GRE Geometry - Everything you need to know Geometry... · GRE Geometry - Everything you need to...
Transcript of GRE Geometry - Everything you need to know Geometry... · GRE Geometry - Everything you need to...
GRE Geometry - Everything you need to know
www.GreenlightTestPrep.com
This slideshow features screenshots from Greenlight Test Prep’s entire Geometry module (consisting of 42 videos). It covers every key concept you need to know about GRE Geometry. It also includes 32 practice questions.
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Note: since these slides are just snippets of a full-length video course, there may be times when you’re unable to glean all the relevant information from a particular screenshot. If, at any time, you’d like to watch the entire video on a certain topic, just click on the link at the top of that page, and you’ll be taken that that particular video.
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GRE Geometry - Everything you need to know
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GRE Geometry - Everything you need to know
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Lines and Angles
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Lines and Angles
l
line: a straight path that extends without end in both directions
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Lines and Angles
l
A
B
AB: line segment
AB: length of line segment AB (e.g., DE=7)
line: a straight path that extends without end in both directions
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Lines and Angles
55
A
BC
55
55
ABC
CBA
55x
x
angle: intersection of 2 lines
: measured in degrees or radians
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Lines and Angles
180
Angles on a line add to 180°
a cb
180a b c
70x
70 180
110
x
x
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Lines and Angles
90
right angle: angle of 90 degrees
P
PQ is perpendicular to AB
BA Q
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Lines and Angles
bisect: cut or divide into 2 equal pieces
J
JK bisects AB
BA
A
BC
bisect s ABC
bisector is the of ABC
line l is the perpendicular bisector of AB
BA
K
l
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Lines and Angles
ac
x
x
b
d
- a and c are vertical angles
- a and c are opposite angles
- a and c are vertically opposite angles
- b and d are opposite angles
Opposite angles are equal
y
y
Aside: 180x y
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Lines and Angles
w 50
yx
50x
Opposite angles are equal
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Lines and Angles
w 50
yx
50x 50 180
130
w
w
130y
Opposite angles are equal
Angles on a line add to 180°
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Lines and Angles
1
2
If two lines do not intersect, they are parallel
1 2
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Lines and Angles
1
2
If two lines do not intersect, they are parallel
y
y
y
y
x
Note: 180x y
x
x
x
1 2
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Lines and Angles
Opposite angles are equal
Angles on a line add to 180°
1
2
1 2
y
y
y
y
x
x
x
x
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Practice Question
A) 10
B) 17.5
C) 22
D) 35
E) 42.5
If l1 and l2 are parallel, then x =
1
2
3 5x
15x
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Practice Question
A) 10
B) 17.5
C) 22
D) 35
E) 42.5
If l1 and l2 are parallel, then x =
1
2
3 5x
15x 3 5x
15 3 5 180
4 10 180
4 170
42.5
x x
x
x
x
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Triangles – Part I
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Triangles – Part I
A
B Cw x
y180w x y
Angles in a triangle add to 180°
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Triangles – Part I
A
B C21
44180w x y
Angles in a triangle add to 180°
w
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Triangles – Part I
A
B C21
44180w x y
Angles in a triangle add to 180°
w
180
180
1
2 4
5
4
1
1
65
w
w
w
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Triangles – Part I
A
B Cw x
y
The longest side is opposite the largest angle
The shortest side is opposite the smallest angle
A
B
C
a
b
c
If then a b c A B C
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Triangles – Part I
1
The sum of the lengths of any two sides of a triangle must be greater than the third side.
2 4
1 2
1 42
4
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Triangles – Part I
If a triangle has sides with lengths 3 and 7, what lengths are possible for the third side?
7
third side 73 37
3 4
rd difference between other 2 sides 3 side sum of other 2 sides
Given lengths of sides A and B
rd 3 sideA B A B
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Triangles – Part I
Given lengths of sides A and B
rd 3 sideA B A B
Angles in a triangle add to 180°
A
B
C
a
b
c
If then a b c A B C
The sum of the lengths of any two sides of a triangle must be greater than the third side.
