1. What term describes a transformation that does not change a figure s size or shape?
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- Eleanore O’Neal’
- 7 years ago
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1 1. What term describes a transformation that does not change a figure s size or shape? () similarity () isometry () collinearity (D) symmetry For questions 2 4, use the diagram showing parallelogram D. E H I F D G 2. reflection across EG carries parallelogram D onto itself. 3. rotation of 90 about I carries parallelogram D onto itself. 4. rotation of 180 about I carries parallelogram D onto itself. 5. Which of these is equivalent to a translation? () a reflection across one line () a composition of two reflections across intersecting lines () a composition of two reflections across parallel lines Page 1 of 47 Revised 11/13/2012
2 6. regular polygon with n sides is carried onto itself by a positive rotation about its center that is a multiple of 60, but less than 360. Which could NOT be the value of n? () 3 () 4 () 5 (D) 6 7. In the diagram, g h and lies on line g. 10 cm 5 cm g The figure is reflected across line g, and its image is reflected across line h. What is the distance from line g to the final image of point? () 5 cm () 15 cm () 20 cm (D) 25 cm 8. What is the image of the point ( 4, 6) under the transformation T x, y y, x () (6, 4) () ( 6, 4) () (4, 6) (D) ( 4, 6) h? Page 2 of 47 Revised 11/13/2012
3 9. figure is rotated about the origin by 180, then is translated 4 units right and one unit up. Which describes the results of the two transformations? () xy, x 4, y 1 () xy, x 4, y 1 () xy, y 4, x 1 (D) xy, y 4, x The point (4, 3) is rotated 90 about the origin. In which quadrant is '? () I () II () III (D) IV 11. figure is reflected across the line y = 2, then reflected across the line y = 4. Which single transformation results in the same image? () a reflection across the line y = 3 () a reflection across the line y = 6 () a translation 2 units up (D) a translation 4 units up 12. Point is the image of point under a transformation T. Line is the perpendicular bisector of at point M. Which describes the transformation T? () a reflection across () a 90 rotation about M () a translation by the vector from to M (D) a dilation about M with scale factor Page 3 of 47 Revised 11/13/2012
4 For questions 13 16, determine if the described transformation(s) is/are an isometry. 13. reflection is an isometry. 14. composition of two reflections is an isometry. 15. dilation is an isometry. 16. composition of a rotation and a dilation is an isometry. 17. In, M is the midpoint of and N is the midpoint of. For which type of triangle is 1 MN? 2 () equilateral only () isosceles only () scalene only (D) any triangle Page 4 of 47 Revised 11/13/2012
5 18. Use the diagram. m k Which series of reflections would result in a rotation of 44 about? () reflect across k then reflect across () reflect across then reflect across k () reflect across then reflect across m (D) reflect across m then reflect across 19. fter a figure is rotated, P P. Which statement(s) could be true? () The center of rotation is P. () The angle of rotation is a multiple of 360. () Either or or both. (D) Neither nor Page 5 of 47 Revised 11/13/2012
6 For questions 20 23, use the diagram where is the reflection of across PQ. P Q 20. P = P 21. PQ 22. Q = Q 23. PQ Page 6 of 47 Revised 11/13/2012
7 For questions 24 26, use the diagram where D is a quadrilateral with D and D. Diagonals and D intersect at E. E D 24. E E 25. DE E 26. DE E Page 7 of 47 Revised 11/13/2012
8 27. Use the figure. y x transformation T is defined as x, y x, y. Which shows the image of figure under T? () y () y x x () y x Page 8 of 47 Revised 11/13/2012
9 For questions 28 29, a transformation S is defined as xy, 3 x, y The pre-image of 3, 6 under S is 9, S is an isometry Given point is located at (1, 3). What is the final image of after this series of transformations? (1) Reflect across the y axis. (2) Translate the image such that xy, x 4, y 2. () ( 1, 3) () ( 3, 5) () ( 3, 1) (D) ( 5, 5) 31. Which transformation does NOT preserve the orientation of a figure? () dilation () reflection () rotation (D) translation Page 9 of 47 Revised 11/13/2012
