GMAT Geometry - everything you need to know

GMAT Geometry - Everything you need to know
This slideshow features screenshots from GMAT Prep Now’s
entire Geometry module (consisting of 42 videos).
It covers every key concept you need to know about GMAT
Geometry. It also includes 27 practice questions.
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Lines and Angles (watch the entire video here)

Lines and Angles
l
line: a straight path that extends without
end in both directions
(watch the entire video here)

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

Lines and Angles


55
A


B
C
55
55
ABC
CBA
 
 

55x 
x
angle: intersection of 2 lines
: measured in degrees or radians

Lines and Angles

180
Angles on a line add to 180°
a cb
180a b c  
70x
70 180
110
x
x
 


Lines and Angles
90
right angle: angle of 90 degrees

P
PQ is perpendicular to AB
 BA 
Q

Lines and Angles
bisect: cut or divide into 2 equal pieces
J
JK bisects AB
 BA


A


B
C
bisects ABC
bisectoris the of ABC
line l is the perpendicular bisector of AB
 BA
K

l

Lines and Angles
a
c
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 


Lines and Angles
w 50
yx

Lines and Angles
w 50
yx
50x  50 180
130
w
w
 

130y 

Lines and Angles
1
2
If two lines do not intersect, they are parallel
1 2

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

Lines and Angles
1
2
1 2
y
y
y
y
x
x
x
x

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 
Note: Figure not drawn to scale

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
   
 


 
 
 
 
Practice Question (watch the entire video here)

Triangles – Part I (watch the entire video here)

Triangles – Part I
A
B C
w x
y
180w x y  
Angles in a triangle add to 180°

A
B C
21
44
180w x y  
w

A
B C
21
44
180w x y  
w
180
180
1
2 4
5
4
1
1
65
w
w
w
  
 


A
B C
w 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 thena b c A B C   

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

If a triangle has sides with lengths 3 and 7, what lengths
are possible for the third side?
3 7

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   

rd
A
B
C
a
b
c

Is w > x? Q
P
w x
y
R
2) 3QR 
1) 6PQ 
Practice Question
A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
C) BOTH statements TOGETHER are sufficient, but NEITHER statement
ALONE is sufficient
D) EACH statement ALONE is sufficient
E) Statements (1) and (2) TOGETHER are NOT sufficient

Q
P
w x
y
R
1) 6PQ 
A
B
C
a
b
c
2) 3QR 
3
1&2)
6
rd
3 9PR 
E
6 3 36PR   
Is w > x?
Is ?PR PQ
INSUFFICIENT
INSUFFICIENT
INSUFFICIENT

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
22
43
Practice Question

(watch the entire video here)Practice Question
A) 65
B) 21
C) 22
D) 21
E) 22
y
y
y
y
y





x
y
22
43
a
43 180ya   
43 22 180xa    



43 22 43
43 22 43
22
22
a x y
x y
x y
x y
a     
   
 
 
Solution #1

Solution #2
(watch the entire video here)Practice Question
A) 65
B) 21
C) 22
D) 21
E) 22
y
y
y
y
y





x
y
22
43
 158 x
  180
158 18
1
0
22
22
22
58y
y x
y x
y x
y x
x 
  
 
 
 

1
22 180
1
58
58
x
x
c
c
c x
  
 
  158 180
22
y x
y x
  
 

Assumptions and Estimation (watch the entire video here)

Assumptions and Estimation
120

• Lines that appear straight can be assumed to be straight

120

60


• Do not make assumptions about angle measurements
x

y

• y+x =180
• Both angles are greater than zero degrees
x



• Do not make assumptions about parallelism
1
2
1 2

Problem Solving Questions
• Figures are drawn to scale unless stated otherwise
• Estimate to confirm calculations and guide guesses

x
40
O
BE
A) 40
B) 50
C) 60
D) 70
E) 80
C
DA
If is the center of the circle,
and , what is the value of ?
O
AB CD x

Data Sufficiency Questions
• Figure conforms to information in question
• Figure does not necessarily conform to information in statements
• Avoid visual estimation

• Angles are greater than zero degrees
• Do not make assumptions about angle measurements
• Do not make assumptions about parallelism
• Use visual estimation sparingly

Geometry Strategies – Part I (watch the entire video here)

Geometry Strategies – Part I
• 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