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Practice Question
Quantity A Quantity B
w, x and y are the lengths of the 3 sides of a certain triangle
2w x y
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
Quantity A Quantity B
w, x and y are the lengths of the 3 sides of a certain triangle
2w x y
Given the lengths of sides A and B in a triangle:
rd 3 sideA B A B
ww x xy 2w wx x
A
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Practice Question
What is the value of x in terms of y ?
A) 65
B) 21
C) 22
D) 21
E) 22
y
y
y
y
y
x
y
2243
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Practice Question
What is the value of x in terms of y ?
A) 65
B) 21
C) 22
D) 21
E) 22
y
y
y
y
y
x
y
2243
a
43 180ya
43 22 180xa
43 22 43
43 22 43
22
22
a x y
x y
x y
x y
a
Angles in a triangle add to 180°
Solution #1
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Practice Question
What is the value of x in terms of y ?
A) 65
B) 21
C) 22
D) 21
E) 22
y
y
y
y
y
x
y
2243
158 x
180
158 18
1
0
22
22
22
58y
y x
y x
y x
y x
x
Angles on a line add to 180°
1
22 180
1
58
58
x
x
c
c
c x
158 180
22
y x
y x
Solution #2
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Assumptions in Geometry
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120
• Lines that appear straight can be assumed to be straight
Assumptions in Geometry
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120
60
• Lines that appear straight can be assumed to be straight
Assumptions in Geometry
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• Do not make assumptions about angle measurements
x
Assumptions in Geometry
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y
• y +x =180
• Both angles are greater than zero degrees
x
Assumptions in Geometry
• y is greater than x
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• Do not make assumptions about parallelism
1
2
1 2
Assumptions in Geometry
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Assumptions in Geometry
A
BC
ABC
• Unless otherwise indicated, ABC refers to the smaller angle.
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Assumptions in Geometry
A
BC
• Unless otherwise indicated, ABC refers to the smaller angle.
ABC
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• Not necessarily drawn to scale
• Avoid visual estimation
Assumptions in Geometry
Geometric Figures
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• Lines that appear straight can be assumed to be straight
• Angles are greater than 0 degrees
• Do not make assumptions about angle measurements
• Do not make assumptions about parallelism
• Unless otherwise indicated, ABC refers to the smaller angle
Assumptions in Geometry
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Geometry Strategies
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Geometry Strategies
• Redraw figures
• Add all given information
• Add all information that can be deduced
• Add/extend lines
• Assign variables and use algebra
•
• Drawn to scale estimate to confirm calculations and guide guesses
• Drawn to scale estimate measurements to confirm or guess
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Triangles – Part II
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Triangles – Part II
Isosceles triangle
• 2 equal sides, 2 equal angles
A
B
C
a
b
c
If then a b c A B C
40 40
100
x
x
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Triangles – Part II
38
38
104
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Triangles – Part II
38
38
104
40
x
x
40 180
2 40 180
2 140
70
x x
x
x
x
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Triangles – Part II
A
B
C
Equilateral triangle
• 3 equal sides, 3 equal angles
60 60
60
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Triangles – Part II
A
B C
base heightArea
2
10
48
Area
1Area base height
2
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Triangles – Part II
A
B C
base heightArea
2
10Area
15
3
2
10
34
8
altitude height
Area
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Triangles – Part II
10
4
8
A B
C
7.5
base heightArea
2
7A
.re
5a
15
4
2
Area
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Triangles – Part II
A
B
C60 60
60
2
3 sideArea
4
6 6
6
2
3Area
4
3 36
4
9 3
6
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Triangles – Part II
60 60
60
The altitudes of isosceles triangles and equilateral triangles bisect the base.
