ACTM Geometry Exam State 2010

Transcription

1 TM Geometry xam State 2010 In each of the following select the answer and record the selection on the answer sheet provided. Note: Pictures are not necessarily drawn to scale. 1. The measure of in the given figure is. 5 o. 88 o. 78 o o 2. In the figure, a linear pair of angles is., F., F Problem #1 12., F., F Problem #2. The area of the polygon on the grid is. 9½ ½. 10 Problem #. circle is drawn inside a rectangle so that it is tangent to three sides of the rectangle. If the dimensions of the rectangle are 2 and then the ratio of the circumference of the circle to the perimeter of the rectangle is. : 5. : 6. : 10. Not enough information 5. The diameter,, of the circle is 10. What are the possible lengths of and? 6. 5, 5., 21., 5 Problem #5., 2 26

2 6. tangent to a circle is always perpendicular to the radius of the circle drawn to the point of tangency. What is the slope of the tangent to the circle whose equation is (x 2) 2 + (y 1) 2 at the point (6,)? The equation of the line that is parallel to 2x y = 6, with a y-intercept that is four more than the y-intercept of the given line is. 2x y = 10. 2x y = 2. 2x y = 6. x 2y = 8. The points = ( 1, ), = (, 1), = (7, ) and = (2, 1) are the vertices of a parallelogram. The point of intersection of the diagonals of is. (, 0). (0, ). (0,). (, 0) 9. is the perpendicular bisector of. Which of the following statements is/are true? P: Q: R: Problem #9. P, Q. P, R. Q, R. P, Q, R 10. In the figure l m n and s and t are transversals. If = 2x +, = 6x + 1, = x + and F = x + 2 then the ratio of to is n m l s F Problem #10 t

3 11. onsecutive angles of the quadrilateral PQRS are supplementary. Which of the following always describes PQRS?. PQRS is a rectangle. PQRS is an isosceles trapezoid. PQRS is a kite. PQRS is a parallelogram 12. quadrilateral is inscribed in a circle. Which of the following statements is true?. is a parallelogram. Opposite pairs of angles are complementary. is an isosceles trapezoid. is a kite 1. PQRSTUVWX is a regular polygon. The measure of the angle PRQ is. 10 o. 0 o. 20 o. 6 o 1. n equilateral triangle and a regular hexagon are inscribed in a circle. The ratio of the perimeter of the hexagon to the perimeter of the triangle is. 2 : 1. 2 :. : 2. Not enough information 15. In the figure l m. pair of complementary angles is. 1, 5. 2,., 7., 8 l m Problem #

4 16. In the figure if l m, m 2 = x o, m = y o, and m = 8x y) o, then x = l o. 15 o. 5 o. 108 o m Problem # Given that and 1. Which of the following statements is/are true? 2 5 F P: 2 Q: F bisects R: and 5 are supplementary 1 Problem #17. P, Q. P, R. Q, R. P, Q, R 18. In the figure m QPS = 12 o and m PRS = 20 o. Then m QXS = P Q S X R T. 52 o. 20 o. 6 o Problem #18. 0 o 19. In and are altitudes. If = 0, = 7, and = 12, then, to the nearest hundredth, = Problem # In the circle if is a diameter, = 15 and the measure of the arc is 50 o, then = Problem #20

5 21.,, and are four distinct points on a line. Using these points, how many segments can be named? The number of regular solids that have triangular faces is prism and a pyramid have the same number of edges. If the base of prism is an octagon then the number of vertices in the pyramid is F 2 The right prism in the figure has a regular hexagonal base. If a side of the base is 6 and the height of the pyramid is 5, then = F' ' ' '. Not enough information 25. In the figure, = 25 and = 10. If = 15 then = ' ' Problem # Problem #25 e sure that you have answered all 25 of the preceding questions before attempting the tie-breaker problems. They will only be used to break ties for first, second and third place. They will be used in the order they are given.

6 Tie reaker Problems Name School Tie-reaker #1 is an inscribed quadrilateral and, and are collinear. Prove: [e sure to state clear reasons for each statement in the proof.] Tie reaker #1 Tie-reaker #2 In, and are medians intersecting at P. etermine the following ratio. Justify your conclusion. R( P) R( P) P Tie reaker #2

7 Name School Tie-reaker # In the figure RST is a right triangle, m SUT = 0 o, m SRT = 20 o and ST = 5. If UV RS, find UV. S V R 20 0 U Tie reaker # T

8 Geometry 2010 Key

9 KY Tie reaker Problems Name School Tie-reaker #1 is an inscribed quadrilateral and, and are collinear. Prove: [e sure to state clear reasons for each statement in the proof.] Tie reaker #1 Since opposite angles of an inscribed quadrilateral are supplementary then and are supplementary [Students may use an inscribed angle argument here] Since and are a linear pair of angles they are supplementary. Therefore,, since both are supplementary to. Tie-reaker #2 In, and are medians intersecting at P. etermine the following ratio. Justify your conclusion. R( P) P R( P) Since and are midpoints of their respective sides. Tie reaker #2 So P P and P P by alternate interior angle theorem. Thus, P P. Since medians intersect at a point that is two-thirds of the distance from the vertex to the midpoint of the opposite side, P = 2P. So the proportionality constant between the triangles is 2. Therefore, R( P) 1. R( P) V S Tie-reaker # In the figure RST is a right triangle, m SUT = 0 o, R 20 0 U m SRT = 20 o and ST = 5. If UV RS, find UV. Tie reaker # tan 20 o and tan 0 o. Thus, RT and UT o o RT UT tan 20 tan 0. RU = RT UT and UV = RU sin 20 o. Therefore, UV = ( Therefore, UV = o tan o tan 0 T )sin 20 o.