Triangles – Part II (watch the entire video here)

Triangles – Part II
Isosceles triangle
• 2 equal sides, 2 equal angles
A
B
C
a
b
c
40 40
100
x
x

38

38
38
104

38
38
104
40

38
38
104
40
x
x
40 180
2 40 180
2 140
70
x x
x
x
x
  
 



38
38
104
40
70
40 180
2 40 180
2 140
70
x x
x
x
x
  
 


70

A
B
C
Equilateral triangle
• 3 equal sides, 3 equal angles
60 60
60

A
B C
10
4 8
Area
- ft2
- cm2
- m2

A
B C
base height
Area
2


10
4 8
Area
1
Area base height
2
 

A
B C
base height
Area
2


10
Area
15
3
2



10
3
4 8
altitude height
Area

10
4
8
A B
C
7.5
base height
Area
2


7
A
.
re
5
a
15
4
2



Area

A
B
C
60 60
60
 
2
3 side
Area
4



A
B
C
60 60
60
 
2
3 side
Area
4


6 6
6
 
2
3
Area
4
3 36
4
9 3
6





60 60
60
The altitudes of isosceles triangles and
equilateral triangles bisect the base.

• 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 height
Area
2


 
2
3 side
Area
4


• The altitudes of isosceles triangles and equilateral triangles bisect the base

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

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
  
 


Right Triangles (watch the entire video here)

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 2
a b c 
a
b
c
2 2 2
a b c 
a
bc 2 2 2
a b c 
a
bc

Right Triangles
8
6
x

Right Triangles
2 2 2
a b c 
a
bc
8
6
x
2 2 2
2
2
8 6
64 36
100
100
10
x
x
x
x
x
 
 




Right Triangles
2 2 2
a 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
4
x

Right Triangles
2 2 2
a 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 2
a b c 
a
bc
6
4
x
2 2 2
2
2
4 6
16 36
20
20
2 5
x
x
x
x
x
 
 


 4 5
2 5
x
x



Right Triangles
• 3-4-5
4
35
• 5-12-13
12
13
5
• 8-15-17
2 2 2
3 4 5 
2 2 2
5 12 13 
• 7-24-25
Pythagorean triples: A set of 3 integers that can be the sides of
a right triangle

Right Triangles
8x 17
15
• 8-15-17
2 2 2
15 17x  
2 2 2
a b c 

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
. . .

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
7
25 Enlarged
by factor
of 2
40
30
48
14
2 corresponding sides required to use Pythagorean triples
. . .

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
4
x
. . .

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 2
a b c 
2 2 2
2
2
3 4
9 16
7
7
x
x
x
x
 
 


. . .

Right Triangles
2 2 2
a b c 
a
bc
• Watch out for Pythagorean triples (and their multiples)
3-4-5
5-12-13
8-15-17
7-24-25

Practice Question
A
A) 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

Practice Question
A
A) 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 2
a 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

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

A) 21
B) 9
C) 2 21
D) 149
E) 3 21
B
A


8
6
7
10
x
7
A
B
10
x
2 2 2
a b c 
2 2 2
2
2
10 7
100 49
149
149
x
x
x
x
 
 


Solution #1

Practice Question
A) 21
B) 9
C) 2 21
D) 149
E) 3 21
A
B
w
x
y
2 2 2
AB w x y  
2 2 2
8 6 7
64 36 49
149
AB   
  

B
A


8
6
7
Solution #2

Special Right Triangles
45-45-90 triangle
1
45
2
2 1.4
45
1
leg : leg : hypotenuse
1 : :
x : :
1
x 2x
2
30-60-90 triangle
1
30
60
2
3
3 1.7
3
leg : leg : hypotenuse
1 : : 2
3xx : : 2x

12
30
x
y

12
30
x
y 30
60
1
2
3
60
enlargement factor: 6

12
30
x
y 30
60
1
2
3
60
 61
6
x 

 3
6 3
6y 


5 2
x
5 2

5 2
x
5 2
45
1
2
45
1
45
45
enlargement factor:
 
 
2
5 4
5
2
2
10
5x 



5 2

45
45
60 60
30
Square Equilateral Triangle
Watch out for special right triangles “hiding”
in squares and equilateral triangles