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Triangles – Part II
• An isosceles triangle has 2 equal sides and 2 equal angles
• An equilateral triangle has 3 equal sides and 3 equal angles (60° each)
base heightArea
2
2
3 sideArea
4
• The altitudes of isosceles triangles and equilateral triangles bisect the base
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Practice Question
A) 27.5
B) 55
C) 62.5
D) 70
E) 125
If AB and CD are parallel, and AB = BC, then x =
A
B
C
D
x
55
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Practice Question
A) 27.5
B) 55
C) 62.5
D) 70
E) 125
If AB and CD are parallel, and AB = BC, then x =
A
B
C
D
x
5555
55
180
110 18
5 5
70
5 5
0
x
x
x
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Right Triangles
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Right Triangles
leg1
• Right triangle: triangle with right (90°) angle
• The hypotenuse is the longest side
leg2
2 2 2
1 2leg leg hypotenuse
2 2 2a b c
a
bc
2 2 2a b c
a
bc 2 2 2a b c
a
bc
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Right Triangles
2 2 2a b c
a
bc
8
6
x
2 2 2
2
2
8 6
64 36
100
100
10
x
x
x
x
x
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Right Triangles
2 2 2a b c
a
bc
8
6
x
2 2 2
2
2
8 6
64 36
100
100
10
x
x
x
x
x
6
4x
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Right Triangles
2 2 2a b c
a
bc
8
6
x
2 2 2
2
2
8 6
64 36
100
100
10
x
x
x
x
x
2 2 2a b c
a
bc
6
4x
2 2 2
2
2
4 6
16 36
20
20
2 5
x
x
x
x
x
4 5
2 5
x
x
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Right Triangles
• 3-4-5
4
35
• 5-12-13
12
135
• 8-15-17
2 2 23 4 5
2 2 25 12 13
• 7-24-25
Pythagorean triples: A set of 3 integers that can be the sides of a right triangle
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Right Triangles
8x 17
15
• 8-15-17
2 2 215 17x
2 2 2a b c
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Right Triangles
• 3-4-5
• 5-12-13
• 8-15-17
• 7-24-25
6-8-10 9-12-15 12-16-20
10-24-26
4
35
4 7 28
5 7 35 213 7x
. . .
. . .
. . .
2 corresponding sides required to use Pythagorean triples
. . .
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Right Triangles
• 3-4-5
• 5-12-13
• 8-15-17
• 7-24-25
6-8-10 9-12-15 12-16-20
10-24-26
. . .
. . .
. . .
50
4
35
Enlarged by factor
of 10
50
24
725 Enlarged
by factor of 2
40
30
48
14
2 corresponding sides required to use Pythagorean triples
. . .
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Right Triangles
• 3-4-5
• 5-12-13
• 8-15-17
• 7-24-25
6-8-10 9-12-15 12-16-20
10-24-26
. . .
. . .
. . .
3
x
4
2 2 2a b c
2 2 2
2
2
3 4
9 16
7
7
x
x
x
x
. . .
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Right Triangles
2 2 2a b c
a
bc
• Watch out for Pythagorean triples (and their multiples)
3-4-5
5-12-13
8-15-17
7-24-25
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Practice Question
AA) 2 3
B) 2 5
C) 30
D) 4 3
E) 4 5
B
The height of this rectangle is twice its width. If the distance
between points A and B is , what is the rectangle’s height? 60
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Practice Question
AA) 2 3
B) 2 5
C) 30
D) 4 3
E) 4 5
x
2x
22 2
2 2
2
2
2 60
4 60
5 60
12
12
4 3
2 3
x x
x x
x
x
x
x
x
B
60
2 2 2a b c
2
4 3
2 2 3x
The height of this rectangle is twice its width. If the distance
between points A and B is , what is the rectangle’s height? 60
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Practice Question
A) 21
B) 9
C) 2 21
D) 149
E) 3 21
If the rectangular box shown here is 6 inches wide, 8 inches long and 7
inches high, what is the distance, in inches, between points A and B ?
B
A
8
6
7
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Practice Question
A) 21
B) 9
C) 2 21
D) 149
E) 3 21
B
A
8
6
7
If the rectangular box shown here is 6 inches wide, 8 inches long and 7
inches high, what is the distance, in inches, between points A and B ?
10
x7
A
B
10
x
2 2 2a b c 2 2 2
2
2
10 7
100 49
149
149
x
x
x
x
Solution #1
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Practice Question
A) 21
B) 9
C) 2 21
D) 149
E) 3 21
If the rectangular box shown here is 6 inches wide, 8 inches long and 7
inches high, what is the distance, in inches, between points A and B ?
