March 31, 2009
2009 State Math Contest
Wake Technical Community College
Geometry Test
1. A popular platform in skateboarding is the half-pipe pictured to the right.
Assume the side decks are 2 yards deep, the flat bottom is 8 feet wide,
and the radius of the pipe is 8 feet (see cross section below). If the width
of a half-pipe is 12 feet, what is its surface area to the nearest tenth of a
square foot?
a. 541.6 sq ft
b. 542.3 sq ft
c. 445.6 sq ft
d. 448.8 sq ft
e. 469.6 sq ft
2. Ted’s hamster has a circular exercise wheel of radius 3.5 inches. One day Ted observed his hamster
exercise and noticed it took a second for the wheel to make a complete turn. What is the hamster’s
average speed in miles per hour to the nearest hundredth? (Note: There are 63360 inches in a mile).
a. 2.15 mph
b. 1.35 mph
c. 1.75 mph
d. 1.25 mph
e. 1.15 mph
3. In quadrilateral ABCD, E is the midpoint of AB , F is the midpoint of BC , G is the midpoint of CD ,
and H is the midpoint of DA . Which of the following must be true?
a. ∠FEH = ∠FGH
b. ∠FEH = ∠EHG
d. both a and c
c. ∠FEH + ∠EHG = 180°
e. both b and c
4. Jackie is fencing in a square area of 576 square feet. Fence posts, which are needed every three feet,
cost $32 each. The fencing costs $4.50 per foot. What is the total cost of the fencing material?
a. $1328
b. $1520
c. $1488
1
d. $1424
e. $1456
2009 State Math Contest
Wake Technical Community College
Geometry Test
5. The Washington Monument in Washington, DC, is 555. 46 feet tall and is casting a shadow 80.8 ft. At
the same time a person is casting a shadow 9.5 inches long. To the nearest tenth of an inch how tall is
the person?
a. 64.3 inches
b. 65.3 inches
c. 63.5 inches
d. 63.4 inches
e. 65.4 inches
6. Five murder suspects, including the murderer, are being interrogated by the police. Results of a
polygraph indicate two of them are lying and three are telling the truth and are displayed below. If the
polygraph results are correct, who is the murderer?
Suspect A:
Suspect B:
Suspect C:
Suspect D:
Suspect E:
a. A
“D is the murderer.”
“I am innocent.”
“It wasn’t E.”
“A is lying.”
“B is telling the truth.”
b. B
c. C
d. D
e. E
7. A 30 cm by 40 cm page of a book includes a 2-cm margin on each side. What percentage of the page do
the margins occupy?
a. 18%
b. 20%
c. 22%
d. 24%
e. 26%
8. What is the measure of the acute angle formed by the hands of a clock when the clock shows the time
12:15 am?
a. 90°
b. 88.5°
c. 82.5°
d. 80°
e. 86.5°
9. A circle is divided into four equal sectors by four radii, as shown. What is the ratio of the
perimeter of one such sector to the circumference of the circle?
a.
1
4
b.
1 4
+
4 π
c.
1 2
+
4 π
2
d.
1
4+π
e.
1 1
+
4 π
2009 State Math Contest
Wake Technical Community College
Geometry Test
10. Susie loved her granddad, because he made everything into a math problem. One day she asked him,
“How old are you?” Granddad replied, “Behold this solid cube! If you multiply the number of edges by
five, add to this four times the number of faces, and subtract twice the number of its vertices, you have
my age.” What is Granddad’s age?
a. 68
b. 64
c. 60
d. 56
e. 48
11. The perimeter of a rectangle is 36 ft and a diagonal is 170 ft. What is its area in square feet?
a. 78 sq ft
b. 76 sq ft
c. 75 sq ft
d. 77 sq ft
e. 72 sq ft
12. A cone has a circular base with a radius of 4 cm. A slice is made parallel to the base of the cone so that
the new cone formed has half the volume of the original cone. What is the radius of the base of the new
cone?
a. 23 2 cm
b. 23 4 cm
c. 2 2 cm
d. 2cm
e. 1cm
13. In hexagon ABCDEF, all interior angles are 120°. If AB = CD = EF = 50 and BC = DE = FA = 100, find
the area of the triangle bounded by BE, CF, and AD to the nearest tenth.
a. 1082.5
b. 1082.9
c. 1083.3
d. 1083.5
e. 1083.9
14. The radii of the circles on the target pictured to the right are 1, 3, and 4 inches.
What is the probability that a random shot that hits the target will be outside the
inner circle, but inside the second circle (the white area)?
a.
1
4
b.
1
3
c.
1
2
d.
3
1
2π
e.
1
4π
2009 State Math Contest
Wake Technical Community College
Geometry Test
15. In square ABCD let M be the midpoint of DC. Let R be the intersection point of diagonal AC and line
segment BM. Find the ratio of the area of triangle CRM to the area of quadrilateral ADMR.
a.
1
4
b.
1
5
c.
1
2
d.
1
3
e.
