6th FGCU Invitationdl Math Competition
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1 6th FGCU nvitationdl Math Competition Geometry ndividual Test Option (E) for all questions is "None of the above." 1. MC = 12, NC = 6, ABCD is a square. 'h What is the shaded area? Ans ~ (A) 8 (C) An equilateral triangle, a square, a regular pentagon, and a regular hexagon all have one vertex in common. What is the subtended angle indicated? Ans : 3. A length of cloth of uniform width is y yards long. f n people each get 10 yards of cloth, there are 8 yards left over. f n - 2 people each get 12 yards of cloth, there are 6 yards left over. What is the length of the cloth? Ans (A) 120 yards (B) 128 yards (C) 138 yards (D) 143 yards 4. n problem 3, what is the largest number of people who can get 6 yards of cloth, with or without any left over? Geom lndiv Page 1
2 5. To prove PQ parallel to FG, you might use all of the following EXCEPT -. Ans ~ (A) Extend B1 to intersect A0 at Z (B) Measure of angle AZB = 75' (C) Angles AZB and BCD are supplementary adjacent interior angles (D) Lines perpendicular to parallel lines are themselves parallel 6. f all of the following statements are true, what may be concluded? Ans L statement : f Linda plans well enough, she will finish her work on time. statement : f Linda finishes her work on time, she will go to the concert. statement ll: Linda did not go to the concert.! (A) Linda planned well enough. (B) Linda did not finish her work. (C) Linda needed to do more work. (D) We can not conclude anythin$. The following three statements are for use in problems 7 and 8. statement : f Linda plans well enough, she will finish her work on time. ~ statement : f Linda did not finish her work on time, then she did not plan well enough. statement ll: f Linda finished her work on time, then she planned well enoug 7. Statement is the of statement. Ans 1 (A) contrapositive (B) conditional (C) disjunction (D) converse 8. Statement ll is the of statement. Ans 1 (A) contrapositive (B) conditional (C) disjunction (D) converse Geom ndiv, Page 2
3 9. Given that the two lines are parallel, what is a - P? Ans (A) 30' (c) o0 (B) (18017)~ (D)- (18017) 10. nscribed angle D = 184 Ans inscribed angle C = 31'. What is the measure of inscribed angle A? (A) 104' (C) 93' (B) 100' (D) 82' A 11. Central angle 0 = 60, central angle 0' = 120. What is the ratio of r to r'? Ans (A) All of the following are valid shadows cast by a cube EXCEPT. Ans (A) square (B) rectangle (C) hexagon (D) rhombus 13. What is the minimum number of colors needed to paint a regular dodecahedron, 1 color per face, so that no two adjacent faces are the same color? (B) 4 (Dl What is the ratio of the area AB'C' to area ABC? The hexagon is regular. ~ (A) 6 (c) 3 A c c ' (0) 9/4 (D) can not be determined 15. One side of a triangle is 7; another side is 18. Which is a possible length Ans i of the third side? 7 Geom ndiv Page 3
4 16. The altitude from the vertex angle of an isosceles triangle and the angle bisector of one of the base angles intersect at point A. Point A is Ans t (A) The center of a circle inscribed in the triangle. (B) The center of a circle which circumscribes the triangle. (C) The centroid of the circle. (D) NOT equally distant from the equal sides of the triangle. 17. All of the following were ancient Greek mathematicians EXCEPT Ans 1 (A) Menaechmus (B) Archytas (C) Eudoxus (D) Thales 18. AB : BC = AD : DC under which of the following 6 conditions? C (A) BC = CD (B) Angle BCA = 60' and angle CAD = 30' D (C) AC is the angle bisector of angle BAD (D) The lengths of AB, AC, and AD are in geometric sequence 19. Which of the following is a valid method of disproof? Ans ~ (A) deduction, (B) induction, (C) comparison (D) example 20. All of the following are valid methods of proof EXCEPT Ans / (A) deduction (B) induction (C) contradiction (D) example 21. n the proof of the Pythagorean theorem given in book of Euclid's Ans 1 Elements, a key theorem used to justify the congruence of certain triangles is (A) SSS (B) ASA (C) HL (D) SAS Geom ndiv Page 4
5 22. A key feature in the proof of the Pythagorean theorem, as given in book of Euclid's Elements, is (A) area AMNl equals area ABED, and area CMNH equals area BCGF (B) AACG ~AH~B, and AACD SAAB (C) triangles with the same base and height have the same area (D) all of the above 23. Line jl passes through the centers of the circles. What is the length of AB? (A) 4 f i (B) 42 (c) 60 (D) can not be determined 24. How many triangles can be constructed from the following information: Ans one side is length 12, one angle of 47' is adjacent to the side of length 12, and the side opposite the 47' angle is length 10. i (B) no triangle (B) 1 (c) 2 (D) can not be determined How many distinct lines are determined by 15 points, no three of which Ans are collinear? ~ (A) 91 (B) 105 (C) 155 (D) How many (true spacial) diagonals does an icosahedron have? Ans ~ (A) 40 (B) 55 (C) 80 (D) 180 Geom lndiv Page 5
6 27. Statement S: f X and Y, then Z. What is the converse of statement S? Ans 1 (A) f X and Y, then not Z. (B) f Z, then X and Y. (C) f X or Y, then Z. (D) f Z, then X or Y. 28. A plane cuts through a square-based pyramid with the apex centered Ans over the base. Which of these cross-sections is NOT possible? (A) rhombus (6) square (C) trapezoid (D) kite A plane cuts through a square-based pyramid with the apex centered over the base. Which of these cross-sections is NOT possible under any conditions? (A) equilateral triangle (B) isosceles triangle (C) scalene triangle (D) right triangle 30. Which sequence is one of "polygonal numbers"? Ans 1 (A) 1, 5, 12, 22, 35,... (C) 1, 6, 13, 22, 32,... (B) 1, 2, 3, 5, 8, 13,... (D) 1,2,3,5,7,11,... Geom ndiv 1 Page 6
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