A
B
wx
y
2 2 2AB w x y
2 2 28 6 7
64 36 49
149
AB
B
A
8
6
7
Solution #2
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Special Right Triangles
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Special Right Triangles
45-45-90 triangle
1
45
2
2 1.445
1
leg : leg : hypotenuse
1 : :
x : :
1
x 2x
2
30-60-90 triangle
1
30
602
3
3 1.7
3
leg : leg : hypotenuse
1 : : 2
3xx : : 2x
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Special Right Triangles
1230
x
y
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Special Right Triangles
1230
x
y 30
601
2
3
60
enlargement factor: 6
61
6
x
3
6 3
6y
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Special Right Triangles
5 2
x
5 2
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Special Right Triangles
5 2
x
5 2
45
12
451
45
45
enlargement factor:
2
5 4
5
2
2
10
5x
5 2
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Special Right Triangles
60 60
60
Square Equilateral Triangle
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Special Right Triangles
45
45
60 60
30
Square Equilateral Triangle
Watch out for special right triangles “hiding” in squares and equilateral triangles
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Special Right Triangles
45
12
451
30
601
2
3
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Practice Question
A) 3 2
B) 2 6
C) 4 3
D) 6 2
E) 6 3 B
A
C
D
If , 6 and 105 , then AD BD AB ABC x
Note: Figure not drawn to scale
x
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Practice Question
A) 3 2
B) 2 6
C) 4 3
D) 6 2
E) 6 3 B
A
C
D
If , 6 and 105 , then AD BD AB ABC x
Note: Figure not drawn to scale
45
45
60
30
x
45
1
451
enlargement factor: ? 6
2
266
2
30
60 2
3
6
2
1
2
12 2
2 2
12 2
2
2
6 2
6x
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Similar Triangles
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Similar Triangles
Similar triangles have the same 3 angles in common
40 20120
40 20
1203 30 1 55.6 60x
enlargement factor: 10
5.6
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Similar Triangles
Similar triangles have the same 3 angles in common
40 20120
40 20
120
With similar triangles, the ratio of any pair of corresponding sides is the same
wa
b c xy
a
w
b c
x y
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Similar Triangles
**
x
5 7
9
6
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Similar Triangles
**
x
5 7
9
With similar triangles, the ratio of any pair of corresponding sides is the same
5
5
63
6
5
3
5
7 9
7
9
x
x
x
x
6
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Similar Triangles
Similar triangles have the same 3 angles in common
40 20120
40 20
120
With similar triangles, the ratio of any pair of corresponding sides is the same
wa
b c xy
a
w
b c
x y
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Practice Question
If , then ABC BCD x
BA
C D
8
10 12
5 x
E
A) 4
25B)
6
C) 6
36D)
5
E) 24
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Practice Question
If , then ABC BCD x
BA
C D
x
E
❤
❤ With similar triangles, the ratio of any pair
of corresponding sides is the same
12
5
5
12 10
12 1
50
50
12
25
6
0
x
x
x
x
x
A) 4
25B)
6
C) 6
36D)
5
E) 24
8
10 12
5
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Quadrilaterals
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Quadrilaterals
Angles in a quadrilateral add to 360°
A
D C
w
x
y360w x y z
B
z
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Quadrilaterals
square
rectangle
trapezoid
parallelogram
rhombus
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Quadrilaterals
parallelogram
opposite sides parallel
rectangle
opposite sides parallel
all angles are 90
rhombus
opposite sides parallel
all sides are equal
square
opposite sides parallel
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Quadrilaterals
trapezoid
2 sides parallel
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Quadrilaterals
Rhombus (and square)
• diagonals are perpendicular bisectors
Rectangle (and square)
• diagonals are equal length
A
D C
B
AC BD
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Quadrilaterals
square rectangle
trapezoid
area base height
base base
height height
base2
base1
height
1 2base basearea height
2
average of bases height
parallelogram rhombus
base
height
base
height
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Quadrilaterals
rhombus
1 2diagonal diagonalarea
2
4
7
area2
28
2
14
4 7
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Quadrilaterals
