Download February Regional Geometry Individual Test

February Regional Geometry Individual Test
The use of a calculator is not permitted!
For all questions, choice E is NOTA, meaning “none of these answers.”
All angles are measured in degrees
Figures are not necessarily drawn to scale
All measurements are given in units, unless otherwise specified.
1. Let’s start our test with a question about the early history of constructions in geometry. Which of the
following problems is solvable using only a compass and an un-marked straightedge?
A) Squaring the Circle
C) Trisecting the angle
E) NOTA
B) Doubling the Cube
D) Constructing a regular heptadecagon
2. Let A be the set of all rhombi and let B be the set of all rectangles. Which of the following sets is
equivalent to A  B ?
A) Quadrilaterals
B) Squares
C) Parallelograms
D) 
E) NOTA
3. Which of the following points of concurrency can exist outside of a triangle?
I. Incenter
III. Orthocenter
II. Circumcenter
IV. Centroid
A) II only
B) III only
C) II and III
D) None
E) NOTA
4. In the following triangle ABC, a line from vertex A is drawn to
point D on BC such that BAD  CAD . AB=4, AC=7, and BC=8.
Find the length of CD.
56
11
7
C)
2
E) NOTA
A)
32
11
32
D)
7
B)
5. Which type of reasoning is used in this argument and what is the type of inference used in this
argument to prove the conditional statement valid?
If Pac-Man eats a power pellet, then he will eat Pinky the ghost.
Pac-Man eats a power pellet.
Therefore, he will eat Pinky.
A) Inductive Reasoning; Modus Ponens
B) Inductive Reasoning; Modus Tollens
C) Deductive Reasoning; Modus Ponens
D) Deductive Reasoning; Modus Tollens
E) NOTA
6. I love pizza. So today, I ordered a pizza from the Pizza Pi Company, but the working mathematician
who calculates the maximum number of slices from pizza was on vacation. Since the Pizza Pi
Company knows that I am a mathematician, they sent me the pizza without any cuts and told me to
cut it myself. I have 8 flimsy knives that I can only use one time each to make a cut. Assuming that I
can only make straight cuts and the pizza is round, what is the maximum number of slices that I can
make from this pizza with only 8 cuts?
A) 16
B) 28
C) 36
D) 37
E) NOTA
7. A square with sides of 4 has two quarter-circles centered at
opposite vertices, as seen in the diagram to the right. What is the
area of the shaded region?
A) 16  4
C) 4  8
E) NOTA
B) 8 16
D) 32  8
Use the following diagram to answer questions 8 and 9. AB=3, BC=5, CD= x, DE=6,
mAE  118 and mBD  40 .
8. What is the value of x in the diagram?
A) 2.5
C) 8
E) NOTA
B) 4
D) 10
9. What is the measure of angle C?
A) 39°
B) 78°
C) 79°
D) 20°
E) NOTA
Use the diagram of the transversal to parallel lines l, m, and n, to answer questions 10 and 11.
10. Angles 3 and 11 are congruent because they are...
A) Vertical angles
C) Alternating exterior
Angles
E) NOTA
B) Alternating interior angles
D) Corresponding angles
11. Angles 1 and 12 are congruent because they are…
A) Vertical angles
C) Alternating exterior
Angles
E) NOTA
B) Alternating interior angles
D) Corresponding angles
12. The answer to this problem is the number of books that Euclid of Alexandria wrote for Elements, or
the number of sides that a convex polygon has with the sum of the interior angles of 1980°.
A) 11
B) 12
C) 13
D) 14
E) NOTA
13. Don’t tell anybody else, but I know the sequel to Snakes on a Plane. It’s going to be called Angles on
a Clock and it stars G. Imatree. The general plot involves a clock store and possessed grandfather
clocks. The time on the clock determines the strength of these possessed clocks; the larger the
smallest angle measure between the hands on the clock, the stronger the clocks are. Given the above
information, at which of the following times would the possessed clocks be strongest?
A) 12:04 am
B) 9:27 pm
C) 7:30 am
D) 2:03 pm
E) NOTA
14. Given the diagram to the left, triangle ABC is a right triangle with point D on
AC with BD  AC . AD=x+8, CD=x-2, and BD=x. What is the area of the
triangle?
