First Round : March 27, 2010

Second Round: April 24, 2010 at The University of Alabama

GEOMETRY EXAMINATION
Construction of this test directed
by
Scott H. Brown, Auburn University Montgomery
and
and Zhiqin Shi, The University of Alabama

INSTRUCTIONS
This test consists of 50 multiple choice questions. The questions have not been arranged in order of difficulty.
For each question, choose the best of the five answer choices labeled A, B, C, D, and E.
The test will be scored as follows: 5 points for each correct answer, 1 point for each question left unanswered,
and 0 points for each wrong answer. (Thus a perfect paper with all questions answered correctly earns a
score of 250, a blank paper earns a score of 50, and a paper with all questions answered incorrectly earns a
score of 0.)
Random guessing will not, on average, either increase or decrease your score. However, if you can eliminate
one of more of the answer choices as wrong, then it is to your advantage to guess among the remaining
choices.
All variables and constants, except those indicated otherwise, represent real numbers.
Diagrams are not necessarily to scale.
We use the following geometric notation:
If A and B are points, then:
AB is the segment between A and B

AB is the line containing A and B

AB is the ray from A through B

AB is the distance between A and B

If A is an angle, then:
m A is the measure of angle A in degrees
If A and B are points on a circle, then:

AB is the arc between A and B

m AB is the measure of AB in degrees

Editing by Zhijian Wu, The University of Alabama

Printing by The University of Alabama

1. The angle of a sector is 96 and its arc is 24 in. Find the area of the circle (in square inches).
(A)

2025

1800

(B)

(C)

900

(D)

450

(E)

225

2. In the triangle ABC, AB = 16 in, AC = 9 in and the median from C is 11 in. Find the perimeter
of the triangle.
(A) 40 in

(B)

42 in

(C) 44 in

(D) 46 in

(E) 48 in

3. In the figure shown, each side of right triangle ABC is the diameter of a semicircle. The area of semicircle M is 100 and the area of semicircle N is 36.
Find the perimeter of ABC.

(A) 100 2
(B) 72 2
(C) 64 2
(D) 56 2
(E) 48 2

N
A

C
P
B

4. In the right triangle ABC with legs of length 5 and 12 and

the hypotenuse of length 13. A semicircle is inscribed in the
triangle. Find the radius of the semicircle.
17
(A)
3

(B)

13
5

10
3

(C)

12
(D)
5

(E)

7
3

13
A

5
C

12

5. The area of a right triangle is 64. What is the shortest possible length of the hypotenuse?

(A) 6 6
(B) 16
(C) 12
(D) 4 3
6. In the figure shown, two small circles are inscribed and tangent to the large
circle at points A and D. Let points B and C be the centers of the circles,
where AB = BC = CD = 1. Find the area of the shaded region.

5
3
13
3
7
3
(A)

(B)

(C)

12
2
24
2
12
2

11
3
11
3
(D)

(E)

24
2
12
2

(E) 6

E
A

7. The interior angles of a polygon are in arithmetic progression; the least angle is 120 and the common
difference is 5 . Find the greatest number of sides that this polygon can have.
(A) 20

(B)

16

(C) 12

(D) 9

(E) 7

8. Determine the equation of the line passing through the points of intersection of the circles (x + 2)2 +
(y + 1)2 = 9 and (x + 3)2 + (y + 4)2 = 16.
(A)

2x + 6y = 13

(C) 2x + 6y = 7

(B) 10x 6y = 7
(D) 10x + 6y = 25

(E) 2x 6y = 13

AE
G

9. You have three inscribed squares, with corners of each square at the 1/5 point
along the sides of its outer square. The area of the largest square is 625 in2 .
What is the area of the smallest square?
(A) 109 in2

(B) 125.2 in2

(C) 225 in2

289 in2

(D)

F
H

(E) 325.2 in2

C
Ellipse

10. A cylinder is sliced by a plane to form the figure shown. The base edge of
the figure is a circle of radius 4 in. The top edge is an ellipse. The highest
point on the ellipse is 10 in above the base and the lowest point on the
ellipse is 6 in above the base. Find surface area of this figure.
(A) 32 in2 (B) 64 in2 (C) 72 in2 (D) 90 in2 (E)

96 in2

11. In the diagram shown, a semicircle and a circle are placed inside a square with side
length of 4. Determine the radius of the circle.

(B) 6 2 3
(C) 8 4 3
(D) 2 + 3
(E) 3 + 3
(A) 1 + 3
A

12. In the trapezoid ABCD, where mDAB = 90 , a circle is inscribed as

shown. The circles point of tangency divides line segment CD into lengths
of 4 and 16. Find the area of the trapezoid.
(A) 576

(B)

288

(C) 144

(D) 72

(E) 36
B

C
D

13. An equilateral triangle ABC and a semicircle are attached in thefigure shown.
Let point D be halfway along the arc of the semicircle. If BD = 2 3, determine

the length of the arc AC.