Angles in a quadrilateral add to 360°
parallelogram
opposite sides parallel
rectangle
opposite sides parallel
all angles are 90
rhombus
opposite sides parallel
all sides are equal
square
opposite sides parallel
trapezoid
2 sides parallel
area base height
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Polygons
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Polygons
Polygon: Closed figure formed by 3 or more line segments
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Polygons
“polygon” “convex polygon” (all interior angles less than 180°)
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Polygons
b
a
180a b c
Triangle
Quadrilateral
Pentagon
c
b
a
c
d360a b c d
b
a
cd 540a b c d e
e
Hexagon
b
a
cd 720a b c d e f
ef
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Polygons
The sum of the interior
angles in an N-sided polygon
is equal to 180 2N
6
1
23
4
5
Octagon
sum of angles 180 2
8180 2
180
10
6
80
N
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Polygons
Regular polygon: equal sides and equal angles
regular pentagon
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Polygons
• Polygon: Closed figure formed by 3 or more line segments
• “polygon” “convex polygon” (all interior angles less than 180°)
Triangle Quadrilateral
Pentagon Hexagon
• Regular polygon: equal sides and equal angles
The sum of the interior
angles in an N-sided polygon
is equal to 180 2N
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Circles
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Circles
Circle: set of points that are equidistant from a given point
center
A
B
C
E
D
diameter u2 radi s
arc
- “arc CDE ”
- “minor arc CE ”
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Circles
circumference 2 radius
2 r
Circumference 3.14
3
22
7
circumference diameter
d
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Circles
circumference 2 r
Circumference
circumference 2
16 feet
16 3
48 f
8
eet
8 ft
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Circles
Area
2area r
2
2
area
6
8
4 ft
8 ft
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Circles
center
A
B
C
E
circumference diameter
circumference 2 radius
3.14
3
2area r
arc
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Practice Question
Quantity A Quantity B
The area of circle A is twice the area of circle B.
The radius of circle A The diameter of circle B
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Practice Question
Quantity A Quantity B B
The area of circle A is twice the area of circle B.
circle A's radiusa
The radius of circle A The diameter of circle B
circle B's radiusb
2 2
2 2
2 2
2
circle A area 2 circle B area
2
2
2
2
2
b
b
a
a
a
b
a
b
b
a
a 2b
2 b
2 2
2area r
divide by b
2b
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Practice Question
A) 9
B) 12
C) 15
D) 18
E) 36
If is the center, 45 , and 6,then the area of the circle isO OBC BC
C
B
O
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Practice Question
A) 9
B) 12
C) 15
D) 18
E) 36
If is the center, 45 , and 6,then the area of the circle isO OBC BC
CO
45
4590
B
With similar triangles, the ratio of any pair of corresponding
sides is the same
6
2area r
2
area
36
2
18
6
2
r
12
6
2
6 r
r
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Pieces of Pi
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Pieces of Pi
C
E
1of circumference
4
90of circumference
360
CE
90
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Pieces of Pi
119
C
E
119of circumference
360CE
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Pieces of Pi
x
C
E
of circumference360
2360
CEx
xr
arc length 2360
xr
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Pieces of Pi
x
C
E 2
of area circ of sect le's area3
r60
o
360
Ox
x
C
r
E
O
2sector area360
xr
360
x
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Pieces of Pi
O
160
6
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Pieces of Pi
O
160
2area360
xr
26area360
436
9
16
160
6
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Pieces of Pi
x
C
E
2360
xCE r
x
C
EO
2area360
xr
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Practice Question
20A)
3
25B)
3
25C)
2
40D)
3
50E)
3
C
B
O
O is the center of the circle with radius 30. If x – w=20, what is the length of arc CDE ?
A
E
Dw
xy
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Practice Question
20A)
3
25B)
3
25C)
2
40D)
3
50E)
3
C
B
O
O is the center of the circle with radius 30. If x – w=20, what is the length of arc CDE ?
