rkansas ouncil of Teachers of Mathematics 2012 State ompetition Geometry Exam For questions 1 through 25, mark your answer choice on the answer sheet provided. (Figures may not be drawn to scale.) fter completing items 1 through 25, answer each of the tiebreaker items in sequential order (do #1 first, followed by #2, and then #3 last). e sure that your name is printed on each of the tiebreaker pages. ongratulations for being selected to participate in the TM State ontest! 1. If l and m are parallel lines and t is a transversal, what is the value of x in the given figure?. 22 t. 28 l (5x-41) 3. 23 m (2x+25) 7 Problem #1. 1 35 7 2. triangle in which the medians and angle bisectors coincide is a(n). scalene triangle. isosceles triangle. right triangle. equilateral triangle 3. To construct a circle that is tangent to the three sides of a triangle one must first construct the. altitudes of the triangle. angle bisectors of the triangle. perpendicular bisectors of the sides of the triangle. medians of the triangle 4. The bases of an isosceles trapezoid have lengths of 10 and 16. The median of the trapezoid has a length of. 26. 12. 14. 11
5. Four angles of a pentagon have degree measures of 100, 108, 115, and 120. What is the measure of the fifth angle?. 108 o. 97 o. 73 o. 78 o 6. In the circle with diameter = 15, the inscribed angle has a measure of 72 o. The length of the minor arc is. 6π. 12 π. 3 π. 5 π 72 15 Problem #6 7. In the figure m E = 30 o and the minor arc, and are all equal. The degree measure of the minor arc is. Not enough information. 30 o. 105 o. 45 o Problem #7 8. n interior angle of a regular polygon measures 140 o. The polygon is a(n). octagon. decagon. heptagon. hexagon 9. 20-foot rope and a 15-foot rope are tied to the top of a 12-foot wall. The ropes are then tied to the ground on opposite sides of the wall in a line that is perpendicular to the base of the wall. The distance between the points the ropes are tied to the ground is 30 E. 25 ft. 7 ft. 24 ft. 9 ft 10. In the adjacent figure is an equilateral triangle and quadrilateral E is a square. The measure of E is. 15 o. 30 o. 10 o. 25 o E Problem #10
11. In trapezoid, and. If m = 110 o and m = 30 o then m is. 80 o. 40 o. 100 o. 110 o 110 Problem #11 30 12. In, m = 40 o and the altitude is 12. If = 20, determine to the nearest hundredth.. 14.30. 5.70. 15.66. 4.34 40 12 20 Problem #12 13. is a right triangle with right angle at. is the midpoint of. If the length of the median from is 6 then =. 12. 10. 9. Not enough information 14. Let = (3, 5) and = (-2, 10) be points in the coordinate plane. The distance is.... 15. The diameter of the circle whose equation is x 2 + y 2 6x = 7 is. 10. 5. 8. 4
16. The slope of the line t that is tangent to the circle at the point (-3,4) is t (-3,4) (0,0) (5,0).... Problem #16 17. is a right triangle with right angle at. If = 12, bisects and m = 30 o then the area of is... 36. Not enough information Problem #17 30 18. In if m = 30 o, m = 45 o and the altitude = 10 then the area of is.... 10 45 30 Problem #18 19. is a right triangle with right angle at. If is an altitude, = 8 and = 10 then =... 6. 12 10 Problem #19 8 20. If, = 3E and the area of is 90 square feet, then the area of is. 270 sq. ft.. 30 sq. ft.. 810 sq. ft.. 10 sq. ft.
21. If the diameter of a circle is increased by units then the circumference of the circle is increased by? units..... 22. If the diameter of a circle is 4 times the reciprocal of its circumference, then the area of the circle is. π. 2. 1. 23. The volume of a right circular cylinder is the same numerical value as the lateral surface area of the cylinder. The radius of the cylinder is. Not enough information. 4. 1. 2 24. The perimeter of a semicircular region, measured in inches, is numerically equal to its area measured in square inches. The radius of the semicircle, measured in inches is. π.. 2π + 2. 2 + E.None of these 25. Using the given diagram, which of the following statements is/are always true? The triangle is to be considered a general triangle. P: y < x Q: x < z R: x < w z x y w Problem #25. P only. Q and R only. Q only. P, Q, and R
Name [Please Print learly] School [Please Print learly] Tiebreaker Questions Your solutions should be written clearly. ll work leading to your final answer must be included. The questions will be used in sequential order to resolve ties for first, second, and/or third place. Tiebreaker #1. In the figure the line makes a 60 o angle with the positive x-axis. What are the exact coordinates of, the point of intersection between the line and the circle in the first quadrant? y 60 (10,0) x Tie reaker # 1 Tiebreaker #2. 2. In, =, = E and E = F. If m = 72 o find the measure of EF. 72 F Tie reaker # 2 E
Name [Please Print learly] School [Please Print learly] Tiebreaker #3. In XYZ, P is the midpoint of, Q is the midpoint of and R is the midpoint of. If the area of QXR is 27 units squared, what is the area of XYZ? X Q P Y R Tie reaker # 3 Z
rkansas ouncil of Teachers of Mathematics 2012 State ompetition Geometry Exam Key 1. 14. 2. 15. 3. 16. 4. E 17. 5. 18. 6. 19. 7. 20. 8. E 21. 9. 22. 10. 23. 11. 24. 12. 25. E 13.
rkansas ouncil of Teachers of Mathematics 2012 State ompetition Geometry Exam Tiebreaker Key 1. In the figure the line makes a 60 o angle with the positive x-axis. What are the exact coordinates of the point of intersection between the line and the circle in the first quadrant? x = 10cos(60 o ) y = 10sin(60 o ) ( ) y 60 (10,0) x Tie reaker # 1 2. In, =, = E and E = F. If m = 72 o find the measure of EF. Since is isosceles m = m = 54 o. Since E is isosceles m E = m E = 63 o. Since FE is isosceles m FE = 63 o. Thus m EF = 180 o 63 o 63 o = 54 o. 72 Tie reaker # 2 E F 3. In XYZ, P is the midpoint of, Q is the midpoint of and R is the midpoint of. If the area of QXR is 27, what is the area of XYZ? rea( PQR) = rea( RQY) rea( PRY) = rea( PRX) rea( XRY) = rea(xrz) Let rea( RQY) = a. Then rea( QRX) = 3a = 27. So a = 9. Thus, rea( XRY) = 36. So rea( XYZ) = 72. Y Q P X R Tie reaker # 3 Z