10 For questions 32 35, determine if the mapping is an isometry. 32. xy, x, y x, y x, y 34. x, y y, x 35. x, y 2 x, y is an isometry. is an isometry. is an isometry. is an isometry. 36. figure is transformed in the plane such that no point maps to itself. What type of transformation must this be? () dilation () reflection () rotation (D) translation Page 10 of 47 Revised 11/13/2012
11 For questions 37 38, determine the truth of the statements about rotations. 37. Rotations preserve the orientation of a figure. 38. Under a rotation, no point can map to itself. For questions 39 41, point P is located at (6, 0) and undergoes a transformation. 39. rotation of 90 about P results in P = P. 40. translation by the vector 6, 0 results in P = P. 41. reflection about the x axis results in P = P Page 11 of 47 Revised 11/13/2012
12 For questions 42 43, use the diagram which shows has been reflected across an unknown line, then reflected across line m to produce. y m x 42. The equation of line is x = If were reflected across line m first, then reflected across line to produce, the equation of line would be x = Page 12 of 47 Revised 11/13/2012
13 For questions 44 46, consider a triangle that has been transformed through rigid motions and its image compared to XYZ. Determine if the given information is sufficient to draw the provided conclusion. 44. Given onclusion X Y XYZ Z 45. Given onclusion X Y XYZ YZ 46. Given onclusion X XY XYZ YZ Page 13 of 47 Revised 11/13/2012
14 47. Look at the figure below. Look at these three figures. II I III Which figures are congruent to the first figure? () I only () II only () I and II only (D) I, II, and III 48. Use the diagram. 1 r s m n Which statement would be used to prove lines r and s are parallel? () 1 and 3 are congruent () 2 and 7 are complementary () 4 and 1 are congruent (D) 8 and 6 are supplementary Page 14 of 47 Revised 11/13/2012
15 For questions 49 51, evaluate whether the image of a figure under the described transformation is congruent to the figure. 49. transformation T follows the rule x, y x 3, y congruent to the figure. 50. transformation T follows the rule x, y y, x congruent to the figure. 51. transformation T follows the rule x, y x, 2y to the figure.. The image of a figure under T is. The image of a figure under T is. The image of a figure under T is congruent 52. In the diagram, m n and p q. p q m n x 88 What is the value of x? () 44 () 88 () 92 (D) Page 15 of 47 Revised 11/13/2012
16 For questions 53 54, consider where = and m m m m Use the Venn diagram. Quadrilaterals I II Parallelograms Rhombi Squares Rectangles III IV V Trapezoids VI VII Kites quadrilateral D has 4 lines of symmetry. Identify the area of the diagram in which D resides. () III () IV () V (D) VII Page 16 of 47 Revised 11/13/2012
17 56. On the coordinate plane, draw triangles and such that: (1) = (2) has been rotated In the diagram, m is the perpendicular bisector of at, and rm D E. D m E Prove D E. 58. In the diagram, D is the perpendicular bisector of. D Prove D D. 59. Prove the sum of the measures of the interior angles of any triangle is Page 17 of 47 Revised 11/13/2012
18 60. onstruct an equilateral triangle inscribed in circle O. O 61. Use the figure. y F D E x (a) Transform by reflecting it across the y-axis to produce. (b) Describe a transformation, or composition of transformations, that maps (c) Describe a single transformation that maps to DEF. to DEF Page 18 of 47 Revised 11/13/2012
19 62. Jimmy is building a go-cart to race. Jimmy sits in a seat toward the left rear corner of the cart. To see other carts behind him, he mounts 3 plane mirrors: one on each side of the cart and one in the center. The scale diagram shows a top view of the cart. Front of art center mirror left side mirror right side mirror J Seat 1 foot (a) Jimmy s eyes are at point J. The mirrors are mounted in a way that Jimmy does not block his own view. Show the area behind the cart that Jimmy can see using the mirrors. (b) Describe how Jimmy could improve his visibility behind the cart using the same three mirrors Page 19 of 47 Revised 11/13/2012
20 63. The diagram shows regular hexagon DEF with center O. F O E D For each part, fill in the blank. (a) is the image of under a 60 rotation about O. (b) The figure is reflected across D. The pre-image of E is. (c) F is the image of D when the figure is rotated about O. (d) is first reflected across F. That image maps to F when reflected across. 64. Draw the image of when reflected across line m. m Page 20 of 47 Revised 11/13/2012
