February Regional Geometry Individual Test

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1 Calculators are NOT to be used for this test. For all problems, answer choice E, NOTA, means none of the above answers is correct. Assume all measurements to be in units unless otherwise specified; angle measurements are in degrees. Figures are not necessarily drawn to scale. 1. Find the converse of the inverse of the contrapositive of the following statement: If John studied last night, John will score an A on his exam. A) If John studied last night, John will score an A on his exam. B) If John did not study last night, John will not score an A on his exam. C) If John will score an A on his exam, then John studied last night. D) If John will not score an A on his exam, then John did not study last night. 2. Let n be the angle measure of a regular hexagon. Let k be n/10. Compute the angle measure of a regular polygon with k sides. A) 90 B) 135 C) 140 D) A radio station is built on an extremely large and flat plateau. The radio station can broadcast to receiving units up to a distance of 30 miles from it. However, it cannot broadcast to units that are too close to it, which means the station cannot broadcast within 5 miles of itself. Find the area of the land, in miles, that the radio station can broadcast to. A) 25 B) 25 π C) 875 π D) 895 π 4. A quadrilateral has diagonals that are perpendicular bisectors of each other. Which of the following best describes the quadrilateral? A) rhombus B) parallelogram C) rectangle D) square 5. In the figure to the right, the two hexagons are similar. The smaller hexagon has an area of 6. The length of one side of the larger hexagon is equal to the perimeter of the smaller hexagon. Compute the area of the larger hexagon. A) 1 B) 6 C) 216 D) Classify a triangle with side lengths 6, 7, and 10. A) acute B) right C) obtuse D) degenerate

2 7. Three regular polygons share a vertex, which is shown to the left. If one of the polygons is a square and another is an octagon, how many sides does the last polygon have? A) 6 B) 8 C) 10 D) Cannot be determined by the information given people are in a room. Each person begins trying to shake hands once with every other person in the room. However, Joe leaves the room after 5 handshakes have taken place (these handshakes may or may not have involved Joe). The remaining people shake hands until the remaining people have all shaken hands with each other. Which of the following could be the total number of handshakes that have taken place? A) 275 B) 280 C) 290 D) D is the midpoint of BC inv ABC. E and F lie on AC and AB, respectively, such that DE and DF bisect ADC and ADB, respectively. Find the measure of EDF. A) 60 B) 75 C) 90 D) In the diagram to the right, circle P is inscribed in a quarter-circle of circle O. Circle Q is inscribed in a quarter-circle of circle P. If the radius of circle Q is, find the radius of circle O, labeled r in the diagram. A) B) 1 C) D) A quadrilateral ABCD has 10 units long, find the area of quadrilateral ABCD. AB =, BC = 7, CD = 7, and AD = 18. Given that diagonal BD is 13 A) B) C) D) 12. Triangle ABC has AB = 6, AC = 12, and A = 60. Let AD be the angle bisector of A. Find the length of AD. A) 4 B) C) D) 9

3 13. Define a radial polygon as a regular polygon such that a circle circumscribing the polygon has a radius longer than the side length of the polygon. Which of the following are radial polygons? I. square II. regular hexagon III. regular nonagon IV. regular 100-gon A) II only B) III only C) II and III D) III and IV 14. A circle is drawn in equilateral triangle ABC such that the circle s center lies on one side of the triangle and the circle is tangent to the other two sides of the triangle. Given that the triangle has side length 12, compute the area of this circle. A) 16π B) 27π C) 36π D) 81π 15. An equilateral triangle has apothem 13. Compute the length of one of the sides of this triangle. A) B) 26 C) D) A cyclic quadrilateral with side lengths 6, 7, 8, and 9 has one of its diagonals drawn, creating 2 triangles inside the quadrilateral. Find the distance between the circumcenters of these triangles. A) 2 B) 3 C) 5 D) A right triangle has hypotenuse 10 and perimeter 24. Find the area of the triangle. A) 24 B) 48 C) 72 D) 96

4 18. In the triangle below, BD is the angle bisector of ABC, CE is the angle bisector of ACB, and X is their intersection. Given AB = AC = 10 and BC = 16, compute the area of triangle BXC. A) 64/3 B) 24 C) 128/3 D) Three circles are drawn in the plane such that each circle is externally tangent to the other two circles. Given that the centers of the circles form a triangle with perimeter 20, find the sum of the circumferences of the three circles. A) 10π B) 15π C) 25π D) 40π 20. In quadrilateral ABCD, AB = 10, BC = 6, and CD = 15. Compute the length of AD if the diagonals of ABCD are perpendicular to each other. A) 14 B) 15 C) 16 D) How many angles does regular polygon ABCDEFGHIJKLMOPQRSTUVWXYZ have? A) 24 B) 25 C) 26 D) For any true statement t, which of the following statements will always be true? A) the inverse of t B) the converse of t C) the inverse of the converse of t D) the converse of the contrapositive of the inverse of the converse of t 23. Triangle ABC has AB = 5 and AC = 6. If the angle bisector from vertex B also happens to be a median of the triangle, find the area of triangle ABC. A) 12 B) 18 C) 24 D) 30

5 24. The inradius of triangle ABC is 4. The area of triangle ABC is 8 more than the perimeter of triangle ABC. Compute the area of triangle ABC. A) 4 B) 8 C) 16 D) Which of the following could not be the number of intersections between 3 coplanar circles? A) 0 B) 2 C) 6 D) In quadrilateral ABCD, we have AB = AD and CB = CD. Which of the following could never describe quadrilateral ABCD? A) kite B) parallelogram C) square D) trapezoid 27. A rectangular building has an area of 800 units squared, with side of the building twice as long as the other side. A dog is tied to a corner of this building by a rope. The length of the rope is the average of the side lengths of the rectangular building. Compute the perimeter of the region the dog can travel. A) 50π B) 60+50π C) 60+80π D) 700π 28. There exists a circular building with radius 300 meters. A circular track of width 5 meters is built around it. Joe sprints around the track exactly one time. How far does Joe run, given that he stays in the center of the track throughout his run? All answer choices are in meters. A) 600π B) 605π C) 610π D) 615π 29. Let ABCD be a square with side length 10. Let the midpoint of AB be M and the midpoint of BC as N. Let X be the intersection of MC and ND. Find the area of triangle CNX. A) 5 B) 10 C) 20 D) Which of the following is an undefined term of geometry? A) arc B) coordinate C) plane D) segment

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