Geometry Individual Test
March Regional
1. If the largest angle in a quadrilateral is equal to three times the measure of the smallest angle
and three of the four angles are congruent, what is the smallest possible measure of the largest
angle?
A. 36°
B. 180°
C. 60°
D. 108°
E. NOTA
2. What is the radius of the largest cylindrical peg that can be fit into a triangular hole of
dimensions 12 by 37 by 35?
A. 2
B. 3
C. 5
D. 10
E. NOTA
3. Given triangle XYZ with dimensions XY = 10 cm and YZ = 21 cm. What lengths are
possible for XZ ?
A. 12cm  XZ  30cm
B. 11cm  XZ  31cm
C. 12cm  XZ  30cm
D 11cm  AC  31
E. NOTA
4. Farmer Brown has 132 feet of barb wire fencing that he wants to use to enclose and
subdivide a part of his field as shown below.
If the total area enclosed by the fencing is 576ft², then there are two possible values for the
variable x. The difference of these two values is
A. 10 ft.
B. 20 ft.
C. 25 ft.
D. 30 ft.
E. NOTA
5. A rectangle is inscribed in a triangle such that its upper right vertex bisects the triangle’s side.
What is the ratio of the shaded area to the un-shaded area?
A. 1:2
B. 4:3
C. 5:4
D. 2:3
E. NOTA
Geometry Individual Test
March Regional
6. The last piece of pumpkin pie has been eaten by either Cherry, Ken, Lisa, or Noriko. The
following is known: Each one said they knew who ate the piece of pie; Cherry always tells the
truth; Ken never tells the truth; Noriko said it was either Ken, Cherry or Lisa; Lisa said, “It
wasn’t me!”; Ken claimed that Noriko is a liar; and Cherry responded that Ken was the only liar.
Who ate the last piece of pie?
A. Cherry
B. Ken
C. Lisa
D. Noriko
E. NOTA
7. Three circles each have radius = r. They intersect each other as shown such that each circle
passes through the center of the other two circles. Find the area of the intersection of their
interiors.
A. A.
 r2
6

3 2
B.  
 r
3
4


C.
3 r 2
3
  3  2
D. 
 r
2


E. NOTA
8. Assume pq || rs in the diagram below. Find the measure of angle x.
140
P
Q
x
R
S
22
A. 18
B. 40
C. 62
D. 22
E. NOTA
9. The sides of a triangle measure 10, 10 and 12. What is the radius of its circumscribed circle?
A. 4 3
B. 6.25
C. 6
D. 5.75
E. NOTA
Geometry Individual Test
March Regional
10. A semicircle with a diameter of 12 cm is used to form a conical cup by bending the
semicircle so that its two corners are connected with the circle’s center forming the vertex of the
cone. What is the volume of the conical cup?
A. 9 3
B. 27 3
C. 18 6
D. 2 15
E. NOTA
11. An 18 inch diameter pizza is cut into 16 slices. What is the distance around one slice?
A. 18 
1

16
9
C. 9  
8
B. 18  9
9
D. 18+ 
8
E. NOTA
12. A large window made up of a rectangle and a semicircle, as shown
has a width of 80 cm and an overall area of 16000 sq cm. How high is the window?
A. 200 cm
B. 200-10 cm
C. 240-10 cm
D.
240 
40