21 65. Draw the image of when rotated 70 about O. O Page 21 of 47 Revised 11/13/2012
22 66. is the image of when a reflected about line. onstruct line. ' ' ' Page 22 of 47 Revised 11/13/2012
23 67. is the image of when rotated about point O. onstruct point O and compute the angle of rotation. ' ' ' Page 23 of 47 Revised 11/13/2012
24 68. Use scalene right triangle. Describe and sketch the transformations required so that the union of the triangle and its image forms (a) a kite. (b) an isosceles triangle. (c) a rectangle Page 24 of 47 Revised 11/13/2012
25 69. Right triangle PQR has sides of length 6 units, 8 units, and 10 units. The triangle is dilated by a scale factor of 4 about point Q. What is the area of triangle P'Q'R'? () 96 square units () 192 square units () 384 square units (D) 768 square units 70. In the diagram below DH. D 4 9 H 8 y What is the value of y? () 13 () 18 () The ratio of the side lengths of a triangle is 3:6:8. second triangle is similar to the first and its shortest side measures 8.0 centimeters. What is the length of the longest side of the second triangle? () 3.0 cm () 10.7 cm () 13.0 cm (D) 21.3 cm Page 25 of 47 Revised 11/13/2012
26 72. In the diagram, a student has placed a mirror on level ground, then stands so that the top of a nearby tree is visible in the mirror. 2 m 3 m 36 m What is the height of the tree? () 24 m () 35 m () 41 m (D) 59 m 73. In the diagram, JG QR. Q J 7 R x 4 G 8 P What is the value of x? () 11 () 6 () 5 (D) Page 26 of 47 Revised 11/13/2012
27 74. Which figure contains two similar triangles that are NOT congruent? () () () (D) 75. Sally constructs a triangle where two of the angles measure 50 and 60. Tom constructs a triangle where two of the angles measure 50 and 70. What is true about the two triangles? () The triangles cannot be similar. () The triangles could be similar. () The triangles must be similar. 76. Triangle has vertices 2, 2, 5, 5, and 5, 3 point 1, 1 with scale factor 4. What is the location of '? () 8, 8 () 10, 10 () 11, 5 (D) 14, 6. The triangle is dilated about the Page 27 of 47 Revised 11/13/2012
28 77. Use the diagram. y m x Dilate line m about the origin with scale factor 2. What is the equation of the line s image? () y = 2x + 2 () y = 2x + 4 () y = 4x + 2 (D) y = 4x Which is NOT a criterion for triangle similarity? () angle-angle () angle-side-angle () side-angle-side (D) side-side Page 28 of 47 Revised 11/13/2012
29 79. Use the diagram. m y P x Dilate line m by a factor 1 2 about P. Which shows the result of the dilation? () y m' () y m' P x P x () m' y (D) y m' P x P x Page 29 of 47 Revised 11/13/2012
30 80. In the diagram, segments and D intersect at E, F lies on, and m E 60. D E F 60 The two segments are dilated about F with scale factor 1. What is m 2 E? () 30 () 60 () 90 (D) Page 30 of 47 Revised 11/13/2012
31 For questions 81 83, use the diagram. E D G F Let the figure be dilated with scale factor k, where k 0 and k m E k m E. 82. G' is between ' and F'. 83. D k D Page 31 of 47 Revised 11/13/2012
32 84. In the diagram, D is dilated with center O to produce '''D', and 1. 3 ' O D' ' ' D What is O? () 1 3 () 1 2 () 2 (D) Page 32 of 47 Revised 11/13/2012
33 85. Use the figure below. y x Which shows the figure dilated about the origin with scale factor 1.5? y y () () x x () y x 86. One vertex of a polygon is ( 4, 5). If the polygon is dilated about the point (0, 2) with scale factor 2, what is the location of? () ( 8, 8) () ( 8, 10) () (8, 10) (D) (8, 4) Page 33 of 47 Revised 11/13/2012
34 87. J (5, 7) is the image of J(3, 3) after a dilation with scale factor 3. Where is the center of dilation? () ( 3, 9) () (0, 0) () (2, 1) (D) (4, 5) 88. Fred stands at corner of a rectangular field shown below. He needs to get to corner. 9 m D 12 m What is the shortest distance from to? () 9 m () 13 m () 15 m (D) 21 m 89. Use the right triangle. What is the value of x? x 15 8 What is the value of x? () 7 () 161 () 7 (D) Page 34 of 47 Revised 11/13/2012
35 90. Use the diagram. b h a x D y c Which is equal to h? () () () (D) ay bx xy ab 91. Use the diagram. E F D Given: D E Prove: E D Page 35 of 47 Revised 11/13/2012