A
E
D
x
arc length 2360
yr
y
30
20x w
180x w
2 160
80
w
w
80 80
arc length 2360
260
9
8
4
3
00
0
3
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Circle Properties
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Circle Properties
A
B
x
“x is an inscribed angle holding/containing chord AB ”
“x is an inscribed angle holding/containing arc AB ”
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Circle Properties
A
B
x
x
Inscribed angles holding the same chord/arc are equal
x
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Circle Properties
A
B
x
C
D
x
Inscribed angles holding chords/arcs of equal length are equal
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Circle Properties
An inscribed angle holding the diameter is a right angle
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Circle Properties
A
B
x
O
“Angle AOB is a central angle holding chord AB”
2x
A central angle is twice as large as an inscribed angle holding the same chord/arc
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Circle Properties
The line from the center to the point of tangency is
perpendicular to the tangent line
“line l is tangent to the circle”
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Circle Properties
**
*
*
x
2x
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Practice Question
Note: Figure not drawn to scale
C
x
20
D
O
B
A
A) 40
B) 50
C) 60
D) 70
E) 80
If is the center and , then O AB CD x
E
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Practice Question
Note: Figure not drawn to scale
C
xD
O
B
A
A) 40
B) 50
C) 60
D) 70
E) 80
90
If is the center and , then O AB CD x
A10
2090
90
80
80
E
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Volume & Surface Area
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Volume & Surface Area
1 ft1 ft
1 ft31 ft
2 ft3 ft
5 ft
Volume length width height
3
Volume 2 3 5
30 ft
Volume
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Volume & Surface Area
r
height h
2Volume r h
3
2Volume r h
10
2
3Vo 1lume
90
0
Volume
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Volume & Surface Area
Surface Area
face
• 6 faces
• 12 edges
• 8 vertices
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Volume & Surface Area
Surface Area
• 6 faces
• 12 edges
• 8 vertices
edge
edge
edge
edge
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Volume & Surface Area
Surface Area
• 6 faces
• 12 edges
• 8 vertices
vertex
vertex
vertex
vertex
vertex
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Volume & Surface Area
8 cm
Surface Area
4 cm
5 cm
2
surface area 40 40 32 32 20 20
184 cm
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Volume & Surface Area
Surface Area
8 cm
4 cm
5 cm
2
surface area 40 40 32 32 20 20
184 cm
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Volume & Surface Area
Surface Area
2area r 2area r
h
2 r
area 2
2
r h
rh
2 2
2
total area 2
2 2
2
r r rh
r rh
r r h
r
h
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Volume & Surface Area
length
volume length width height
width
height
r
2volume r h
2 2
2
surface area 2
2 2
2
r r rh
r rh
r r h
surface area sum of areas of all 6 sides
h
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Units of Measurement
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Units of Measurement
• If conversion is required, relationship will be given
- e.g., (1 kilometer = 1000 meters)
- e.g., (1 mile = 5280 feet)
• Note: Relationships not given for units of time
- e.g., (1 hour = 60 minutes)
Conversions
- e.g., (1 day = 24 hours)
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Geometry Strategies – Part II
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• Redraw figures
• Add all given information
• Add any information that can be deduced
• Add/extend lines
• Assign variables and use algebra
• Problem solving questions drawn to scale:
• Circle:
• Break areas/volumes into manageable pieces
• Two or more triangles and length required
• Right triangle:
- use Pythagorean Theorem to relate sides
- watch for Pythagorean Triples and special triangles
- beware of circle properties (inscribed/central angles, tangent lines)
- look for isosceles triangles
- estimate to confirm calculations and guide guesses
- look for similar triangles
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Practice Question
Quantity A Quantity B
Area of an equilateral triangle with each side
having length 4k
Area of a circle with diameter 2k
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
Quantity A Quantity B
2
3 sidearea
4
Area of an equilateral triangle with each side
having length 4k
Area of a circle with diameter 2k
2area r
2
3 4
4
k
23 16
4
k
23 4k
(radius = k)
2k
4 32divide by k
A
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Practice Question
A) 1
4B)
3
3C)
2
5D)
3
5E)
2
60
1x 4 3x
What is the value of x ?
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Practice Question
A) 1
4B)
3
3C)
2
5D)
3
5E)
2
30
601
2
3
60
1x 4 3x
What is the value of x ?
