2013 ACTM Regional Geometry Exam


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1 2013 TM Regional Geometry Exam In each of the following choose the EST answer and record your choice on the answer sheet provided. To insure correct scoring, be sure to make all erasures completely. The tiebreaker questions at the end will only be used to resolve ties in first, second and/or third place. They will be used in the order given. omplete the first 25 questions before attempting the tiebreaker questions. Figures are not necessarily drawn to scale. 1. onsider the statement If Jacob studies geometry, then he will appreciate logic. Which of the following statements are logically equivalent to this statement?. Jacob doesn t study geometry or he will appreciate logic.. Jacob studies geometry but he doesn t appreciate logic.. If Jacob appreciates logic, then he will study geometry.. If Jacob doesn t study geometry, then he will not appreciate logic. 2. line contains the points (2,1) and (5,3). The slope of the line is The line 3x + 2y = 6 is perpendicular to a second line in the plane. The slope of the second line is Which of the following is not an abbreviation for theorems used to prove the congruence of triangles?. SSS. S. S. SS 5. In the figure bisects and E E. Which triangle congruence theorem assures that. SS. SSS. S. SS E Problem #5
2 6. Suppose and intersect at O and. Which of the following is false?. O and EO are supplementary angles.. OE and O are complementary angles.. O and OE are congruent angles.. O and O are adjacent angles. E. ll are false. O E Problem #6 7. is a trapezoid with m = 139 o. Then m is. 31 o. 51 o. 41 o o Problem #7 8. The supplement of an angle is five times the complement of the angle. What is the measure of the angle?. 23½ o. 22½ o. 67½ o. 112½ o 9. The centroid of a triangle is determined by. angle bisectors of the angles of the triangle.. the intersections of the midlines of the triangle.. the intersection of the perpendicular bisectors of the sides of the triangle.. the intersection of the altitudes of the triangle. 10. G is the centroid of the. If line segments,, and are drawn parallel to the sides of the triangle, then E + FH + JK is. onehalf the perimeter of.. threefourths the perimeter of.. twothirds the perimeter of.. cannot be determined.. K F G H J Problem #10 E
3 11. n Xscale and a Yscale are marked adjacently as shown. reading of 72 on the Yscale would correspond to what reading on the Xscale X 170 Y 100? Problem # The area of a rectangle and the area of a square are equal. The length of the rectangle is twice the width of the square. If the perimeter of the rectangle is 40 inches, then the perimeter of the square is. 40 inches. 32 inches. 44 inches. 64 square inches 13. The distance between (1, 2) and (6, 10) in the coordinate plane is Find the shortest distance across the river shown at the right.. 52 ft. 65 ft. 48 ft. ft river Problem # point on the xaxis that is a distance of 5 units from (1,4) is. (2, 0). (0,0). (3,0). (4, 0) 16. The altitude from the right angle in divides the hypotenuse into segments of lengths 4 and 9. The length of the altitude is. Not enough information Problem #16 4
4 17. In right triangle, bisects. If the measure of = 50 o then m =. 115 o. 80 o. 65 o. 40 o Problem # In a circle the inscribed angle measures 40 o. If the diameter of the circle is 18, then the length of the arc() is.... E. Not enough information 19. is a parallelogram with m = (3x 4y) o and m = (x + y) o. The value of the fraction is. Not enough information... (3x  4y) (x + y) Problem # circle in the coordinate plane has = (3, 4) and = (9, 2) as endpoints of its diameter. The center of the circle is. (6, 3).. (3, 3). (3, 1) 21. In, = 12, m = 60 o and m = 45 o. Find the area of Problem # In right triangle m = 48 o, = 10 and bisects. Then, to the nearest thousandth, = Problem # If the corresponding sides of two triangles are proportional, which of the following are not proportional in the triangles?. orresponding angle bisectors. orresponding altitudes. orresponding medians. reas of the triangles
5 24. In right triangle with right angle at, is a median. If = 10 and = 16 then = Problem # is a square. E = F = G = H = ¼. If a point is chosen at random in the square, what if the probability it lies in either of the four corner triangles? H G F.... Not enough information E Problem #25
6 Tie reaker Questions Name Please Print School The following three questions will used to break ties for first, second and/or third place. The will be used in the order they are given. e sure you have answered the previous 25 questions before proceeding to the tiebreaker questions. Your answers should be written clearly with reasonable justification of your work. 1. In the figure, and m = 120 o. If = 5, and = 8, find the perimeter of the quadrilateral. Hint: Extend the segments and to intersect at E. E 5 Tiebreaker #
7 Tie reaker Questions Name Please Print School 2. plane intersects a sphere and is perpendicular to a diameter of the sphere at a point that is 8 inches from the center of the sphere. The diameter of the sphere is 20 inches. What is the area of the circle that is the intersection of the plane and sphere?
8 Name Please Print School 3. is a square with a side of 12 inches. P is a point is in the interior of the square such that P = P = PQ. etermine P. P Q Tiebreaker #3
9 Tie reaker Questions Name Key Please Print School The following three questions will used to break ties for first, second and/or third place. The will be used in the order they are given. e sure you have answered the previous 25 questions before proceeding to the tiebreaker questions. Your answers should be written clearly with reasonable justification of your work. 1. In the figure, and m = 120 o. If = 5, and = 8, find the perimeter of the quadrilateral. Hint: Extend the segments and to intersect at E. E = 60 o, E = 30 o, E = 60 o. E =, and E = 10. In, E = 18, and = so E =. Therefore, = = = + 13 = 13( + 1) (optional) E Tiebreaker # plane intersects a sphere and is perpendicular to a diameter of the sphere at a point that is 8 inches from the center of the sphere. The diameter of the sphere is 20 inches. What is the area of the circle that is the intersection of the plane and sphere? Since the diameter of the sphere is 20 inches, then O = 10 and O = 8. Thus = 6. Therefore the area of the circle of intersection between the plane and the sphere is 36 sq. in O
10 3. is a square with a side of 12 inches. P is a point is in the interior of the square such that P = P = PQ. etermine P. R P is isosceles so bisects. So Q = 6. lso, PR = Q = 6. Thus, If P = x, then R = x and R = 12 x. Therefore, (P) 2 = (12 x) Hence, x 2 = x + x Therefore, 24x = 180. So x = 7.5. Thus, P = 7.5. Q Tiebreaker #3 Key (25 Questions) E E E
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