E. NOTA
13. The bases of a trapezoid are 7 cm and 13 cm. A line which is twice as far from the shorter
base as it is from the longer base divides the trapezoid into two parts. If the line is parallel to the
two bases and the area of the smaller part is 16 sq. cm, find the area of the larger part of the
divided trapezoid.
A. 55 sq. cm
B. 24 sq. cm
C. 70 sq. cm
D. 34 sq. cm
E. NOTA
14. (2k +1) evenly spaced points on a circle are numbered 0 through 2k in succession.
Determine the measure of the angle formed at point k by connecting the point 0 to point k to
point 2k?
A.
180
k
B.
360
k
C.
180
2k  1
D.
360
2k  1
E. NOTA
15. Find the area of a circle that circumscribes a triangle whose sides measure 20, 21 and 29.
A.
π
B. 420π
C. 625π
D. 145π
E. NOTA
Geometry Individual Test
March Regional
16. An arbelos is the region bordered by three mutually tangent circles with co-linear centers
(see the shaded region in the figure below). If the area of the shaded region is 42π and the radii
of the two smaller circles have a ratio of 2:3, what is the radius, x,
of the largest circle?
A.
B. 5
C. 7
D. 35
E. NOTA
17. An isosceles triangle is cut away from each corner of a square piece of
paper so that only a rectangle remains. The shaded area in the figure below
represents what was removed from the square. If the sum of the areas of the
triangles that were removed is 200 square units, what is the length of
diagonal “e”?
B. 14
A. 12
C. 18
D. 20
e
E. NOTA
18. How many faces does a polyhedral solid with 18 vertices and 32 edges have?
A. 14
B. 16
C. 18
D. 28
E. NOTA
19. Starting at 12:00 noon, how long will it take the hands of a typical analog clock to form a
90° angle?
A. 16
4
minutes
11
B. 16
1
minutes
3
C. 15 minutes
D. 16
5
minutes
12
E. NOTA
20. You are given four sticks to form a quadrilateral; 2 sticks are 5 inches long and 2 are 8 inches
long. If no more than one of the angles formed is to be a right angle, how many non-congruent
quadrilaterals can be formed?
A. 1
B. 2
C. 3
D. 4
E. NOTA
21. In obtuse triangle ABC, point R is located on side AC such that it is
equidistant from vertices A and B. If angle ACB measures 28°, what
is the measure of angle BAC?
A. 28°
B. 54°
C. 62°
D. 108°
A
R
E. NOTA
28
B
Geometry Individual Test
March Regional
22. The side of an equilateral triangle inscribed in a circle measures
units. What is the
length of the diagonal of a square circumscribed about the same circle?
A. 15
C. 30
B. 30
D. 30
E. NOTA
23. Joe Smith has purchased a new gazebo. The gazebo is shaped like a regular octagon with a
side measure of 4 feet. In order to tile the floor, Joe needs to know the total area. What is the
total area of the floor of Joe’s gazebo?

A. 16 2  2


B. 32 1  2

C. 32
 2


D. 16 1  2
E. NOTA
E
24. In this figure, the area of square ABCD is 100 square units and the area of
isosceles triangle DEC is 10 square units. Find the distance from A to E.
A.
244
B. 12
C.
146
D. 13
C
D
E. NOTA
A
B
B
25. In the figure to the right, line segments AF and CF partition
pentagon ABCDE into a rectangle and two triangles. For
A
which of the following can the value be determined?
I. a° + b°
II. b° + c°
III. a°+ b° + c° + d°
A. II only
III only
B. I and II only
E. NOTA
C. II and III only
D. I and
C
a
d
b
E
c
F
D
Geometry Individual Test
March Regional
26. What is the sum of 35th and the 50th term of the sequence:
X
Y
1
0
2
1
A. 170
3
3
4
6
5
10
B. 1085
…..
…..
35
…..
…..
C. 1820
50
D. 695
E, NOTA
27. Consider the conditional statement: If a pentagon is regular, then it is equiangular. Which
of the following is true?
A.
B.
C.
D.
E.
Only the conditional is true
Only the conditional and the contrapositive are true.
Only the conditional, converse and inverse are true.
The conditional, converse, inverse and contrapositive are all true.
NOTA
28. A chord is the perpendicular bisector of a radius of length 12 units in circle O. What is the
length of the chord?
A. 3 3
B. 12 3
C. 6 3
D. 27
E. NOTA
29. Given two intersecting circles  x  2    y  1  4 and  x  3   y  4   9,
2
2
2
2
what is the equation of the line passing through the points of intersection?
A. y 
1
17
x
3
6
1
17
B. y   x 
3
6
C. y 
1
5
x
3
2
1
5
D. y   x 
3
2
E. NOTA
30. What is the area of a quadrilateral with 3 of its vertices spaced evenly around a circle of
radius 6 and the 4th vertex located at the center of the circle?
A. 18 3
B. 9 6
C. 36
D. 36 3
E. NOTA