45
1
2
45
1
30
60
1
2
3

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 , thenAD BD AB ABC x    
x

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 , thenAD BD AB ABC x    
45
45
60
30
x
45
1
45
1
enlargement factor: ?
6
2
26
6
2
30
60 2
3
6
2
1
2
12 2
2 2
12 2
2
2
6 2
6
x 
 

 






Similar Triangles (watch the entire video here)

Similar Triangles
Similar triangles have the same 3 angles in common
40 20
120
40 20
120
With similar triangles, the ratio of any pair
of corresponding sides is the same
w
a
b c x y
a
w
b c
x y
 

Similar Triangles

*
*
x
5 7
9
6

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

Practice Question
If , thenABC BCD x   
BA
C D
8
10 12
5 x
E
A) 4
25
B)
6
C) 6
36
D)
5
E) 24

Practice Question
If , thenABC 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
25
B)
6
C) 6
36
D)
5
E) 24
8
10 12
5

Quadrilaterals (watch the entire video here)

Quadrilaterals
Angles in a quadrilateral add to 360°
A
D C
w
x
y
360w x y z   
B
z

Quadrilaterals
square
rectangle
trapezoid
parallelogram
rhombus

Quadrilaterals
parallelogram




opposite sides parallel
rectangle
opposite sides parallel
all angles are 90


rhombus

all sides are equal




square

Quadrilaterals
trapezoid
2 sides parallel

Quadrilaterals
Rhombus (and square)
• diagonals are perpendicular bisectors
Rectangle (and square)
• diagonals are equal length
A
D C
B
AC BD

Quadrilaterals
square rectangle
trapezoid
area base height 
base base
height height
base2
base1
height
1 2base base
area height
2
average of bases height
 
  
 
 
parallelogram rhombus
base
height
base
height

Quadrilaterals
rhombus
1 2diagonal diagonal
area
2

4
7
area
2
28
2
14
4 7




Quadrilaterals
Angles in a quadrilateral add to 360°
parallelogram




rectangle
all angles are 90


rhombus

all sides are equal




square
trapezoid
2 sides parallel
area base height 

Polygons (watch the entire video here)

Polygons
Polygon: Closed figure formed by 3 or more line segments





Polygons
“polygon” “convex polygon” (all interior angles less than 180°)



Polygons
b
a
180a b c  
Triangle
Quadrilateral
Pentagon
c
b
a
c
d
360a b c d   
b
a
c
d 540a b c d e    
e
Hexagon
b
a
c
d 720a b c d e f     
ef

Polygons

 




The sum of the interior
angles in an N-sided polygon
is equal to  180 2N 
6
1
2
3
4
5
Octagon
 
 
 
sum of angles 180 2
8180 2
180
10
6
80
N 
 



Polygons
Regular polygon: equal sides and equal angles
regular pentagon
 



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 

Circles (watch the entire video here)

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”



Circles


 circumference 2 radius
2 r


  

Circumference
3.14
3
22
7
 


circumference diameter
d


 




Circles


circumference 2 r
Circumference
 
 
circumference 2
16 feet
16 3
48 f
8
eet






8 ft

Circles
Area


2
area r
 
2
2
area
6
8
4 ft




8 ft

Circles
 center

A
B

C

E

circumference diameter 
 circumference 2 radius  
3.14
3
 

2
area r
arc



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

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
2
area r
2
area
36
2
18
6
2



 
  
 
 
  
 

r
12
6
2
6 r
r



Pieces of Pi (watch the entire video here)

Pieces of Pi


C
E
1
of circumference
4
90
of circumference
360
CE 

90

Pieces of Pi

119

C
E
119
of circumference
360
CE 

Pieces of Pi

x

C
E
 
of circumference
360
2
360
CE
x
x
r


 arc length 2
360
x
r

Pieces of Pi
O

C
E
 2
ofarea circof sect le's area
3
r
60
o
360
O
x
x
C
r
E



?

