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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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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Lines and Angles (watch the entire video here)


http://www.gmatprepnow.com/module/gmat-geometry/video/858

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

BC

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

BA Q




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




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




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 Cw x

y180w x y

Angles in a triangle add to 180°





A

B C21

44180w x y


w





A

B C21

44180w x y


w

180

180

1

2 4

5

4

1

1

65

w

w

w





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





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 3 sideA B A B


A

B

C

a

b

c

If then 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

If then a b c A B C

2) 3QR

3

1&2)

6


rd 3 sideA B A B

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

2243

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

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


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

2243

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

40O

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 ?

OAB 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

If then 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

48

Area

- ft 2

- cm 2

- m 2





A

B C

base heightArea

2

10

48

Area

1Area base height

2





A

B C

base heightArea

2

10Area

15

3

2

10

34

8

altitude height

Area





10

4

8

A B

C

7.5

base heightArea

2

7A

.re

5a

15

4

2

Area





A

B

C60 60

60

2

3 sideArea

4





A

B

C60 60

60

2

3 sideArea

4

6 6

6

2

3Area

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 heightArea

2

2

3 sideArea

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 2a b c

a

bc

2 2 2a b c

a

bc 2 2 2a b c

a

bc




Right Triangles

8

6

x




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




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




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




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




Right Triangles

8x 17

15

• 8-15-17

2 2 215 17x

2 2 2a 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

725 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

. . .

. . .

. . .

34

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 2a b c

2 2 2

2

2

3 4

9 16

7

7

x

x

x

x

. . .




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




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


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




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

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



Practice Question

A) 21

B) 9

C) 2 21

D) 149

E) 3 21



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



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





1230

x

y





1230

x

y 30

601

2

3

60

enlargement factor: 6





1230

x

y 30

601

2

3

60


61

6

x

3

6 3

6y





5 2

x

5 2





5 2

x

5 2

45

12

451

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

12

451

30

601

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 , then AD 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 , then AD BD AB ABC x


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




Similar Triangles (watch the entire video here)



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




Similar Triangles

**

x

5 7

9

6




Similar Triangles

**

x

5 7

9


5

5

63

6

5

3

5

7 9

7

9

x

x

x

x

6




Similar Triangles

Similar triangles have the same 3 angles in common

40 20120

40 20

120


wa

b c xy

a

w

b c

x y




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


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




Quadrilaterals (watch the entire video here)



Quadrilaterals

Angles in a quadrilateral add to 360°

A

D C

w

x

y360w x y z

B

z




Quadrilaterals

square

rectangle

trapezoid

parallelogram

rhombus




Quadrilaterals

parallelogram

opposite sides parallel

rectangle


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 basearea height

2

average of bases height

parallelogram rhombus

base

height

base

height




Quadrilaterals

rhombus

1 2diagonal diagonalarea

2

4

7

area2

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

d360a b c d

b

a

cd 540a b c d e

e

Hexagon

b

a

cd 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

23

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

2area r

2

2

area

6

8

4 ft

8 ft




Circles

center

A

B

C

E

circumference diameter

circumference 2 radius

3.14

3

2area 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

2area 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

1of circumference

4

90of circumference

360

CE

90




Pieces of Pi

119

C

E

119of circumference

360CE




Pieces of Pi

x

C

E

of circumference360

2360

CEx

xr

arc length 2360

xr




Pieces of Pi

O

C

E 2

of area circ of sect le's area3

r60

o

360

Ox

x

C

r

E

?




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




Pieces of Pi

O

160

6




Pieces of Pi

O

160

2area360

xr

26area360

436

9

16

160

6




Pieces of Pi

x

C

E

2360

xCE r

x

C

EO

2area360

xr




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


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




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 , then O AB CD x

E


Practice Question


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




Volume & Surface Area (watch the entire video here)



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





r

height h

2Volume r h

3

2Volume 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

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





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




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

Cx

• 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

xC

D





1) 10AC

2) 30x


A

B E

xC

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 2Are lines l1 and l2 parallel?

2) b da

bc

d

e

1) 180e b







Practice Question

2) b d

1) 180e b

1 2

a

bc

d

e

180

1 2

SUFFICIENT

SUFFICIENT

D

Are lines l1 and l2 parallel?




Practice Question

Note: Figure not drawn to scale A) 1

4B)

3

3C)

2

5D)

3

5E)

2

60

1x 4 3x

What is the value of x ?


Practice Question


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




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°


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




Practice Question

B

A C D

2) AC CD

1) 5BC

If 12, does 90 ?AC ACB







Practice Question

2) AC CD

1) 5BC INSUFFICIENT

B

If 12, does 90 ?AC ACB

A C D12

INSUFFICIENT

1 & 2)

12

5

INSUFFICIENT

E




Practice Question

What is the area of triangle ?ABC

60

5

12


B) 15 3

5 119C)

2

D) 32.5

E) 36

A

BC


Practice Question

What is the area of triangle ?ABC

60


B) 15 3

5 119C)

2

D) 32.5

E) 36


12

3 6 36h

6 3

5

A

BC

base heightarea

2

5 6 3area

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


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

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




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

harea2

bh

Which triangle has the longest base? INSUFFICIENT

SUFFICIENT

B




Practice Question


21) 14 48 0y y

A C

B

55

y

hat is the area of ?W ABC

22) 16 60 0y y







Practice Question


21) 14 48 0y y

A C

B

55

y

hat is the area of ?W ABC

22) 16 60 0y y

21) 14 48 0

6 8 0

6, 8

y y

y y

y

area 12

area 12

SUFFICIENT

h

22) 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

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


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




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

area2

bh

12 16area

9

2

6

h

area2

bh

20

2

96 10

.6

9

9

6h

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

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





For additional practice questions, see the bottom of our Geometry module

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http://www.gmatprepnow.com/module/gmat-geometry

http://www.gmatprepnow.com/module/gmat-geometry


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