# GEOMETRY. Part I. _,_ c. August 2010

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1 GEOMETRY Part I nswer all 28 questions in this part. Each correct answer will receive,',"eliits. No partial credit will be allowed. For each question, write on the "pnce provided the numeral preceding the word or expression that best coljlpletes the statement or answers the question. V [56] \> I. [n the accompanying diagram, MC ~ MYz. X Which two statements identify corresponding l:ongruent parts for these triangles? ( I) ~.xy and L C :::= L Y y _,_ c (2) :::= yz and LC ~ LY (3) C ==.xy and L :::= LY Z (4) C :::= yz and L :::= LY 2. support beam between the floor and ceiling of a house forms a 90 angle with the floor. The builder wants to make sure that the floor and ceiling are parallel. Which angle should the support beam form with the ceiling? (1) 45 (2) 60 (3) 90 (4) In the accompanying diagram, the vertices of /)J)EF are the midpoints of the sides of equilateral triangle C, and the perimeter of MC is 36 cm. What is the length, in centimeters, of EF? (I) 6 (2) 12 (3) 18 (4) 4 c 3 4. What is the solution ofthe following system of equations? y= (x + 3)2-4 y=2x+5 (1) (0, -4) (3) (-4, -3) and (0,5) (2) (-4, 0) (4) (-3, -4) and (5,0) 4 5. One step in a construction uses the endpoints of to create arcs with the same radii. The arcs intersect above and below the segment. What is the relationship of and the line connecting the points of intersection of these arcs? (1) collinear (2) congruent (3) parallel (4) perpendicular 5 6. If MC ~ LlZXY, ml = 50, and mlc = 30, what is mly? (1)30 (2)50 (3)80 (4)

2 50 GEOMETRY 7. In the accompanying diagram of G L MGE and!lold, L.GE ~ L.LOD, and E ~ OD. To prove that MGE and!lold are congruent by SS, what other information is needed? (1) GE ~ LD (3) L.GE ~ L.OLD (2) G ~ OL (4) LEG ~ L.ODL E o D Point is not contained in plane l. How many lines can be drawn through point that will be perpendicular to plane I? (1) one (2) two (3) zero (4) infinite 8 9. The equation of a circle is x 2 + (y - 7)2 = 16. What are the center and radius of the circle? (1) center = (0, 7); radius = 4 (3) center = (0, -7); radius = 4 (2) center = (0, 7); radius = 16 (4) center = (0, - 7); radius = 16 9 _ 10. What is an equation of the line that passes through the point (7, 3) and is parallel to the line 4x + 2y = 1O? (1)y=~x-~ (2)Y=-~x+ Ii (3)y=2x -I1 (4)y=-2x II. In MC, = 7, C = 8, and C = 9. Which list has the angles of MC in order from smallest to largest? (1) L., L., L.C (3) L.C, L., L. (2) L., L., L.C (4) L.C, L, L Tangents P and P are drawn to circle 0 from an external point, P, and radii O and O are drawn. IfmL.P = 40, what is the measure of L.O? (1) 140 (2) 100 (3) 70 (4) What is the length of the line segment with endpoints (-6, 4) and (2, -5)? (1) Jlj (2) J0 (3) m (4) Ms 13 GEOMETRY In which polygon does the sum of the measures of the interior angles equal the sum of the measures ofthe exterior angles? (1) triangle (2) hexagon (3) octagon (4) quadrilateral In the accompanying diagram of circle 0, chords and CD intersect at E. IfCE = 10, ED = 6, and E = 4, what is the length of E? o (1)15 (3)6.7 (2) 12 (4) In the accompanying diagram of M~C' medians D, E, and CF intersect at G. F G E If CF = 24, what is the length of FG? (1) 8 (2) 10 (3) 12 (4) 16 0 c If a line segment has endpoints (3x + 5, 3y) and (x - 1, -y), what are the coordinates of the midpoint of? (I) (x + 3, 2y) (2)(2x + 2, y) (3) (2x + 3, y) (4)(4x + 4, 2y) If the surface area of a sphere is represented by 144n, what is the volume in terms ofn? (1) 36n (2) 48n (3) 216n (4) 288n Which transformation of the line x = 3 results in an image that is perpendicular to the given line? (1)rxoax is (2)rJcaxis (3)rv~x (4)rX ~ In the accompanying diagram of regular pentagon CDE, E is drawn. What is the measure of LE? E( 08 (1) 36 (3) 72 (2) 54 (4) c 14. The lines represented by the equations y + ~x = 4 and 3x + 6y = 12 are (1) the same line (3) perpendicular (2) parallel (4) neither parallel nor perpendicular transformation of a polygon that always preserves both length and orientation is (1) dilation (2) translation (3) line reflection (4) glide reflection MC is similar to WEF. The ratio of the length of to the length of DE is 3:1. Which ratio is also equal to 3:1? (I) ml ml.d (3) area of MC area of WEF (2) ml. (4) per~meter of MC m L.F penmeter ofwef 23