(B) (5 + 3)
(A) (3 3)

(C) (7 2 3)
(D) (3 + 3)
(E) (5 3)

19
M (B)
9

25
M (C)
9

44
52
M (D)
M (E)
9
9

65
M
9

B
A

14. Two tangents to a circle form an angle equal to 50 . If the radius of

the circle is 10M, determine the length of the smaller of the intercepted
arcs.
(A)

10M
B

O
C

15. A regular hexagon has sides 8in long. Find the area of the triangle formed by connecting alternate
vertices.

(B) 48 3 in2
(C) 54 3 in2
(D) 6o 3 in2
(E) 72 3 in2
(A) 36 3 in2
16. What the exact area of the triangle whose vertices are (0, 2), (1, 4) and (2, 3) ?
(A) 3

(B) 2

1
2

(C)

1
2

(D) 1

(E)

1
2

17. In the ABC, OB bisects ABC, and AB = AC. If mABO = 2x + 20 and mCBO = 4x + 12,
what is mBAC ?
(A) 90

(B)

68

(C) 56

(D) 38

(E) 26

18. A chord is perpendicular to a diameter of a circle at a point which divides the diameter into segments
having the ratio 1 : 3. In what ratio does the chord divide the circumference ?
(A) 1 : 4

(B) 2 : 4

(C) 3 : 5

(D)

1:2

(E)

3:4

19. Hannah bought a box

in the
shape of a rectangular prism. She found the lengths of the diagonals of
the faces to be 20, 2 61, 2 89. Determine the volume of the box.
(A)

1920 U 2

(B) 1440 U 2

(C) 960 U 2

(D) 480 U 2

(E) 320 U 2
B

20. In the figure shown, four chords are drawn so that two intersects at point C
and chord AE is the diameter of the circle. If AB = 6, BC = 8, CD = 4 and
DE = 12, then what is the area of the circle ?
(A) 27 U 2

(B) 64 U 2

81 U 2

(C)

(D) 96 U 2

(E) 108 U 2

6
C 8
4 O

12

21. If six golf balls are packed tightly in a cylindrical can, then what fraction of the can is unoccupied ?
(A)

1
6

(B)

2
3

(C)

1
4

(D)

1
3

3
4

(E)

22. In value, the area of an equilateral triangle is equal to its perimeter. What is the length of the side
of the triangle ?

(A) 4 3 U
(B) 2 3 U
(C) 6 3 U
(D)
3U
(E) 3 3 U
23. Find the area of a trapezoid in which the bases are 17 F T and 42 F T and the legs are 15 F T, and
20 F T.
(A) 340 F T 2

(B)

354 F T 2

(C) 455 F T 2

(E) 708 F T 2

(D) 595 F T 2

24. Find the area of the shaded region shown in the figure where the petals are formed
by the semicircles.
(A)

32 64

(B) 32 32

(C) 32 16

(D) 2 8

(E) 2 4

25. The volume of the truncated cone in the figure shown is

(A) 36

(B) 64

(C) 81

(D) 144

6
(E)

None of these

12
9

26. What is the area of the shaded region shown in the figure if the diameter RP
of the circle is 8 ?
(A) 1.5

(B)

(C) 24

(D) 72

(E) 100

P
45
S

27. Rounded to the nearest hundred cubic meters, what is the total capacity (cone
and cylinder) of the storage container shown in the figure ?
(A)

1, 400

(B) 2, 000

(C) 5, 700

(D) 8, 100

8m

(E) 9, 400

6m
10m

28. In the figure shown, mC = 90 , AD = DB, DE AB, AB = 20

and AC = 12. Find the area of the quadrilateral ADEC.
(A) 75

(B)

58 21

(C) 37 21

(D) 48

(E) 36
A
C

29. In the triangle ABC, AE, BF, CD are medians. F H is parallel and
equal in length to AE. BH and HE are drawn, and F E is extended
and meets BH in G. Which one of the following statement is not
necessarily correct ?
(A) AEHF is parallelogram
(D) F G =

3
4 AB

(B)

HE = HG

H
E

(B) 10

(E) F G is a median of triangle BF H

(C)

15

(D) 12.5

D
A
E

(E) 20
B

31. In the figure shown, mP = 40 and P AB is formed by three

tangents to the circle O. Find mAOB.
(A) 45

(B) 50

(C) 55

(D) 160

(E)

70

B
S

40
P

A T
E

32. In the figure shown, CE and DE are equal chords of a circle with center O.

Arc AB is a quarter of the circle. Find the ration of the area of CED to the
area of AOB.