30
With similar triangles, the ratio of any pair of corresponding
sides is the same
1 4 3
1 2
2 1 1 4 3
2 2 4 3
2 2 3
5 2
5
2
x x
x x
x x
x
x
x
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Practice Question
Quantity A Quantity B
Perimeter of ABC 10
A
B
C
3
4
ABC is an isosceles triangle
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
Quantity A Quantity B D
Perimeter of ABC 10
A
B
C
3
4
ABC is an isosceles triangle
3BC 10 10=
4BC 11 10
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Practice Question
If is tangent to the circle with center , then AC O DBC
D
O
B CA
40
A) 50°
B) 55°
C) 60°
D) 65°
E) 70°
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Practice Question
If is tangent to the circle with center , then AC O DBC
D
O
B CA
40
A) 50°
B) 55°
C) 60°
D) 65°
E) 70°
50
130
25
25
65
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Practice Question
Quantity A Quantity B
5x2x y
7x y
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
Quantity A Quantity B
7x y
5 2 90 180
7 0 18
90
9 0
7
x x
x
x
90
2 90x
5x2x y
90
90
90
B
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Practice Question
What is the area of triangle ?ABC
60
5
12
A) 15
B) 15 3
5 119C)
2
D) 32.5
E) 36
A
BC
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Practice Question
What is the area of triangle ?ABC
60
A) 15
B) 15 3
5 119C)
2
D) 32.5
E) 36
enlargement factor: 6
12
3 6 36h
6 3
5
A
BC
base heightarea
2
5 6 3area
2
30 3
2
15 3
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Which of the following could equal the length of side ?KM
Indicate all possible values.
[A] 1
[B] 2
[C] 3
[D] 4
[E] 5
[F] 6
[G] 7
K
L
M
2 4
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(watch the entire video here) Practice Question
Which of the following could equal the length of side ?KM
Indicate all possible values.
[A] 1
[B] 2
[C] 3
[D] 4
[E] 5
[F] 6
[G] 7
K
L
M
2 4
Given lengths of sides A and B in a triangle:
rd 3 sideA B A B
4 2 4 2
2 6
2 6
KM
KM
KM
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Practice Question
Quantity A Quantity B
Volume of spherewith radius 2
32
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
Quantity A Quantity B
Volume of spherewith radius 2
32B
2
4
4
4
sphere volume
sphere
cube volume
4 volume
sphere volu
4 4
me 64
-64
32 64
3 6
2
42
64
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If is a parallelogram, then what is its perimeter?ABCD
A B
CD
3 3x y
4 2 2y x
6x y 2 6 13x y
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Practice Question
A) 22
B) 24
C) 26
D) 28
E) 30
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Practice Question
If is a parallelogram, then what is its perimeter?ABCD
A) 22
B) 24
C) 26
D) 28
E) 30
perimeter 3 3 2 6 13 4 2 2
1
6
4 4 18
4 18
4 18
22
x y x y y x x y
x
x
y
y
A B
CD
6 2 6 13
5 7
x y x y
x y
6x y 2 6 13x y 3 3 4 2 2
5
1
5 5
x y y x
x
x
y
y
4 2 2y x
3 3x y
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Practice Question
In the regular octagon shown here, what is the value of x?
x degrees
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Practice Question
The sum of the interior
angles in an N-sided polygon
is equal to 180 2N
In the regular octagon shown here, what is the value of x?
x 135 degrees
sum of all angl8 8
180 6
es 180 2
180 6eac
6h angle 1
880
8
3180
4
135
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Practice Question
The diagonal of square has length 3.23ABCD
Area of square ABCD 3.23
Quantity A Quantity B
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
The diagonal of square has length 3.23ABCD
Area of square ABCD 3.23
Quantity A Quantity B B
x
x
3.23 2 2 2a b c
a
bc
2
2 2
2
2
3.23
2 3.23
3.23
2
x x
x
x
3.23
23.23
2x
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Practice Question
What is the value of ?x
Note: Figure not drawn to scale
155 3x
6 30x
4 70x
A) 5
B) 7
C) 15
D) 21
E) 25
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Practice Question
What is the value of ?x
Note: Figure not drawn to scale
155 3x
6 30x
4 70x
A) 5
B) 7
C) 15
D) 21
E) 25
180 4 70x
180 155 3x
6 30x
180 4 70 180 155 3 6 30 180
110 4 25 3 6 30 180
105 5 180
5 75
15
x x x
x x x
x
x
x
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Practice Question
A
B
C
D
E
Quantity A Quantity B
x y
2
3
4
5
ky
x
k
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
Quantity A Quantity B
x y
AB DE
x y k
A
B
C
D
E
2
3
4
5
xy
ky
k
B
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Practice Question
x
y
The above circle is tangent to the x- and y-axes. It has radius 4, and its center is at O.
