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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