C
B
Y
C
W
Z
W
X
Y
X
17) Fill in the blanks:
a) The definition of an isosceles triangle specifies that two _______be congruent.
b) A quadrilateral with exactly one pair of parallel sides is a _________.
c) The diagonals of a quadrilateral are congruent and they are perpendicular bisectors of each other. The
quadrilateral can best be described as a _________.
d) The vertex angle of an isosceles triangle is
50 o. Each base angle is ____ degrees.
e) A parallelogram with congruent diagonals must be a __________.
f) A quadrilateral with four congruent sides may be a _______, but must be a ______.
g) The diagonals of a quadrilateral are not congruent, but they bisect each other. The quadrilateral must be a
________, but it cannot be a __________.
18) What value of x makes ∆ABC similar to ∆EDC?
A
E
5
x
B
page 4
12
C
11
D
19) Solve for x:
20)
AB || CD
Name the triangles that are similar and solve for x and y:
A
9
C
x
4
y
4
E
7
3
5
6
x
B
D
21) The sides of a triangle are 5, 9, and 10. A similar triangle has a perimeter of 60. Find each side of the second
triangle.
22) Solve for x:
23) Given:
AB = x+4
BC = x
DE = x+2
EF = 3
A
E
10
x–1
C
24) Solve for x:
F
E
C
B
D
b) Find the length of AB.
5
D
c) What are the restrictions on AC ?
25) Find the
45°
26
15
A
a) Solve for x.
x–1
B
ΔABC ~ ΔDEF
tan∠A .
34
30
A
x
26) sin∠A =
5
9
. Solve for x.
27) Find the lengths of
P
x
4
PM and MT .
5 3
S
6 3
A
45°
M
page 5
60°
T
28) sin < C =
5
B
6
18
a) Find the length of AB .
b) Find the length of AC .
c) Find cos <C.
d) Find tan <C.
A
C
29) Evaluate each of the following... your answer must include a justification.
a)
cos60 o
b)
tan60 o
c)
sin 45 o
30) If A = an acute angle and tanA = 1, then the measure of A is________?
31) Rhombus ABCD has AB=12 and m<A=60, find AC and BD.
32) Find the altitude of an isosceles triangle whose base is 14 and whose perimeter is 50.
33) Solve for x for the following trapezoid:
x
120°
135°
9
34) The radius of a circle inscribed in a square with a perimeter of 20 feet is:
35) The radii of two concentric circles are 3 and 21. Find the length of the chord of the larger circle that is
tangent to the smaller circle.
page 6
36) PSRQ is an inscribed quadrilateral.
S
P
PSR = 97o . Find the measure of ∠S .
R
Q
37) In circle Q, ∠ SQR = 48° and RQ = 70 cm.
38) TX is tangent to circle Y at point Z.
Find the LENGTH of minor arc SR.
If XY = 41 and XZ = 9. Find
PX.
S
Y
R
Q
P
T
X
Z
39) How far from the center of a circle with radius 50 is a chord of length 80?
40) In circle A,
∠ LTA = 60º. If AL = 15, find the length of the chord TH .
T
A
L
H
A
41) O is the center of the circle.
70o and Arc DE = 40o .
o
<EDC = 35 .
Arc AC =
Find all arcs and angles that you can!
B
O
70°
35°
D
40°
C
E
42) UY is tangent to circle S.
Find the area of the circle if SU is 41 and YU is 40.
Also find ZU.
43) Find arc BC.
U
A
4x
Z
S
3x
Y
x
C
page 7
B
44) The radii of two concentric circles are 9 and 15. Find the length of the chord of the larger circle that is
tangent to the smaller circle.
45) AB and CD are chords of a circle intersecting at X, and X is the midpoint of AB. If CX = 8 and XD = 18, find
the length of AB.
46) Solve for x and y.
47) Two concentric circles have radii 10 and 16.
Find the length of a chord of the larger circle that is tangent to the smaller
circle.
53°
y
x
48) The radius of circle O is 7. Find:
a.
measure of arc BD
b. measure of arc BDC
c.
m<BDC
d.
m<DBC
e.
the perimeter of ΔBDC
f.
g.
60°
B
C
E
O
D
the length of altitude BE
the perimeter of ΔBED
49) ABCD is a quadrilateral inscribed in a circle. If arc AB = 70°, arc BC = 110°,
and arc CD = 90°, find the size of each angle of the quadrilateral.
50) If the point C (2,3) is the center of a circle and the circle passes
through the point (–6, –8), what is the diameter of the circle?
51) Find the length of the arc of a sector with a 40° central angle in a circle of radius of 10.
52) Find the area of the sector of a circle with a 40° central angle in a circle of radius of 10.
53) The bases of a trapezoid are 8 feet and 15 feet. The area of the trapezoid is 115 square feet. The height of
the trapezoid is:
54) A regular hexagon whose perimeter is 300 is inscribed in a circle. Find the area between the circle and the
hexagon.
page 8
55) The legs of a right triangle are 16 and 20. Find the length of the radius of the circle that circumscribes the
triangle.
56) A triangle has sides 8, 8 and 12. Find the area.
57) Find the area of an equilateral triangle with a side of 20.
58) A rhombus has sides of length 10 and one of its angles has a measure of 60o .
Find the area of the rhombus.
