Geometry Exam
2009 University of Houston Math Contest
Name: ____________________________
School: ____________________________
Please read the questions carefully and give
a clear indication of your answer on each
question.
There is no penalty for guessing.
Judges will use written comments and/or
calculations to settle ties.
Good luck.
Name: ________________________________ School: _______________________________
University of Houston
High School Mathematics Contest
Geometry Exam Spring 2009
Directions:
You have 60 minutes to complete this exam. Calculators are not permitted. Choose the correct
answer for each question, and write the letter (A, B, C, D or E) corresponding to that answer in
the blank to the right of each question. There is no penalty for guessing. In the case of a tie,
students work will be used to determine the winner so show all work clearly on either your
exam or on scrap paper. Write your name and school on top of each page in the blanks provided.
Note: Geometric figures in the problems may not be drawn to scale.
1.
2.
Find the center and radius of the circle defined by the equation ( x + 2 ) + ( y 7 ) = 9 .
2
(A)
Center ( 2, 7 ) , Radius 3
(B)
Center ( 2, 7 ) , Radius 9
(C)
Center ( 2, 7 ) , Radius 3
(D)
Center ( 2, 7 ) , Radius 9
(E)
Center ( 2, 7 ) , Radius 81
1. ________
If a circle has a circumference of 16 inches, find its area.
2. ________
(A) 8 in 2
3.
(C) 32 in 2
(D) 64 in 2
(E) 256 in 2
W hat can you conclude about about ABC and DEF , given that AB DE ,
BC EF , and A D ?
(A)
(B)
(C)
(D)
(E)
4.
(B) 16 in 2
The two triangles are similar.
The two triangles are congruent.
The two triangles are both scalene
The two triangles are both acute.
There is not enough information to draw a conclusion.
3. ________
Find the ratio of the edge of a cube to the cubes diagonal.
4. ________
(A)
(B)
2
2
1
(C)
3
(D)
(E)
3
3
Name: ________________________________ School: _______________________________
5.
Given that the statement If p, then q is false, which of the following statements must
always be correct?
I.
II.
III.
IV.
(A)
(B)
(C)
(D)
(E)
6.
If not p, then not q is false.
If p, then not q is true.
If q, then p is true.
If not q, then not p is false.
5. ________
I and II only
III only
I and III only
IV only
II and IV only
RT is an altitude of QRS , and mQRS = 90D . If ST = 5 and QT = 7 , find the length
of RS .
R
7.
(A)
35
(B)
2 15
(C)
2 21
(D)
74
(E)
2 6
6. ________
A chord of length 24 cm is drawn in a circle of radius 13 cm. Find the distance from the
chord to the center of the circle.
7. ________
(A) 25 cm
8.
(B) 11 cm
(C) 12 cm
(D) 5 cm
(E) 13 cm
Find the measure of an angle satisfying the following:
Six times the complement of an angle is five less than the supplement of the angle.
(A)
(B)
(C)
(D)
(E)
25D
71D
73D
35D
26 3 7 D
University of Houston
High School Mathematics Contest
8. ________
Geometry Exam
Spring 2009
Name: ________________________________ School: _______________________________
9.
Isosceles triangle TRI has base angles T and I and median RM . If the perimeter of
TRI is 54 inches and TM = 4 inches, find the length of TR .
(A)
(B)
(C)
(D)
(E)
10.
11.
8 inches
38 inches
23 inches
25 inches
46 inches
9. ________
EF is the median of trapezoid ABCD, with measures listed as below. Find the length of
AB .
(A)
(B)
13
(C)
21
4
(D)
17
(E)
21
x+8
10. ________
2x + 7
5x 4
C
In the circle below, chords AC and DB intersect at point E. If m p
AD = 73D and
p = 55D , find mAED .
mBC
D
36.5D
55D
64D
73D
128D
(A)
(B)
(C)
(D)
(E)
73
11. ________
A
E
12.
55
Let A represent the line which passes through the point ( 8, 9 ) and is perpendicular to
2
the line y = x + 6 . Find the x-intercept of A .