2
5
16. A point (a, b ) is a lattice point if both a and b are integers. A lattice point is called visible if the line
segment from (0,0 ) to it does NOT pass through any other lattice points. Which of the following lattice
points is visible?
a.
(28,14)
b. (28,15)
c. (28,16 )
d. (28,18)
e. (28,21)
17. What is the measure of an interior angle of a regular 15-sided convex polygon?
a. 156°
b. 144°
c. 160°
d. 136°
e. 165°
18. What is the volume of a tetrahedron of side length 4 inches to the nearest tenth of a cubic inch?
a. 7.9 in³
b. 8.1 in³
c. 7.3 in³
d. 8.5 in³
e. 7.5 in³
19. Three faces of a rectangular box have a common point, which is a corner of the box. The centers of
these faces are the vertices of a triangle with sides of length 4, 5, and 6 cm. What is the volume of the
box to the nearest cubic centimeter?
a. 78 cm³
b. 110 cm³
c. 220 cm³
4
d. 125 cm³
e. 170 cm³
2009 State Math Contest
Wake Technical Community College
Geometry Test
20. In triangle PQR, S is a point on side PQ so that PR = 35, PS = 11 and RQ = RS = 31. What is the length
of SQ?
a. 10
b. 11
c. 12
d. 13
e. 14
21. Quadrilateral ABCD is a trapezoid inscribed in a circle. In addition, AD is a diameter, AD is parallel to
BC , and m∠ADC = 75° . If AD = 9 inches, how long to the nearest tenth of an inch is arc BC?
a. 8.9 in
b. 9.4 in
c. 8.8 in
d. 9.1 in
e. 9.0 in
22. Granddad’s cake is round with icing all over, including down the side. Some weight-conscious family
members demand “inside” pieces that only have icing on the top. If Betty makes exactly six straight
cuts across the cake, what is the maximum number of “inside” pieces she can produce?
a. 6
b. 7
c. 8
d. 9
e. 10
23. The diagram shows parts of two circles of radius 1 unit. One has
a center at (0,0 ) . The other has center at (1,0 ) . What is the area
of the shaded region? That is the area inside the circle with
center (1,0 ) above the x-axis, but outside the other circle.
a. 1 sq unit
b.
π
3
sq units
c.
π
6
−
3
sq units
4
5
d.
π
6
+
3
sq units
4
e.
π
3
+
3
sq units
4
2009 State Math Contest
Wake Technical Community College
Geometry Test
24. Summit Plummet at Disney World is billed as the most thrilling freefall waterslide in the country. It is
120 feet high and the slide is 351 feet long. Daredevil Drop, the freefall waterslide at Emerald Point, is
76 feet high. What would the length of the slide need to be if Daredevil Drop is similar to Summit
Plummet?
a. 222.3 ft
b. 235.4 ft
c. 228.6 ft
d. 236.2 ft
e. 228.5 ft
25. Thom was bored one snow day and decided to make up what he called Phone Math. He told his Mom
that BAT + NET = TON. She said the two legs of a right triangle are AS and DO. Using your
knowledge of the phone which of the following words would represent the hypotenuse?
a. AL
b. OK
c. IL
d. EL
6
e. UK
2009 State Math Contest
Wake Technical Community College
Geometry Test
SHORT ANSWER
Place the answer in the appropriate space.
66. A triangle is cut from the corner of a rectangle. The resulting pentagon has sides of length 8, 10, 13, 15,
and 20 inches, although not necessarily in that order. What is the area of the pentagon?
67. A pyramid has a square base 6 m on a side and a height of 9 m. Find the volume of the portion of the
pyramid which lies above the base and below a plane parallel to the base and 3 m above the base.
68. Point O is the center of a pair of concentric circles of radius 1 cm and 4 cm, pictured below. Find the
angle in degrees of the sector such that the area between the inner and outer circle (shaded below) is 1
the area of the larger circle.
O
69. In some military applications, angles are measured in mils. The arc length cut along a circle of radius
1000 meters by a 1 mil central angle is approximately 1 meter. Using this measurement approximate
how many mils are in a 45° angle to the nearest mil?
70. Two trees of heights 20 ft and 30 ft have ropes running from the top of one to the bottom of the other.
How high above the ground do the ropes intersect?
30 ft
20 ft
7
6
2009 State Math Contest
Wake Technical Community College
Geometry Test
1. a
2. d
3. d
4. e
5. b
6. e
7. c
8. c
9. e
10. a
11. d
12. b
13. a
14. c
15. b
16. b
17. a
18. e
19. c
20. d
21. b
22. e
23. d
24. a
25. c
66. 270 sq in
67. 76 cu m
68. 64°
69. 785 mils
70. 12 ft
8
2009 State Math Contest
Wake Technical Community College
Geometry Test
1. (6 + 4π + 8 + 4π + 6) *12 = 541.6 sq ft
2.
7π
3600
*
≈ 1.25 mph
63360
1
3. EF is parallel to AC and HG , because the transversals are cut into proportional segments. Similarly,
FG is parallel to BD and EH . This makes quadrilateral EFGH a parallelogram. Hence, both a and c
are true.