3 52 GEOMETRY 24. What is the slope ofa line perpendicular to the line whose equation is 2y = -6x + 8? I (1) -3 (2) - (3) t (4) In the accompanying diagram., ofcircle C, m QT = 140 and mlp = 40. What is mrs? (1) 50 (3) 90 (2) 60 (4) GEOMETRY 53 Part II nswer all 6 questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, grapbs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [12] 29. The accompanying diagram shows \ isosceles trapezoid CD with II DC and D =: C. IfmLD = 2x and mlcd = 3x + 5, find mld. D (3x + 5) C 26. Which statement is logically equivalent to "If it is warm, then I go swimming"? (I) If! go swimming, then it is warm. (2) Ifit is warm, then I do not go swimming. (3) If! do not go swimming, then it is not warm. (4) Ifit is not warm, then I do not go swimming In the ~ol~anying diagram C of MCT, E II T. IfC = 3, C = 10, and CE Z = 6, what is ~E S the length of ET? (1) 5 (3) 20 (2) 14 (4) 26 T right circular cone has a base with a radius of 15 cm, a vertical height of 20 cm, and a slant height of 25 cm. Find, in terms of1t, the number ofsquare centimeters in the lateral area of the cone. III 28. Which geometric principle is used in the construction shown below? (l) The intersection of the angle bisectors of a triangle is the center of the inscribed circle. (2) The intersection of the angle bisectors of a triangle is the center of the circumscribed circle. (3) The intersection of the perpendicular bisectors ofthe sides ofa triangle is the center ofthe inscribed circle. (4) The intersection of the perpendicular bisectors ofthe sides ofa triangle is the V)... <'" I"'C center of the circumscribed circle In the accompanying diagram of MlQP, side HP is extended through P to T, mlqpt = 6x + 20, mlhqp = x + 40, and mlphq = 4x - 5. Find mlqpt. Q (6x + 20)"/ (4x - 5) T P H (Not drawn to scale)

4 54 GEOMETRY 32. On the line segment below, use a compass and straightedge to construct equilateral triangle C. [Leave all construction marks.] GEOMETRY 55 Part III nswer all 3 questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [12] - -S 35. In the accompanying diagram of quadrilateral CD with diagonal D, ml = 93. mld = 43. mlc= 3x + 5, m L DC = x + 19, and ml DC = 2x + 6. Determine if is parallel to DC. Explain your reasoning. D 1 93 ' 43' '(x + 19)' \2x + 6)' - (3x + 5)' c 33. In the diagram below, car is parked 7 miles from car. Sketch the points that are 4 miles from car and sketch the points that are 4 miles from car. Label with an X all points that satisfy both conditions. Car Car 36. The coordinates of the vertices of!'!,c are (1, 3), (-2, 2), and qo, -2), On the grid below, graph and label!'!,""c", the result of the composite transformation D 2 0 T J,_2' State the coordinates of ", ", and C". I I I I y 34. Write an equation for circle 0 shown on the accompanying graph. 13=l=8.~~_~ b=, t±tttmrlii ~tl,±t x + ~-t+ r L! I I ; I--l-~-j I I 8:t-t-±+±+±±ti±ttH±tt-1

5 ~ I 56 GEOMETRY 37. In the accompanying diagram,!l.rst T is a right triangle. The altitude, h, to the hypotenuse has been drawn. Determine the length ofh. ~ R. 5 ~ S Part IV nswer the question in this part. correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, diagrams, graphs, charts, etc. correct numerical answer with no work shown will receive only 1 credit. [6] 3S. Given: Quadrilateral CD has vertices (-5, 6), (6, 6), C(S, -3), and D(-3, -3). Prove: Quadrilateral CD is a parallelogram but is neither a rhombus nor a rectangle. [The use of the grid below is optional.] UJ

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