5
(A)
58
5
(B)
University of Houston
High School Mathematics Contest
29
2
(C) 11
12. ________
(D) 15
(E)
22
5
Geometry Exam
Spring 2009
Name: ________________________________ School: _______________________________
13.
A spherical balloon has volume 240 in 3 . If 20% more air is blown into the balloon (and
it still retains its spherical shape), find the radius of the enlarged balloon.
13. ________
(A) 2 3 9 in.
14.
(B) 6 in.
(C) 6 2 in.
(D)
36 in.
(E)
6 3 180
in.
5
Find the area of a hexagon with apothem 6 inches.
14. ________
(A) 36 3 in 2
15.
(C) 72 3 in 2
(D) 216 3 in 2
(E) 144 in 2
Below is a net that can be used to create a solid, formed by three congruent rectangles
and two congruent equilateral triangles. How many vertices and edges does the solid
have?
(A)
(B)
(C)
(D)
(E)
16.
(B) 72 in 2
10 vertices and 14 edges
18 vertices and 18 edges
6 vertices and 14 edges
9 vertices and 9 edges
6 vertices and 9 edges
15. ________
A construction can be found below with all steps shown except for any final straightedge
marking(s). Which of the following constructions is being performed?
(A)
(B)
(C)
(D)
(E)
Angle bisector
Equilateral triangle
Segment bisector
Line through a point parallel to a given line
Construct a circle given 3 points
A
University of Houston
High School Mathematics Contest
16. ________
B
Geometry Exam
Spring 2009
Name: ________________________________ School: _______________________________
17.
In spherical geometry, a line is defined to be a great circle of the surface of a sphere. Two
lines are parallel in spherical geometry if they never intersect.
Given line A on a sphere along with the above definitions, how many lines on the sphere
are parallel to line A ?
17. ________
(A) 0
18.
(B) 1
(C) 2
(D) 4
(E) An infinite number
In the diagram below, CD & EF , m1 = ( x 7 ) , m2 = ( 3x ) , and m3 = ( 9 x + 19 ) .
D
Find the value of x.
G
D
1
2
18. ________
3
E
(A)
19.
161
12
(B) 14
(C)
84
5
(D)
3
2
(E)
168
13
Given regular octagon ABCDEFGH, remove vertex G and use the remaining vertices to
form heptagon ABCDEFH. Find the measure of F .
(A)
45 D
(B)
157.5 D
(C)
128 4 7 D
(D)
135D
(E)
112.5 D
University of Houston
High School Mathematics Contest
19. ________
Geometry Exam
Spring 2009
Name: ________________________________ School: _______________________________
20.
AC = 120D and
BE is a secant of circle G and BF is tangent to circle G at point C. If m p
the diameter of circle G is 10 cm, find the length of BC .
(A)
(B)
(C)
(D)
(E)
5 3 cm
5 2 cm
10 cm
5 cm
30 cm
20. ________
D
B
C
F
21.
Quadrilateral ABCD is inscribed in a circle, with mC = 70D . Find the measure of A .
21. ________
(A) 70D
22.
23.
(B) 110D
(C) 145D
(D) 140D
(E) Can not be determined
CE and BD are angle bisectors of ABC which intersect at point F. If mBFC = 110D ,
find the measure of A .
(A)
40D
(B)
55D
(C)
70D
(D)
110D
(E)
Cannot be determined
22. ________
o
110
F
A sector with central angle 80D and radius 12 has been removed from a figure, as shown
below. Find the area of the portion of the circle which remains.
(A)
(B)
(C)
(D)
(E)
56
3
56 + 72
3
32
128
112
University of Houston
High School Mathematics Contest
23. ________
12
80o
Geometry Exam
Spring 2009
Name: ________________________________ School: _______________________________
24.
Triangle ABC has medians AE , BF , and CD which intersect in point G as shown
below. Given that AF = DB , BC = 10 , and FG = 11 , find the length of AE .
B
(A)
(B)
(C)
(D)
(E)
4 6
5 3
3 11
21
3 19
24. ________
D
G
A
F
C
25.
Circle D is inscribed in right triangle ABC with right angle C, and AB is tangent to the
circle at G. If CB = 12 and GB = 7 , find AC .