36 92. In the diagram, is a dilation of. ' ' ' (a) Find the center of dilation O. (b) ompute the scale factor of the dilation. 93. Prove that any two isosceles right triangles are similar. 94. In the diagram, is a right triangle with right angle, and D is an altitude of. b a d e c D Use the fact that D D to prove a b c. 95. Triangle One has vertices (2, 4), (2, 8), and (5, 4). Triangle Two is similar to Triangle One and has two of its vertices at ( 1, 1) and ( 7, 1). (a) Draw Triangle One and Triangle Two on the coordinate plane and label the vertices. (b) Draw and label a third triangle that is similar to Triangle One, has two vertices at ( 1, 1) and ( 7, 1), but is not congruent to Triangle Two Page 36 of 47 Revised 11/13/2012
37 96. Triangle has sides of length 3, 6, and 8. Triangle DEF is similar to triangle and has one side with length 24. (a) What are the possible scale factors of the dilation of triangle to triangle DEF? (b) ompute one possible area of triangle DEF. 97. The diagram shows a surveyor s map. 950 m m The surveyor is trying to measure the direct distance between points and, which are on opposite sides of a lake. From point, point is 950 meters away in a direction 20 west of north. From point, point is 880 meters away in a direction 50 east of north. Which represents the distance between and? 2 2 () cos70 () () sin50 sin sin 70 sin onsider a triangle. Which statement is true? () () () (D) c a b 2abcos c a b 2abcos c a b 2abcos c a b 2abcos Page 37 of 47 Revised 11/13/2012
38 99. small airplane flies due north at 150 kilometers per hour. wind is blowing towards the direction 60 east of north at 50 kilometers per hour. Which figure represents the final speed and direction of the airplane? N N N () () () (D) N 100. Use the diagram What is cos? () () () (D) Page 38 of 47 Revised 11/13/2012
39 For questions , consider a triangle and each given set of measurements. 101.,, and m are sufficient to solve the triangle using the Law of Sines. 102.,, and m are sufficient to solve the triangle using the Law of Sines. 103.,, and are sufficient to solve the triangle using the Law of Sines The diagram shows a parallelogram D D What is the parallelogram s area? () () 15 () 15 3 (D) Page 39 of 47 Revised 11/13/2012
40 105. In the diagram, is a non-right triangle. a b c Which describes the area of the triangle? () 1 2 ab () absin () 1 sin 2 ab (D) 1 cos 2 ab 106. Given: cos and sin What is the approximate value of cos 154? () 0.90 () 0.44 () 0.44 (D) Page 40 of 47 Revised 11/13/2012
41 For questions use the statement below. Given: n angle measures k, where k > sin k cos 90 k 108. sin k sin 180 k 109. cosk cos 180 k 110. Let cos m. What is the value of sin? () m () 1 m () 2 1 m (D) 1 m Page 41 of 47 Revised 11/13/2012
42 111. In where is a right angle, 7 sin. What is cos? 4 () () () 3 4 (D) Use the diagram Which statement is true? () () () 13 sin 5 12 cos tan Page 42 of 47 Revised 11/13/2012
43 113. Use the diagram x Which is the value of x? () () () (D) x 41cos35 tan35 x x cos35 41 x tan Use the diagram. 5 2 d 45 What is the value of d? () 5 () 5 2 () 10 (D) Page 43 of 47 Revised 11/13/2012
44 For questions , let cos x m cos 180 x 116. cos 90 x 117. sin 90 x = m = m = m 118. Let a cos28. Which statement is true? () a cos 62 () a cos 152 () a sin 62 (D) a sin What is cos 1 3 2? () 30 () 45 () 60 (D) Page 44 of 47 Revised 11/13/2012
45 120. In the diagram, < D and D = D. D Which statement is true? () cos sin D () cos sin D () cos sin D 121. What is tan 60? () () () (D) What is tan 1? () 30 () 45 () 60 (D) Page 45 of 47 Revised 11/13/2012
46 123. In GHI, the sine of angle G equals 1 2. GHI is a dilation of GHI about G with a scale factor of 2. What is the sine of angle G'? () 1 4 () () 2 (D) In triangle, m 25, a = 6.2, and b = 4. Find all possible measures of the remaining two angles and the third side In triangle, m 25, a = 6.2, and c = 4. (a) Find all possible measures of the remaining two angles and the third side. (b) Find all possible areas of the triangle Page 46 of 47 Revised 11/13/2012
47 126. The diagram shows a model of a closet floor on which Kim is laying carpet. (Measurements are approximate.) (a) What is the area of the closet? (b) The carpet Kim is using is cut by the carpet store in rectangular pieces from a 4-foot wide roll. What is the shortest length of carpet Kim would need to cover the closet floor in a single piece? Justify your answer Page 47 of 47 Revised 11/13/2012