136
9
8
C)
3
A)
B)
264
9
D) Not enough information
E) NOTA
15. Using all of your special right triangle rules and the Law
of Sines, plus the diagram to the right, evaluate sin105 .
A)
1 3
4
B)
6 2
4
C)
1 2
4
D)
3 2
4
E) NOTA
Use the following information to answer questions 16 and 17: A triangle ABC is placed on a
Cartesian Coordinate plane, with vertices at A(-2,1), B(3,5), and C(8,6).
16. At what point on the coordinate plane is the centroid of the triangle located?
A) (6,4.5)
B) (4,3)
C) (4.5, 6)
D) (3,4)
E) NOTA
C) 17.5
D) 35
E) NOTA
17. What is the area of triangle ABC?
A) 7.5
B) 15
18. Given the side lengths of 7, 8, and 16, the triangle with these side lengths could be described as…
A) Acute
B) Right
C) Obtuse
D) Scalene
E) NOTA
19. What is the set of all points in a plane at a given distance from a given point in the plane?
A) Sphere
B) Circle
C) Ellipse
D) Cone
E) NOTA
20. What is the ratio of the areas of the circumscribed circle to the inscribed circle in a regular hexagon
with side length 6?
A)
4
3
B)
3
4
C)
3
2
D)
2 3
3
E) NOTA
21. Triangle ABC is drawn, with point D on AB , point E on
AC and DE BC . AD=x+1, BD=x+3, DE=x-2, and
BC=x+5. What is BC?
36
5
14
D)
5
A) 3  22
B)
C) 8  22
E) NOTA
Use the following information to answer questions #22 and 23. A rhombus has diagonals of length 16
and 30 units.
22. Find the sum of the area and the perimeter of the rhombus.
A) 240
B) 308
C) 548
D) 572
E) NOTA
23. Suppose a circle is inscribed in the rhombus, such that the circumference touches the four sides. What
is the circumference of the inscribed circle?
A) 16
B) 17
C)
120
17
D)
240
17
E) NOTA
24. The Cameron Kim Telecommunications network is rather large and continuously expanding, which
requires a direct cable link with every company CK Telecommunications is working with. Also, each
company in the network needs to be connected with every company in the network. If there are 30
companies that CK Telecommunications is connected with, how many cables are required so that
every company can communicate with each other?
A) 405
B) 434
C) 435
D) 465
E) NOTA
25. Find the annular volume between the circumscribed sphere and the inscribed sphere of a cube with
side length of 5 .
A)
15 15  5 5
6
B)
5 10
3
C)
20 5
3
D)
40 10  20 5
3
E) NOTA
26. Which of the following theorems cannot be used to prove two triangles similar?
A) AAA
B) SSA
C) SAS
D) ASA
E) NOTA
27. What is the lateral surface area of a cone with a radius of 5 and a height of 12?
A) 60
B) 65
C) 75
D) 80
E) NOTA
28. I have an orange (which for the purposes of this problem is a perfect sphere) and I cut a segment of
the sphere, such that there is a circular cross-section with an area of 20 . The distance from the
center of the sphere to the center of the circular cross section is 4. What is the original volume of the
orange, before the segment was cut off?
A) 144
B) 216
C) 288
D) 864
E) NOTA
D) 36
E) NOTA
29. Given trapezoid ABCD, with AB CD and
diagonals AC and BD intersecting at point
E. AE=6, BE=4, CE=16, and DE=x. Given
that DEC has an area of 64, then what is
the area of AEB ?
A) 4
B) 16
C) 24
30. What is the converse of the contrapositive of the converse of the inverse of the contrapositive of the
following conditional statement?
“If I do not get a perfect score on this test, then pigs will fly.”
A) If I get a perfect score on this test, then pigs will not fly.
B) If I do not get a perfect score on this test, then pigs will fly.
C) If pigs fly, then I did not get a perfect score on this test.
D) If pigs do not fly, then I did get a perfect score on this test.
E) NOTA