C
(A)
(B)
(C)
(D)
(E)
23
27
30
35
37
26.
25. ________
D
Two pyramids are similar. The lateral area of the first pyramid is 240 cm2, and the lateral
area of the second pyramid is 540 cm2. If the larger pyramid has a volume of 810 cm3,
find the volume of the smaller pyramid.
(A)
640
cm3
9
(B)
540 cm3
(C)
360 cm3
(D)
720 cm3
(E)
240 cm3
University of Houston
High School Mathematics Contest
26. ________
Geometry Exam
Spring 2009
Name: ________________________________ School: _______________________________
27.
A company manufactures dice in large quantities and ships them to other game
companies. The manufacturer gets a cube-shaped cardboard box ready to send to a
customer, and fills it with 512 dice. If the dice are stacked neatly in rows in such a way
that they fill the entire volume of the box, how many dice are touching at least one face
of the cardboard box?
(A)
(B)
(C)
(D)
(E)
28.
296
216
256
384
512
27. ________
A cube is shown in Figure 1 below. In Figure 2, vertices A, B, C, and D of the same cube
are shown as vertices of a right triangular pyramid. If the volume of the cube is 54 cm3,
find the volume of the right triangular pyramid.
(A)
18 cm3
(B)
12 cm3
(C)
36 cm
(D)
9 cm
(E)
27 cm3
28. ________
A
D
Figure 1
29.
D
Figure 2
Rectangle ABCD is shown below with CD = 10 and AD = 8 . If E is the midpoint of
BC , G is the midpoint of AB , F is a point on CD , and the area of EFG is 18, find the
length of DF .
F
(A)
(B)
(C)
(D)
(E)
8
7
4
5
6
University of Houston
High School Mathematics Contest
29. ________
E
B
Geometry Exam
Spring 2009
Name: ________________________________ School: _______________________________
30.
A building is made from stacked cubes, and views from three different perspectives are
illustrated below. There are exactly ten stacks that are one unit high.
Top View:
Right View:
Right
Front View:
Front
30. ________
How many cubes are needed to construct this building?
(A) 22
31.
(C) 25
(D) 28
(E) 33
Which of the following polygons will tessellate the Euclidean plane?
I.
II.
III.
IV.
(A)
(B)
(C)
(D)
(E)
32.
(B) 24
Every pentagon
Every triangle
Every hexagon
Every quadrilateral
31. ________
II only
III only
IV only
II and IV only
None of the above
Circle A and Circle B have common internal tangent FG which intersects AB at E. If
circle A has radius 10, circle B has radius 8, and AB = 27 , find the length of BE .
(A)
(B)
(C)
(D)
(E)
4
8
9
10
12
F
B
32. ________
E
A
University of Houston
High School Mathematics Contest
Geometry Exam
Spring 2009
Name: ________________________________ School: _______________________________
33.
A square pyramid is made of foam and has base edge 12 cm and height 8 cm. A knife
slices through the apex perpendicular to the base and through two opposite vertices of the
base (cutting along one of the diagonals of the base), forming two distinct congruent
polyhedra. Find the total surface area of one of the resulting polyhedra.
(A)
(B)
(C)
(D)
(E)
34.
(312 + 96 2 ) cm
(192 + 24 17 ) cm
( 264 + 60 2 ) cm
(192 + 48 2 ) cm
33. ________
2
Triangle ABC exists on a coordinate plane, and the midpoints of its sides are located at
the points ( 2, 1) , ( 3, 7 ) , and ( 5, 4 ) . Find the area of triangle ABC.
(A) 10
35.
240 cm 2
(B) 15
(C) 20
(D) 30
(E) 36
34. ________
In the diagram below, an equilateral triangle with side length 4 inches is shown with both
its inscribed and circumscribed circles. Find the area of the gray shaded region.
(A)
(B)
(C)
(D)
(E)
4
in 2
3
8 4 3 2
+
in
3
9
16 2 3 2
+
in
3
9
8 3 8 2
in
9
3
35. ________
8 3
in 2
3
END OF EXAM
(Make sure all answer blanks are filled in with the letter of the correct answer.)
University of Houston
High School Mathematics Contest
Geometry Exam
Spring 2009