48 KEY Selected Response Key # Question Type Unit ommon ore State Standard(s) DOK Level Key 1 M 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O M 1 G.O M 1 G.O M 1 G.O.4 2 D 8 M 1 G.O M 1 G.O M 1 G.O.5 1 D 11 M 1 G.O.5 2 D 12 M 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O M 1 G.O.10 1 D 18 M 1 G.O M 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O M 1 G.O MTF 1 G.O MTF 1 G.O M 1 G.O.5 1 D 31 M 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O M 1 G.O.2 1 D 37 MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O MTF 1 G.O M 1 G.O M 1 G.O.9 1 Shaded questions do not pertain to the non-honors SS-based Geometry course Page 1 of 12 Revised 10/23/2012
49 KEY Selected Response Key # Question Type Unit ommon ore State Standard(s) DOK Level Key 49 MTF 1 G.O MTF 1 G.O MTF 1 G.O M 1 G.O MTF 1 G.O MTF 1 G.O M 1 G.O R 1 G.O R 1 G.O R 1 G.O R 1 G.O R 1 G.O ER 1 G.O.5, G.MG R 1 G.O.3, G.O R 1 G.O R 1 G.O R 1 G.O R 1 G.O R 1 G.O R 1 G.O M 2.1 G.SRT M 2.1 G.SRT M 2.1 G.SRT.5 2 D 72 M 2.1 G.SRT M 2.1 G.SRT M 2.1 G.SRT.5 1 D 75 M 2.1 G.SRT M 2.1 G.SRT M 2.1 G.SRT.1a 2 78 M 2.1 G.SRT.2 1 D 79 M 2.1 G.SRT.1b 1 80 M 2.1 G.SRT.1a 1 81 MTF 2.1 G.SRT MTF 2.1 G.SRT MTF 2.1 G.SRT M 2.1 G.SRT M 2.1 G.SRT M 2.1 G.SRT.1 1 D 87 M 2.1 G.SRT M 2.2 G.SRT M 2.2 G.SRT M 2.1 G.SRT R 2.1 G.SRT R 2.1 G.SRT R 2.1 G.SRT R 2.1 G.SRT R 2.1 G.SRT R G.SRT.5, G.SRT.9, G.SRT.11 2 Shaded questions do not pertain to the non-honors SS-based Geometry course Page 2 of 12 Revised 10/23/2012
50 KEY Selected Response Key # Question Type Unit ommon ore State Standard(s) DOK Level Key 97 M 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT MTF 2.2 G.SRT MTF 2.2 G.SRT MTF 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT MTF 2.2 G.SRT MTF 2.2 G.SRT MTF 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT.8 1 D 114 M 2.2 G.SRT MTF 2.2 G.SRT MTF 2.2 G.SRT MTF 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT M 2.2 G.SRT.6 2 D 122 M 2.2 G.SRT M 2.2 G.SRT R 2.2 G.SRT R 2.2 G.SRT R 2.2 G.SRT.8, G.MD.3 2 Shaded questions do not pertain to the non-honors SS-based Geometry course Page 3 of 12 Revised 10/23/2012
51 KEY onstructed Response Solutions 56. This question assesses the student s ability to apply a transformation to a figure with certain conditions placed on the image. nswers will vary, but the image of point must be itself, and must have been rotated 90 (counterclockwise). Thus, is the center of rotation. See example at right. = 57. This question assesses the student s ability to do geometric proofs. nswers will vary. Solutions may be paragraph, 2-column, flow, or other type of proof. However, statements must be justified. Either transformation-based or Euclidean-based proofs are acceptable. Here is one example of a transformational geometry based proof: Statement Justification r D E given m m is the perpendicular bisector of given r definition of a reflection m rm definition of a reflection rm D E E is an isometry of D D E the image of a figure defined by a set of points is defined by the images of those points reflections are isometries figures are congruent if one is the image of the other under an isometry 58. This question assesses the student s ability to do geometric proofs. nswers will vary. Solutions may be paragraph, 2-column, flow, or other type of proof. However, statements must be justified. Either transformation-based or Euclidean-based proofs are acceptable. Here is one example of a transformational geometry based proof: Statement Justification D is the perpendicular bisector of given r definition of a reflection D D D r definition of a reflection r D D definition of a reflection r D D the image of a figure defined by a set of points is defined by the D images of those points D is an isometry of D reflections are isometries D D figures are congruent if one is the image of the other under an isometry D D corresponding parts of congruent figures are congruent Page 4 of 12 Revised 10/18/2012
52 KEY onstructed Response Solutions 59. This question assesses the student s ability to do geometric proofs. nswers will vary. The proof can be found in nearly every geometry textbook. 60. This question assesses the student s ability to construct figures. The solution should look something like this. Set compass equal to radius. hoose an arbitrary point on the circle to be the center of an arc and mark an arc on the circle. Use the intersection of the circle and the arc as the center of the next arc. Repeat. The six points are the vertices of a regular hexagon. lternating points are the vertices of an equilateral triangle. O 61. This question assesses the student s ability to apply a transformation to a figure, describe a series of transformations from one figure to another, and compose transformations into a single transformation, if possible. (a) See figure (b) Reflection across the line y = 2.5. y (c) composition of two reflections is a rotation whose center is the intersection of the two lines, and the angle of rotation is twice the angle between the lines. Since y = 2.5 is perpendicular to the y-axis, a rotation of 180 (or 180 ) about (0, 2.5) maps to DEF. F D E x Page 5 of 12 Revised 10/18/2012
