Grissom Math Tournament
Geometry Test
April 11, 2009
1. The measure of an angle is 30º more than twice its complement. Find the measure of the angle.
A. 20º
B. 35º
C. 50º
D. 65º
E. 70º
2. The angles of a quadrilateral have measures 5x, 3x + 20, 4x + 5, and 5x – 5. Which of the following is
equal to the measure of one of the angles?
A. 20º
B. 50º
C. 90º
D. 95º
E. 125º
3. Which of the following is sufficient to prove a given quadrilateral is a parallelogram?
A.
B.
C.
D.
E.
The diagonals are perpendicular.
The diagonals bisect each other.
Two adjacent sides are congruent.
A diagonal forms two congruent triangles.
The quadrilateral has two sets of congruent sides.
4. Find the measure of
STB = 3xº.
STB if
R=80º,
S
S = xº and
A. 20º
B. 40º
C. 60º
B
T
R
D. 80º
E. 120º
5. What is the radius of a circle if the area of the circle is equal to the area of a triangle with a base of length
16 and a corresponding altitude of length 4π?
A. 4
B. 4 2
C. 4 2 π
E. 8π
D. 8 2
6. What is the area of a 30º - 60º - 90º right triangle if the length of the altitude to the hypotenuse is 2 3 ?
A. 4 3
B. 8
C. 8 3
D. 12
E. 16 3
7. For her birthday, Crystal invites 23 girls to a party. Each of the 24 girls sits between her two best friends
around a table. Each girl sends a text message to every other girl at the table and an additional text to each
of her two best friends, and everyone (other than Crystal) sends an additional text to Crystal. How many
texts are sent?
A. 240
B. 252
C. 263
D. 600
A
8. Trapezoid ABCD has median EF . If CD = 24 + 2N
AB
and EF = 21 – N, then find the value of
.
2
A. 21 + 2N
B. 18 – 4N
C. 9 – 2N
E. 623
B
E
F
D
D. 45 + N
C
E. 18 + 2N
9. In circle O, AB = 16, AO CM , and M is the
midpoint of AO . Find the length of chord CD .
C
M
A
A. 4 2
B. 4 3
D. 8 3
E. 16 3
C. 8 2
B. 117º
B
D
A
10. In parallelogram ABCD, if m ADC= 2x – 3,
m DAC= 3x – 8, and m ACD = x + 5, then
find m DAB.
A. 95º
O
C. 121º
B
D
D. 124º
C
E. 144º
B
11.
In ΔABC, BD bisects ABC. The perimeter of ΔABC is 40.
If AD = 9 and DC = 12, find AB.
A
A.
19
7
B.
57
7
C.
19
4
D.
57
4
C
D
E.
68
7
12. Two concentric circles have radii of 2 inches and 6 inches.
Line m is tangent to the smaller circle. How many points
are equidistant from both circles and the line m?
m
A. 1
B. 2
C. 3
D. 4
E. infinitely many
13. Given the following true statements:
(1) All P and X are N.
(2) All N except P are X.
(3) No P are M.
(4) No R are N.
(5) All M are either X or R.
Which of the following must be true?
I. No R are P.
II. Some X are P.
A. I only
B. I and II only
C. I and III only
III. Some X are M.
D. I, II, and III
E. None of these
14. Line l is the perpendicular bisector of the line segment with endpoints (-1, -1) and (9, 1). Line m contains
the point (-1, 3) and is parallel to the line x – 2y = -12. Find the coordinates of the point of intersection of
lines l and m.
A. (4, 0)
B.
3 17
,
7 2
C. (3, 4)
D. (5, 3)
E. (3, 5)
15. A metal ball with radius 6 cm is melted down and the entire amount is recast as a cone with the same radius.
What is the height of the cone (in cm.)?
A. 6
B. 8
C. 12
D. 18
E. 24
A
16. In the figure shown, ABNM, BCON, and CDPO,
are congruent rectangles. Find the ratio of the
area of quadrilateral KNOL to the area of LOPD.
1
3
B.
2
5
C.
4
9
B. 40º
C. 60º
D
O
P
L
D.
1
2
N
E.
17. In circle O with diameter AD and tangent CE ,
the measure of arc AB is 80º and the measure of
CED = 20º. Find the measure of minor arc BC.
A. 30º
C
K
M
A.
B
3
5
C
B
A
D. 70º
O
D
E
E. 100º
18. One of two similar polygons has an area 35% more than that of the other. What is the ratio of the perimeter
of the larger polygon to the perimeter of the smaller?
A. 20:17
B. 400:289
C. 10 35 : 35
D. 3 15 : 10
E. none of these
19. A circle is inscribed in a triangle with sides of length 2, 3, and 4. Find the area inside the triangle and
outside the circle.
A.
3 15 4
4
B.
9 15 5
12
C.
18 15 5
12
D.
27 15 4
36
E.
3 15 4
12
20. Alphatown is due south of Betaville. Kappa City is due east of Alphatown. The distance from Alphatown
to Kappa City is 1 mile less than twice the distance from Alphatown to Betaville. The distance from
Betaville to Kappa City is 7 miles less than 3 times the distance from Alphatown to Betaville. How far is
Alphatown from Kappa City (in miles)?
A. 12
B. 14
C. 15
D. 17
E. 18
21. Which of the following is an equation of the line containing the longer diagonal of the quadrilateral with
vertices at (3, -1), (6, 1), (5, 3), and (4, 4)?
A. 2x – y = 4
B. x – 2y = 5
C. 2x – y = 7
D. 3x – 2y = 16
E. none of these
22. What is the sum of the lengths of the diagonals of a regular hexagon with area 54 3 ?
A. 36 2
3
B. 36 1
3
C. 36 1 2 3
D. 18 2
3
E. 36
D
23. The ratio of the area of TBC to the area of DAT is 1:4.
The measure of ATD = 60º, and the measure of CTB = 30º.
If the length of CD 4 5 , find AB.
C
A
A. 2 2 3
B.8
C. 4 2 3
D. 6 3
B
T
E. 4 4 3
A
24. In the diagram, CD is tangent to circle O. Minor arc AC is
congruent to minor arc BC, AB= 6, AF = 12, and FE = 20. Find ED.
B
O
A. 12
B. 10
D. 6
E. 4
F
E
C. 8
C
D
25. Square MATH is the base of a pyramid with point P as its vertex, and PM = PA=PT=PH=MA. Point X is
the intersection of the diagonals MT and AH of the square. A plane cuts PM, PT , and PX at points Q, R,
and S respectively. If PQ = 3, and PR = 4, compute PS.
A.
5
2
B.
12
7
C.
12 2
7
D.
15
7
E.
15 2
7
TB1: How many different squares with integral side lengths have areas between 1008 and 2009?
TB2: The height of a right cylinder is twice its radius. The radius of a right cone is the same measure as the
radius of the cylinder. Find the height of the cone if the lateral area of the cylinder has the same value as the
volume of the cone.
TB3: How many ordered pairs (x, y) where x and y are positive integers, lie in the interior of the polygon with
consecutive vertices: (4, 0), (8, 4), (6, 8), (5, 6), (4, 8), and (0,4)?