Quantity A Quantity B
's distance from the originO
5
O
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
x
y
The above circle is tangent to the x- and y-axes. It has radius 4, and its center is at O.
Quantity A Quantity B
's distance from the originO
A
O
4
4
d
4
4
3
4
5 5
5 5
5
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Practice Question
K is the surface area of cylinder A. If the radius of cylinder B is twice the radius of cylinder A, and the height of cylinder B is twice that of cylinder A, what is the surface area of cylinder B?
A) 2K
B) 3K
C) 4K
D) 6K
E) 8K
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Practice Question
K is the surface area of cylinder A. If the radius of cylinder B is twice the radius of cylinder A, and the height of cylinder B is twice that of cylinder A, what is the surface area of cylinder B?
A) 2K
B) 3K
C) 4K
D) 6K
E) 8K
2surface area 2 2r rh 1
1
22 2
2 2
1 1 1
4
2
2
2surface area 2 2r rh
2
2 2
8 8
1
2 2 2
6
A
B
K
4K
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2 14 48 0y y
A C
B
y
Quantity A Quantity B
Area of .ABC 12
55
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Practice Question
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
2 14 48 0y y
A C
B
Quantity A Quantity B
Area of .ABC 12
55
2 14 48 0
6 , 80 68
y y
y y y
6y
8y
y4 4
area2
bh
12 12
3
8 3ar 1ea
22
12= 12
=
C
Practice Question (watch the entire video here)
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Practice Question
A
BD
C
E F
If bisects , and bisects , then BD CBE DE BEF w
w
50
A) 25
B) 35
C) 50
D) 55
E) 65
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Practice Question
A
BD
C
E F
If bisects , and bisects , then BD CBE DE BEF w
w
50
xx
yy 180 2y
180 2x
50 180 2 180 2 180
410 2 2 180
230 2 2
23
5
0
11
2
x y
x y
x y
x
x y
y
A) 25
B) 35
C) 50
D) 55
E) 65
180
180
180 x
w x y
w x
y
y
w
11180
65
5
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Practice Question
If is a rectangle, then what is the length of ?ABCD EC
A) 7.8
B) 8
C) 8.4
D) 9
E) 9.6
A
B
D
C
E
12
16
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Practice Question
If is a rectangle, then what is the length of ?ABCD EC
A) 7.8
B) 8
C) 8.4
D) 9
E) 9.6
A
B
D
C
E
B
C
DE
D C
B
E
12
16
16
16
121216
20
area2
bh
12 16area
9
2
6
h
area2
bh
20
2
96 10
.6
9
9
6h
h
h
EC
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Practice Question
Quantity A Quantity B
Area of .ABC 6
In the rectangular coordinate plane, ABC has vertices at (0,-2), (0,1) and (4,b), where b<0.
A: Quantity A is greater
B: Quantity B is greater
C: The two quantities are equal
D: The relationship cannot be determined
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Practice Question
Quantity A Quantity B
Area of .ABC 6
In the rectangular coordinate plane, ABC has vertices at (0,-2), (0,1) and (4,b), where b<0.
4x
base heightArea
2
y
3
4
3 4
2
C
6=
x
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Practice Question
If the both circles have radius 6, and O and P are their centers, what is the area of the shaded region?
A) 24 18 3
B) 24 12 3
C) 18
D) 36 24 3
E) 18 12 3
PO
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Practice Question
If the both circles have radius 6, and O and P are their centers, what is the area of the shaded region?
A) 24 18 3
B) 24 12 3
C) 18
D) 36 24 3
E) 18 12 3
cab
2 660
6360
6
6
6
a b
d e
b c
d f
2sector area360
xr
O
e
P
fd
24
24
24 24
24
24 1
9 3 3
8
9
3
b d
b d b d
a b d e b c d f
a b c d e f
a b c d e f
a b c d e f
a b c d e f
b66
6
2
3 sidearea
4
23b 3
69
4
b d a b c d e f
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For additional practice questions, see the bottom of our Geometry module
500+ videos 600+ practice questions step-by-step study guide All you need. All for free
GRE Geometry - Everything you need to know
500+ videos 600+ practice questions step-by-step study guide All you need. All for free
500+ videos 600+ practice questions step-by-step study guide All you need. All for free
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GRE Geometry - Everything you need to know