53 KEY onstructed Response Solutions 62. This question assesses the student s ability to apply transformations to real-world situations. The areas that Jimmy can see behind the cart are shown in the diagram below. He has some major blind spots. Front of art center mirror left side mirror right side mirror J Seat student may solve this problem by changing positions of mirrors, moving Jimmy s seat so his eyes are in a different location, rotating mirrors, or any combination of these to improve the areas of visibility behind the cart. They may not change the dimensions of the car or the mirrors. For example: (1) moving the center mirror about 2 left (2) rotating the left mirror counter-clockwise 25. (3) rotating the right mirror clockwise 30 Front of art center mirror left side mirror right side mirror J Seat Page 6 of 12 Revised 10/18/2012
54 KEY onstructed Response Solutions 63. This question assesses the student s knowledge of rigid transformations. (a) (b) (c) 120 or 240 (d) E 64. This question assesses the student s ability to apply rigid transformations. m 65. This question assesses the student s ability to apply rigid transformations. O m O m O m O Page 7 of 12 Revised 10/18/2012
55 KEY onstructed Response Solutions 66. This question assesses the student s ability to apply rigid transformations. ' ' ' 67. This question assesses the student s ability to apply rigid transformations. (rc marks to construct perpendicular bisectors of and left off for clarity.) ' 134 ' O ' Page 8 of 12 Revised 10/18/2012
56 KEY onstructed Response Solutions 68. This question assesses the student s ability to apply rigid transformations. (a) reflect across line (b) reflect across line (c) rotate 180 about midpoint of segment 91. This question assesses the student s ability to do geometric proofs. nswers will vary. Solutions may be paragraph, 2-column, flow, or other type of proof. However, statements must be justified. Statement Justification D is perpendicular to given E is perpendicular to given E is a right angle definition of perpendicular D is a right angle definition of perpendicular E D all right angles are congruent E D reflexive property E D angle-angle similarity postulate 92. This question assesses the student s ability to apply similarity transformations. ' O ' ' O 7.2 cm. O 12.1 cm. The scale factor is approximately 7.2 cm cm Page 9 of 12 Revised 10/18/2012
57 KEY onstructed Response Solutions 93. This question assesses the student s ability to do geometric proofs. Given is an arbitrary isosceles right triangle. Let angle be the right angle. m 90 by the definition of right angle. The triangle sum theorem states m m m 180, so m m 90 Since is isosceles, two angles must be congruent. Since m and m must be greater than zero, neither can measure 90 and be congruent to angle. Thus, m m 45. onsider a second arbitrary isosceles right triangle XYZ with right angle at Z. y a parallel argument, it can be shown that m X m Y 45. XYZ by angle-angle similarity. 94. This question assesses the student s ability to do geometric proofs. Given D, then e a 2, so a ce. a c Given D, then d b, so b c Thus 2 b a b ce cd c d e c c c. cd. 95. This question assesses the student s ability to apply similarity transformations. (a) The lengths of the sides of (right) Triangle One are 3, 4, and 5. The distance between the two vertices given for Triangle Two is 6. Since it is not stated which vertices of Triangle One correspond to Triangle two, there are three possible scale factors: 2, 1.5, and 1.2. If the scale factor is 2, then the sides of Triangle Two have lengths 6, 8, and 10. Since the given side is a leg of length 6, the other leg of length 8 could be one of four segments. The third vertex could be at ( 7, 9), ( 7, 7), ( 1, 9), or ( 1, 7). (See diagram.) ny one of these is a correct answer for the third vertex. If the scale factor is 1.5, the sides have length 4.5, 6, and 7.5, and third vertex could be at ( 7, 5.5), ( 7, 3.5), ( 1, 5.5), or ( 1, 3.5). (Not shown) If the scale factor is 1.2, the sides have length 3.6, 4.8, and 6, and third vertex could be at ( 4.8, 3.8), ( 4.8, 3.5), ( 3.2, 5.5), or ( 3.2, 3.5) to the nearest tenth. (Not shown. Students do not have to compute the location exactly; a reasonable approximation is sufficient.) (b) One of the other scale factors must be chosen and a third vertex located. y x Page 10 of 12 Revised 10/18/2012
