March Regional________________________________________Geometry Individual
For all questions, NOTA means “None of the Above Answers are Correct”.
Please note that all drawings are not to scale.
1. In right triangle ABC with right angle at C, D lies on AB and CD is
perpendicular to AB. If AD = 4 and DB = 9, what is the length of
?
A) 5
B) 6
D)
E) NOTA
C)
2. What is the cosine of a 60 degree angle in a right triangle?
A)
B) 1
C)
D) 0
E) NOTA
3. Which of the following types of lines will never intersect?
I.
Orthogonal
II.
Parallel
III.
Perpendicular
IV.
Skew
A) I, III only
B) II, IV only
C) I, II, IV
D) III only
E) NOTA
4. Kay is an aspiring chef and practices by baking cookies at home. Since Kay especially loves
snickerdoodle cookies, she decides to bake them one day. She uses baking sheets that are rectangular
with dimensions of 18 inches by 12 inches. To make cookies, Kay uses circles of dough with radius 1
inch. If the radius of a cookie expands ½ of an inch during the baking process and Kay doesn’t want any
cookies to be overlapping each other before and after baking (although they can be tangent), what is the
maximum number of snickerdoodle cookies that Kay can make on a single baking sheet?
A) 20
B) 22
C) 24
D) 32
E) NOTA
5. A rectangular piece of 7x10 paper is folded as shown such that the two
opposite corners touch. Find the area of the shaded region.
A)
B)
D)
E) NOTA
C)
6. The radius of circle O is 63. A point P lies outside of circle O such that two segments extending from P
are tangent to the circle at points A and B.
maximum possible integer length of
A) 99
B) 101
does not contain the center of circle O. What is the
?
C) 125
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D) 167
E) NOTA
March Regional________________________________________Geometry Individual
7. A regular octagon has sides of length 1. If the area of the octagon is in the form
, where
is
simplified as much as possible, find the value of
A)
B)
C)
D)
E) NOTA
8. Quadrilateral ABCD is circumscribed around a circle. If AB = 5, BC = 8, and CD = 10, what is the length
of segment AD?
A) 6
B) 7
C) 8
D) 9
E) NOTA
9. If the measure of minor arc AB is 150° and the
measure of ∠E is 45°, what is the measure of ∠
?
(F is not the center of the circle)
A) 20°
D) 45°
B) 25°
E) NOTA
C) 30°
10. Referring to the diagram in question 9, If
A) 4
B) 5
,
C)
, and
D) 7
, then what is the length of
E) NOTA
11. Triangle ABC has sides of length 24, 45, and x. Over the set of integers, how many values can x take on?
A) 47
B) 49
C) 55
D) 69
E) NOTA
12. Peter is currently at the point (3,7) on the Cartesian plane. To travel to his house, he boards the Operation
Express, which flips him across the line
. After getting off the Operation Express, Peter is at the
point where his house is. What are the coordinates of his house?
A) (3,7)
B) (-3,7)
C) (3,-7)
D) (7,3)
E) NOTA
13. What is the sum of the exterior angles of a convex polygon with 27 sides?
A) 180
B) 360
C) 540
D) 1080
E) NOTA
14. Bonquisha likes to keep track of time. She checks her watch and the time is currently 3:14. How many
minutes must she wait until the hands first make a right angle?
A) 16
B)
C)
D)
E) NOTA
15. A square is inscribed within a circle of radius 4. What is the perimeter of this square?
A)
B)
C)
D)
E) NOTA
16. Given 6 points in space, with no 3 points collinear, how many distinct planes can be determined?
A) 15
B) 20
C) 28
D) 36
E) NOTA
17. A triangle with integer side lengths has a perimeter of 36. What is the difference between the largest
possible area and the smallest possible area of this triangle?
A)
B)
C)
D)
E) NOTA
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?
March Regional________________________________________Geometry Individual
18. If
and
are two chords of circle O which intersect perpendicularly at
point E as shown, and
,
, and
, what is the radius of
circle O?
A)
B)
C)
D)
E) NOTA
19. In triangle ABC, the median from vertex A and the median from vertex B intersect at right angles to each
other at point P. If
A)
,
B) 10
, and
, what is the length of
C)
D)
20. Find the area of the triangle formed by connecting the points
A)
B) 7
C) 12
D)
?
E) NOTA
and
E) NOTA
21. Two circles of equal radius lay on a horizontal plane tangent to each other. A line externally tangent to
both circles binds a region with the two circles. A smaller circle is inscribed within this region. Find the
ratio of the radius of the smaller circle to the radius of each larger circle.
A)
B)
C)
D)
E) NOTA
22. Two circles with distinct radii lie in the first quadrant of the xy-plane. The smaller circle is tangent to the
y-axis and the larger circle is tangent to the x-axis. The two circles are externally tangent to each other. If
the centers of both circles lie along the line
, what is the sum of their radii?
A)
B)
C)
D)
E) NOTA
23. Two telephone poles, one 40 feet tall and the other 60 ft tall, stand vertical a distance apart. Telephone
wires extend from the top of each pole to the midpoint of the other as shown. The wires crisscross at a
point a height h above the ground. Find h.
A) 36
B)
C)
D) It is dependent on how far apart the poles are
E) NOTA
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March Regional________________________________________Geometry Individual
24. Segments and
in the diagram are parallel. Using the given angle
measures, what is the measure of angle ?
A)
B)
D)
C)
E) NOTA
25. Equilateral triangle RST has side lengths of
. A point P in the interior of triangle RST has segments
extending from it perpendicular to each side of the triangle, meeting
respectively. If
and
A)
, determine the length of
B)
C)
26. Isosceles trapezoid ABCD has bases
D)
and
,
, and
at points X, Y, and Z,
.
E) NOTA
with lengths of 12 and 13, respectively, and diagonals of
length 16. What is the length of each leg of the trapezoid?
A)
B) 12
C)
D)
E) NOTA
27. Which of the following are always true?
I.
II.
If
and
III.
If
, then
A) I only
B) III only
, then
C) I, II only
D) I, II, and III E) NOTA
28. One day, Kelsey was riding her pony through her pasture located at the point (2,10) when her pony
became very thirsty. However, Kelsey had to reach her home located at the point (10,7) as soon as
possible. The nearest river followed the line
. How far must Kelsey travel if she takes the shortest
possible path first to the river and then home?
A) 10
B) 12
C) 15
D) 18
E) NOTA
29. Marshall is walking his pet poodle through town one day when he realizes he needs to go to the grocery
store. Upon reaching the grocery store, he ties his pet poodle outside the grocery store as shown. The
length of his poodle’s leash is 12 feet. If the size of the poodle is
negligible and the poodle cannot break through walls, what is
the area which the poodle has to roam?
A)
B)
C)
D)
E) NOTA
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March Regional________________________________________Geometry Individual
30. Which famous ancient mathematician/astronomer is attributed to writing the famous collection of 13
books about geometry and number theory known as The Elements?
A) Archimedes
B) Euler
C) Eratosthenes
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D) Euclid
E) NOTA