Pieces of Pi

x

C
E
 2
ofarea circof sect le's area
3
r
60
o
360
O
x
x
C
r
E



O
 2
sector area
360
x
r
360
x

Pieces of Pi
O
160
6

Pieces of Pi
O
160
 2
area
360
x
r
 
 
2
6area
360
4
36
9
16
160






6

Pieces of Pi

x

C
E
 2
360
x
CE r

x

C
EO
 2
area
360
x
r

Practice Question
20
A)
3
25
B)
3
25
C)
2
40
D)
3
50
E)
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
w
x
y

20
A)
3
25
B)
3
25
C)
2
40
D)
3
50
E)
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 2
360
y
r
y
30
20x w 
180x w 
2 160
80
w
w


80 80
  
 
arc length 2
360
2
60
9
8
4
3
0
0
0
3






Circle Properties (watch the entire video here)


Circle Properties
A
B
x
“x is an inscribed angle holding/containing chord AB”

“x is an inscribed angle holding/containing arc AB”

Circle Properties

A
B
x
x
Inscribed angles holding the
same chord/arc are equal
x

Circle Properties

A
B
x


C
D
x
Inscribed angles holding chords/arcs
of equal length are equal

Circle Properties

An inscribed angle holding
the diameter is a right angle

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

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”

Circle Properties
*
*
*
*

x
2x


Practice Question

C
x
20
D
O
B
A
A) 40
B) 50
C) 60
D) 70
E) 80
If is the center and , thenO AB CD x 
E

Practice Question
C
x
D
O
B
A
A) 40
B) 50
C) 60
D) 70
E) 80
90
If is the center and , thenO AB CD x 

A
10
20
90
90
80
80
E

Volume & Surface Area (watch the entire video here)

Volume & Surface Area
1 ft
1 ft
1 ft3
1 ft 
2 ft
3 ft
5 ft
Volume length width height  
3
Volume 2 3 5
30 ft
  

Volume

 r
 height h
2
Volume r h
 3
2
Volume r h
10
 
2
3Vo 1lume
90
0



Volume

Surface Area
face
• 6 faces
• 12 edges
• 8 vertices

Surface Area
• 6 faces
• 12 edges
• 8 vertices
edge
edge
edge
edge

Surface Area
• 6 faces
• 12 edges
• 8 vertices




vertex
vertex
vertex
vertex
vertex

Surface Area
8 cm
4 cm
5 cm
2
surface area 40 40 32 32 20 20
184 cm
     


Surface Area
2
area r 2
area 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

length
volume length width height  
width
height
 r
2
volume 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

Units of Measurement (watch the entire video here)

Units of Measurement
• Metric: kilometers, kilograms, liters, etc.
• English: miles, pounds, gallons, etc.
What is the perimeter of this triangle?
12
13

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)

Geometry Data Sufficiency Questions (watch the entire video here)

Geometry Data Sufficiency Questions
A
B
C
x
• Do not estimate lengths and angles

1) 30x 
2) AD DC
What is the length of AD?
B
C
D

1) 30x 
2) AD DC
A
B
C
D
x
• To find one length requires at least one other length

1) 30x 
2) AD DC
INSUFFICIENT
A
B
C
D
INSUFFICIENT
1&2) 30 &x AD DC 
30
INSUFFICIENT
E

1) 10AC 
2) 30x 
If , what is the length of ?AE EC AB
A
B E
x
C
D

1) 10AC 
2) 30x 
A
B E
x
C
D
• Sketch figure and add information

1) 10AC 
2) 30x 
A
B E
x
C
D
x
10

1) 10AC 
2) 30x 
A
B E
x
C
D
x
10
• Mentally grab and move points and lines

1) 10AC 
2) 30x 
A
B E
x
C
D
10
x

1) 10AC 
2) 30x 
A
B E
C
D
INSUFFICIENT
INSUFFICIENT
30 30
• To find one length requires at least one other length

1) 10AC 
2) 30x 
A
B E
C
D
10
30 30
INSUFFICIENT
INSUFFICIENT
1 & 2) 10 and 30AC x SUFFICIENT
C

• Do not estimate lengths and angles
• To find one length, requires at least one other length
• Sketch diagram and add information

Geometry Strategies – Part II (watch the entire video here)

• 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
Geometry Strategies – Part II (watch the entire video here)

Practice Question
1 2
Are lines l1 and l2 parallel?
2) b d
a
b
c
d
e
1) 180e b 
ALONE is sufficient

Practice Question
2) b d
1) 180e b 
 
 
 
 
1 2
a
b
c
d
e
180 

1 2





SUFFICIENT
SUFFICIENT
D

Are lines l1 and l2 parallel?