59) In parallelogram ABCD, if AB = 15, BC = 8, and angle B = 30°, find the area of ABCD.
60) In isosceles triangle ABC with AB = BC = 10, and angle A = 45°, find the area.
61) An isosceles trapezoid has bases of 18 and 28. Its area is 96. Find its altitude.
62) Find the radius of a circle inscribed in
(a) a square with perimeter of 20
(b) an equilateral triangle with perimeter of 18
(c) a regular hexagon with a perimeter of 60
63) A rhombus has diagonals of 20 and 30. Find its area.
64) An isosceles trapezoid has bases of 8 and 10, with base angles of 60°. Find the area.
65) Find the AREA of the figure shown.
8
15
66) Find the area of an equilateral triangle with a side length of 9.
67) The area of a square is
11u2 . The length of the diagonal of the square is:
68) A square is inscribed in a circle. If the radius of the circle is
find the area of the square.
page 9
6 3,
69) Find the area of the rectangle formed by the the lines y = −9, y = 14, x = –8, x = 21 .
70) Find the area of an isosceles triangle if the legs are 20 and the base is 30.
71) P is the center of the circle. Find the area of the shaded region.
8
12
P
72) Verify that ABCD is a parallelogram and find the area and perimeter of ABCD.
A (4, 9), B (9, 9), C (3, –3), and D (–2, –3).
73) Find the area and perimeter of ΔABC .
A (1, 2), B (1, 6), and C (4, 1)
74) Find the area and the perimeter of
trapezoid ABCD.
5
B
A
45°
D
75) A regular hexagon with side of length 14 is inscribed
in a circle. Find the area of the shaded region.
14
60°
C
14
76) ΔABC has coordinates A (–10, 1), B (7, 1), C (10, 5). Find the area and the perimeter of ΔABC .
77) A circle is inscribed in an equilateral triangle with side length 8 6 .
Find the area of the shaded region.
78) This cone has a height of 15 cm and a radius of 5 cm.
Find its volume.
79) Two spheres have radii 4 and 14. What is the ratio of their volumes?
page 10
80) Find the areas and volumes of each. Please break surface area down into lateral area and area of the base(s) :
(a) A sphere with radius of 4
(b)A cylinder with base diameter of 12 and height of 10
(c) A cone with base radius of 3 and an altitude of 4.
(d) A prism whose bases are equilateral triangles with sides of 8, and whose height is 12
(e) A cube whose faces have diagonals of 9.
(f) A hexagonal prism whose bases have a perimeter of 36 and whose height is 10.
81) Two spheres have radii of 2 and 4. What is the ratio of their volumes? What is the ratio of their surface
areas?
82) Find the total surface area of the rectangular prism:
83) Find the volume of the figure which is
made up of a cylinder with a half sphere on
top.
6 in
12
64
3 in
8 in
16 in
3
2
84) Find the TOTAL SURFACE AREA of a cylinder with
radius of 4 and height of 11.
85) A square base pyramid has a slant height of 17 cm and a height of 15 cm.
Find the volume of the pyramid.
86) A prism has right triangle bases with legs 7 and 24. The height of the prism is 9.
Find the total area of the prism.
87) A regular square pyramid has a slant height of 12 cm and a height of 8 cm.
Find the area and volume.
88) A hemisphere sits on top of a cylinder whose diameter is 8 and whose height is 12. Find the volume and the
surface area of the entire figure.
page 11
89) The area of a trapezoid is 72 and its altitude is 6. Find the lengths of the bases if one is twice as long as the
other.
90) Find the area of a circle whose circumference is 20π.
91) A triangle has vertices A(1, 8), B(–3, 7), and C(–9, –12).
a) Find the equation of the perpendicular bisector of AB.
b) Find the equation of the line parallel to AB that passes through C.
c) Find the equation of the altitude from A.
d) Find the equation of the median from A.
92) Find the perimeter and the area of an equilateral triangle with a height of 9.
93) Given two points A (–4, 3) and B (6, –5), find the equation of the perpendicular bisector of segment AB.
94) Given the equation 3 (y + 4) – 2x = y + x + 1
a) Find the slope of this line
b) Find the slope of any line perpendicular to this line
c) Find the slope of any line parallel to this line
d) Find the x and y-intercepts of this line
95) What is the equation of the line in y = mx + b form with a slope of 2/3 that passes through the point (4, 2)?
96) On a piece of graph paper, sketch each equation:
a) y =
1
2
x − 3 b) y = 3
c) 2y + 3 = –3x + 4
d) y = x
97) What is the equation of the line that is parallel to the x-axis and passes through the point (10,-17)?
page 12
98) What is the equation of the line that is perpendicular to the x-axis and passes through the point (-7, 10)?
99) What is the equation of the line (in point-slope form) that passes through the points (5,1) and (23,-12)? What
is the equation of the same line in slope-intercept form?
100) What is the equation of the line (in slope-intercept form) that has a slope of
4
3
and passes through the
origin?
101) What is the equation of the line (in slope-intercept form) that passes through the points (0,
102) The coordinates of A are (-3,8) and the coordinates of B are (7,-4).
←⎯→
a) Find the equation of the line that is parallel to AB and passes through the point (1, –2).
←⎯→
b) Find the equation of the line that is perpendicular to AB and passes through the point (–5, 6).
page 13
5
2
) and ( −
7
3
, 0)