58 KEY onstructed Response Solutions 96. This question assesses the student s ability to apply similarity transformations and compute the area of a non-right triangle. (a) The possible scale factors are 8, 4, and (b) Let a = 3, b = 6, and c = 8, using the law of cosines, cos. So, cos cos The area of the triangle is 1 absin sin The new triangle would be scaled up by a factor of 3, 4, or 8; the area would then scale by 9, 16, or 64. The resulting area would be about 69, 122, or This question assesses the student s ability to use the law of sines. sin sin 25 sin114 sin 25 Using the law of sines,, so , so c c 4 sin16 sin 25 There is a second possible solution, as this is an ambiguous case: , so c 2.6. c This question assesses the student s ability to use the law of cosines, law of sines, and the general formula for the area of the triangle (a) Using the law of cosines, (b) The area of the triangle is sin sin 25 b cos25, so b 3.1. Using the law of sines,, so sin This question assesses the student s ability to apply the right triangle trigonometry to real-world situations. (a) The area of the closet is equal to the area of a containing rectangle minus the areas of the two shaded right triangles. Using right triangle trigonometry, the smaller right triangle has legs of 1.25 sin and 1.25 cos The larger right triangle has legs of and The length of the rectangle is square 2 2 The area of the closet is feet. (b) The length of the closet is 4.57, so Kim cannot just by a 2 wide piece it will be too short. She will have to buy a 4.57 (about 4 7 ) piece to fit the length of the closet and cut off 2 feet of it so it fits the width Page 11 of 12 Revised 10/18/2012 2
59 KEY Notes on Practice Materials The Geometry Honors Practice Materials are provided to help teachers and students prepare for the SD Semester Exams in Geometry Honors. School choosing to teach regular Geometry using the SS should also use these materials. The questions are representative of the style, format, and type that will be on the exams. They are not, however, completely parallel in construction. That is, practice questions on a particular standard show how that standard may be assessed, but the questions on the actual exam could assess that standard in a different way. Teachers must provide students with opportunities to explore all aspects of a standard, and not simply focus on those addressed by the practice materials. There are 3 types of questions in the practice materials that will appear on the semester exams. M Multiple hoice. This is a traditional selected-response type of question. Each item will have 3 or 4 possible responses. MTF Multiple True/False. These items will have 2 4 true/false questions based on a common lead-in statement or concept. These should remain grouped together when provided for practice. R onstructed response. These items may have multiple parts that address one or more standards. The DOK level is the overall level of the item, though some parts may be at lower levels. Short R items average about 6 8 minutes to complete; longer R items average minutes to complete. fourth type of question is the Extended Response (ER). These will not appear on the semester exam at this time, but are indicative of longer, performance-type tasks that will appear on the Smarter alanced ssessments beginning in Sample solutions are provided for R and ER questions. Student methods may vary and any logical, mathematically correct approach, including different types of proof, should be accepted as correct. It is expected that students will have access to mathematical tools for these practice questions and the semester exam. Tools include compass, protractor, ruler, patty paper, and graph paper. Students may use calculators, including graphing calculators, constructed response questions Page 12 of 12 Revised 10/18/2012
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