gre geometry - everything you need to know geometry... · gre geometry - everything you need to...

GRE Geometry - Everything you need to know

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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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Lines and Angles


(watch the entire video here)


http://www.greenlighttestprep.com/module/gre-geometry/video/858

Lines and Angles

l

line: a straight path that extends without end in both directions





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

50x






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





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





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?

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






Given lengths of sides A and B

rd 3 sideA B A B


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.





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





Practice Question


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





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





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





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

Solution #2





Assumptions in Geometry





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


• y is greater than x





• Do not make assumptions about parallelism

1

2

1 2







A

BC

ABC

• Unless otherwise indicated, ABC refers to the smaller angle.






A

BC

• Unless otherwise indicated, ABC refers to the smaller angle.

ABC





• Not necessarily drawn to scale

• Avoid visual estimation


Geometric Figures






• 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






Geometry Strategies





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





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

104






38

38

104

40

x

x

40 180

2 40 180

2 140

70

x x

x

x

x






A

B

C

Equilateral triangle

• 3 equal sides, 3 equal angles

60 60

60






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

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





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

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

. . .

. . .

. . .

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





Practice Question

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

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






60 60

60

Square Equilateral Triangle






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

Note: Figure not drawn to scale

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





Similar Triangles

Similar triangles have the same 3 angles in common

40 20120

40 20

1203 30 1 55.6 60x


5.6





Similar Triangles


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


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





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





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



is equal to 180 2N





Circles





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


The area of circle A is twice the area of circle B.

The radius of circle A The diameter of circle B





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





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





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

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





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





Circle Properties





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






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






8 cm

Surface Area

4 cm

5 cm

2

surface area 40 40 32 32 20 20

184 cm






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





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 Strategies – Part II





• 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


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



Practice Question


Area of an equilateral triangle with each side

having length 4k

Area of a circle with diameter 2k









Practice Question


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





Practice Question

A) 1

4B)

3

3C)

2

5D)

3

5E)

2

60

1x 4 3x

What is the value of x ?





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





Practice Question


Perimeter of ABC 10

A

B

C

3

4

ABC is an isosceles triangle









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





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


5x2x y

7x y









Practice Question


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





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





Practice Question

What is the area of triangle ?ABC

60

A) 15

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





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


(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


(watch the entire video here) Practice Question



Practice Question


Volume of spherewith radius 2

32









Practice Question


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





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


Practice Question

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

In the regular octagon shown here, what is the value of x?

x degrees





Practice Question



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





Practice Question

The diagonal of square has length 3.23ABCD

Area of square ABCD 3.23










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





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

A

B

C

D

E


x y

2

3

4

5

ky

x

k









Practice Question


x y

AB DE

x y k

A

B

C

D

E

2

3

4

5

xy

ky

k

B





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.


's distance from the originO

5

O









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.


's distance from the originO

A

O

4

4

d

4

4

3

4

5 5

5 5

5





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





2 14 48 0y y

A C

B

y


Area of .ABC 12

55



Practice Question







2 14 48 0y y

A C

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)




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


Area of .ABC 6

In the rectangular coordinate plane, ABC has vertices at (0,-2), (0,1) and (4,b), where b<0.









Practice Question


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





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




http://www.greenlighttestprep.com/module/gre-geometry



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