(A)
2:1
(B)
3:1
(C) 4 : 1
(D) 3 : 1
(E) 2 : 1

33. In the figure shown, ABCD is a rectangle with P , a point on AB, and
P S BD, P R AC, AF BD and P Q AF . What is the sum
P R + P S equal to ?
(A) P Q

(B) AE

(C) P T + AT

(D)

(C) BH = DC

30. In the figure shown, AB = AC, mBAD = 30 and AE = AD. Find x

(which is mEDC).
(A) 7.5

AF

C
A

P
T
Q

(E) EF
D

S
E

R
C

34. Two poles, 6inch and 18inch in diameters, are placed, as in the figure shown,
and bounded together with wire. The length of the shortest write that will go
around them is

(A) 12 3 + 16
(B) 12 3 + 7

(D) 12 12 + 15
(E) 24
(C) 12 3 + 14

18
6
C

35. In the figure shown, A, B, C are points of tangency to the circle

3
with diameter inches. Find X (in inches).
8
(A)

3
16

(B)

1
8

(C)

1
32

(D)

3
32

(E)

X
3/8

1
16

1/2
A

B
60
D

36. On a map whose scale is 400 miles to an inch and a half, a certain estate is represented by a rhombus
3
in. What is the area of the estate in square miles ?
having a 60 angle. The diagonal opposite 60 is 16

2500
1250
5625 3
(A)
(B)
(C) 1250
(D)
(E) 1250 3
2
3
3
C

37. In the ABC, AC = CD and mCAB mABC = 30 . Find

mBAD.
(A) 30

(C) 22 21

(B) 10

(D)

15

(E) 20
B

A
38. In the right triangle shown, the sum of distances BM and M A is equal to
the sum of distances BC and CA. If M B = x, CB = h and CA = d, then x
equals
(A)

hd
2h + d

(B) d h (C)

1
h2 + d2 h
d (D) h + d 2d (E)
2

M
x
B
h
C

39. In the triangle ABC, AB = AC, mA = 40 and point O is within the triangle with OBC = OCA.
Find mBOC.
(A) 140

(B) 35

(C)

110

(D) 55

(E) 70

40. In the triangle ABC, AC = 12. If one of the trisectors of angle B is the median to AC and the other
trisector of angle B is the altitude to AC, find the length of the altitude.

(A) 2 3
(B) 3 3
(C) 4 3
(D) 7 5
(E)
3
41. At what fraction of an hour after 3 oclock are the minute and hour hands of a twelve-hour clock
pointing in the same direction ?
(A)

3
4

(B)

3
7

(C)

3
11

(D)

3
17

(E)

3
19

42. Whats the area of a circle inside of which is inscribed a triangle with side lengths 5, 12, 13 ?
(A) 32.25

(B)

42.25

(C) 44.25

(D) 50.25

(E) 60

43. Circle B passes through the center of circle A and is tangent to it. Circle C passes through the center
of Circle B and is tangent to it. What fraction of the area of Circle A lies inside Circle B but outside
C?
(A)

1
4

(B)

1
8

(C)

1
16

(D)

3
16

(E)

17
32

44. A rhombus has a 60 degree angle. What is the ratio of its area to that of a circle inscribed inside it ?

7 3
7 2
8 3
(A)
(B)
(C)
(D) 4 3
(E) 5
3
3
3
45. In the triangle ABC, angle A is 120 degrees, BC + AB = 21 and BC + AC = 20. Whats the sum
of the three sides ?
(A)

28

(B) 21

(C) 24

(D) 13

(E) 20

46. In the triangle ABC, BC = 2 2, CA = 12 and mACB = 135. If D is the midpoint of AB, what
is the length of CD ?

2 21
2 26
(A)
(B)
21
(C)
(D)
26
(E) 2 29
3
3
47. In the figure shown, if the area of the letter L equals the area of the
triangle,
62 6
what is the length x of the ends of the L ? The right answer is
.
3

3 13
5 + 13
5 13
3 + 13
13
(A)
(B)
(C)
(D)
(E)
3
3
3
3

x
2
x
2

48. What
isthe maximum number of points with integer coordinates which can lie on a circle with center
at ( 3, 5) ?
(A) 0

(B) 3

(C) 2

(D)

(E) 4

49. A cube and sphere have the same surface area. What is the ration of volume of the sphere to that of
the cube ?
r
r
5
2
3
6
(A)
(B)
(C)
(D)
(E) 1

50. What is the radius of a circle inscribed in an isosceles right triangle whose legs have length 2 ?

(A)
2
(B) 2
(C) 2 3
(D) 2 3
(E) 2 2