Practice Question
Note: Figure not drawn to scaleA) 1
4
B)
3
3
C)
2
5
D)
3
5
E)
2
60
1x 
4 3x 
What is the value of x ?

Practice Question
Note: Figure not drawn to scaleA) 1
4
B)
3
3
C)
2
5
D)
3
5
E)
2
30
60
1
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
 

  
  
 



Practice Question
If is tangent to the circle with center , thenAC O DBC 

D
O
B CA

40
A) 50°
B) 55°
C) 60°
D) 65°
E) 70°

Practice Question
If is tangent to the circle with center , thenAC O DBC 

D
O
B CA

40
A) 50°
B) 55°
C) 60°
D) 65°
E) 70°
50
130
25
25
65

Practice Question
B
A C D
2) AC CD
1) 5BC 
If 12, does 90 ?AC ACB  
ALONE is sufficient

Practice Question
2) AC CD
1) 5BC INSUFFICIENT
B
If 12, does 90 ?AC ACB  
A C D
12
INSUFFICIENT
1 & 2)
12
5
INSUFFICIENT
E

Practice Question
What is the area of triangle ?ABC
60
5
12
A) 15
B) 15 3
5 119
C)
2
D) 32.5
E) 36
A
BC

Practice Question
What is the area of triangle ?ABC
60
A) 15
B) 15 3
5 119
C)
2
D) 32.5
E) 36
12
 3 6 36h  
6 3
5
A
BC
base height
area
2


5 6 3
area
2
30 3
2
15 3





Practice Question
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 
A) 22
B) 24
C) 26
D) 28
E) 30

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 

Practice Question
What is the value of ?x
 155 3x
 6 30x 
 4 70x 
A) 5
B) 7
C) 15
D) 21
E) 25

Practice Question
What is the value of ?x
 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
             
           
 



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

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
2
surface area 2 2r rh  
1
1
    
2
2 2
2 2
1 1 1
4
 
 

 
 


2
2
2
surface area 2 2r rh  
    
2
2 2
8 8
1
2 2 2
6
 
 

 
 

A
B
K
4K

Practice Question
2) AC AB
1) 8CB 
C
B
A
x
If the circle has radius 4, is 80?x 

Practice Question
2) AC AB
1) 8CB 
C
B
A
x
If the circle has radius 4, is 80?x 
SUFFICIENT
INSUFFICIENT
A

Practice Question
2) BE EA
1) 30BCE 
If ABCD is a rectangle, is the area of ∆EBC greater
than the area of ∆AEC ?
C B
AD
E

Practice Question
2) BE EA
1) 30BCE 
C B
AD
E
If ABCD is a rectangle, is the area of ∆EBC greater
than the area of ∆AEC ?
B E A
DC
harea
2
bh

Which triangle has the
longest base?INSUFFICIENT
SUFFICIENT
B

Practice Question
2
1) 14 48 0y y  
A C
B
55
y
hat is the area of ?W ABC
2
2) 16 60 0y y  
ALONE is sufficient

Practice Question
2
1) 14 48 0y y  
A C
B
55
y
hat is the area of ?W ABC
2
2) 16 60 0y y  
  
2
1) 14 48 0
6 8 0
6, 8
y y
y y
y
  
  

area 12
area 12
SUFFICIENT
h
  
2
2) 16 60 0
6 10 0
6, 10
y y
y y
y
  
  

area 12
The sum of the lengths
of any two sides of a
triangle must be greater
than the third side.
5 5 10 

SUFFICIENT
D

Practice Question
A
B
D
C
E F
If bisects , and bisects , thenBD CBE DE BEF w  
w
50
A) 25
B) 35
C) 50
D) 55
E) 65

Practice Question
A
B
D
C
E F
If bisects , and bisects , thenBD CBE DE BEF w  
w
50
x
x
y
y 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 


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

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
area
2
bh

  12 16
area
9
2
6


h
area
2
bh

20
2
96 10
.6
9
9
6
h
h
h


 EC

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 

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









ca
b
 2
6
60
6
360
6
6
6
a b
d e
b c
d f
 



   
 
 
 
 2
sector area
360
x
r
O 
e
P
f
d






        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


 


       
       
          
       
      
b
66
6
 
2
3 side
area
4


2
3
b 3
6
9
4

 
b d a b c d e f    

For additional practice questions, see the
bottom of our Geometry module
www.GMATPrepNow.com

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