4. The area has sides of length 24 feet. Hence the fencing will cost 4(24 )(4.5) + 32(32 ) = $1456 .
5.
x
555.46
=
⇒ x = 65.3 inches.
80.8
9.5
6. If suspect A is lying, then suspect D is telling the truth. If suspect A is telling the truth, then suspect D is
lying. Hence, one of the liars must be either A or D. If suspect B is lying, then suspect E is also lying.
This would give 3 liars so suspect B must be telling the truth and so is suspect E. That leaves suspect C
as the other liar. Since C said “It wasn’t E.”, then E must be the murderer.
7.
2(2)(30) + 2(2)(36)
= 0.22 or 22%
30(40)
8. The hour hand moves
360
= 0.5° every minute. Hence, at 12:15 the angle is 90° − 0.5°(15) = 82.5° .
60(12)
9. The circumference of the circle is 2π r and the perimeter of one sector is 2r + 0.5rπ . Hence the ratio is
1 1
+ .
4 π
10. 5(12 ) + 4(6 ) − 2(8) = 68
11. Let x be one side of the rectangle and 18 − x be the other side, then x 2 + (18 − x ) = 170 . This equation
gives x either 7 or 11. Hence the rectangle is 77 square feet.
2
12. Let r be the radius of the new cone, h be the height of the original cone, and h1 be the height of the new
cone. Then
h h1
hr
16hπ 2r 2π hr
and
=
. Solving for r we get 23 4 cm.
= ⇒ h1 =
4 r
4
3
3 4
13. Using properties of parallel lines it can be proven that the triangle bounded by BE, CF, and AD is an
equilateral triangle of side length 50. Hence its area is 1082.5 to the nearest tenth.
14.
9π − π
8π
1
=
= .
16π
16π 2
9
2009 State Math Contest
Wake Technical Community College
Geometry Test
15. Let X be the area of triangle ABR, W be the area of triangle BRC, Y be the
area of triangle CRM, and Z be the area of quadrilateral ADMR. Then we get
the equations X + W = Y + Z , X = 4Y , and X + Z = 3(W + Y ) . Hence, the
relationship between Y and Z is 1 to 5.
16. A lattice point is visible if and only if its coordinates are relatively prime to one another. Thus, (28,15)
is a visible lattice point.
360°
= 24° . So, the interior angle is 180° − 24° = 156° .
15
17. The central angle is
s3 2
. Thus, the volume is approximately 7.5
18. The volume of a tetrahedron is given by the formula V =
12
cubic inches.
19. Let the sides of the box be represented by 2x, 2y, and 2z. We can get the equations x 2 + y 2 = 4 2 ,
5
27
45
; and z 2 =
.
x 2 + z 2 = 5 2 , and y 2 + z 2 = 6 2 . Solving these equations gives x 2 = ; y 2 =
2
2
2
Hence, the volume of the box is 220.45 cubic centimeters.
20. Let 2x be the length of SQ and construct the height of triangle PQR. Then use the Pythagorean Theorem
2
to get 35 2 − x 2 = 312 + (11 + x ) . So the length of side SQ is 13.
21. Arc AC measures 150° so arc CD measures 30° and m∠DAC = 15° . Since AD is parallel to BC we
120(9π )
get that m∠ACB = 15° . Hence arc AB measures 30° and arc BC measures 120°. Thus,
≈ 9.4
360
inches.
22. If Betty makes sure that each new cut crosses every previous cut, then she will have 10 inside pieces.
23. The semicircle centered at (1,0) has area
diagram have area
π
6
π
2
. The sectors drawn in the
and the equilateral triangle has area
the area of the shaded region is
π
2
−
24. Let x be the length of the slide, then
π
6
−
π
6
+
3
. Hence
4
3 π
3
.
= +
4
6
4
351 x
=
. So the length of the slide is 222.3 ft.
120 76
25. AS = 27 and DO = 36 so the hypotenuse must be 45 or IL.
10
2009 State Math Contest
Wake Technical Community College
Geometry Test
66. The sides of the rectangle must be 15 and 20. The only way the other three sides work to give a right
triangle is if the hypotenuse of the triangle cut off is the 13, the side of length 20 had 12 inches cut off
(leaving 8 inches)and the side of length 15 had 5 inches cut off (leaving 10 inches). Hence, the area of
the pentagon is 300 – 30 = 270 square inches.
67. Using similarity the pyramid above the parallel plane has side of length 4 m. Hence, the volume below
36(9 ) 16(6 )
the parallel plane is
−
= 76 cubic meters.
3
3
68. Let x be the measure of the angle in degrees. Then
69. Let x be the number of mils in a 45° angle. Then
16π x(16π − π )
=
so x = 64°.
6
360
45°
x
=
so x is approximately 785 mils.
2000π 360°
70. Let h be the total distance between the two trees and x be the height above the ground where the ropes
2h
x
x 3h 5
5
=
=
or
. Either way x is 12 feet.
intersect. Using similar triangles
20 h
30
h
11