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Transcript
TEST NAME: Math II Geometry GCO TEST ID: 340678 GRADE: 09 10 SUBJECT: Mathematics TEST CATEGORY: My Classroom Math II Geometry GCO Page 1 of 46 11/19/14, Math II Geometry GCO Student: Class: Date: Read the passage 'Quilt Making' and answer the question below: Quilt Making Quilt Making Many years ago quilts were made from scraps of fabric to use on beds for warmth. Sewing the pieces of fabric together was called “piecing” the quilt. The pieces of fabric were frequently squares or triangles. The old pieced quilts are greatly valued today as works of art. Making quilts from the old patterns is a popular hobby. Natalie makes quilts using some of the old patterns. The following figures show the stages for one of the patterns Natalie will use for a new quilt. Figure 1 shows a pattern that begins with a square and four congruent triangles. The square is 4 inches on each side. The triangles are isosceles with a height of 2 inches, and the vertex angle is a right angle. Natalie will cut the square from one color and the four triangles from a different color or pattern. Figure 2 shows four more triangles that are cut from a different color or pattern being added to the first figure. The new triangles are isosceles with a right angle at the vertex. Math II Geometry GCO Page 2 of 46 A completed pattern is called one “square.” Natalie will make 20 squares for her quilt and will stitch them together in a 4by5 array. She adds 0.25 inch to each side of each square to create a seam when she sews the squares together. To finish the quilt, she sews a border that is 4 inches wide around the outside of the entire array. It takes Natalie a long time to make her quilt, but she has something beautiful for her bed when it is complete. 1. Read "Quilt Making" and answer the question. Figure 1 is reflected over and then reflected over Which transformation would result in the same image? A. reflecting Figure 1 over and rotating clockwise around the center of square ABCD B. reflecting Figure 1 over and rotating clockwise around the center of square ABCD C. rotating Figure 1 D. rotating Figure 1 clockwise around the center of square ABCD clockwise around the center of square ABCD Read the passage 'Quilt Making' and answer the question below: Math II Geometry GCO Page 3 of 46 2. Read "Quilt Making" and answer the question. The definition of congruence states two figures are congruent if one figure can be mapped onto the other through a sequence of rigid motions. According to this definition, which rigid motions could Natalie use to prove all four triangles in Figure 1 are congruent? A. Dilate each triangle by a scale factor of 4 around the center of the square ABCD, which results in corresponding sides and angles being congruent. B. Rotate each triangle around the center of the square ABCD, which results in corresponding sides and angles being congruent. C. Translate each triangle vertically and/or horizontally, which results in corresponding sides and angles being congruent. D. Reflect each triangle vertically and/or horizontally, which results in corresponding sides and angles being congruent. 3. Which transformation does not preserve distance? A. B. C. D. 4. Point lies at Point is reflected over the yaxis and then translated 7 units down to produce to produce from Which rule could have been used A. B. C. D. Math II Geometry GCO Page 4 of 46 5. A transformation to the graph of a function, takes the function and translates it left 5 units. Which equation represents A. B. C. D. 6. A function f(x) is described by the ordered pairs If the given function is translated 2 units to the right and then reflected across the x axis, what are the ordered pairs that describe the transformed function? A. B. C. D. 7. Figure D is a regular polygon. If D is rotated not concurrent with D, but if D is rotated the resulting figure is then the resulting figure is concurrent with D. Which shape could D be? A. A 9sided polygon B. An 18sided polygon C. A 20sided polygon D. A 36sided polygon Math II Geometry GCO Page 5 of 46 8. A regular polygon has n sides. Which algebraic expression represents the number of degrees of rotation around its center that would carry the polygon onto itself? A. B. C. D. 9. The figure below shows line segments a and b passing through a regular pentagon. Which statement describes a series of transformations that will result in a pentagon that is mapped onto the one shown? A. rotating the pentagon 72° clockwise about its center and then reflecting it across segment a B. rotating the pentagon 72° clockwise about its center and then reflecting it across segment b C. rotating the pentagon 90° clockwise about its center and then reflecting it across segment a D. rotating the pentagon 90° clockwise about its center and then reflecting it across segment b 10. Which number of degrees could a regular hexagon be rotated around its center to not carry it back onto itself? A. 60° B. 120° C. 150° D. 180° Math II Geometry GCO Page 6 of 46 11. The opposite sides of a certain quadrilateral are congruent to each other and parallel to each other. The midpoints of two opposite sides are connected to form a midsegment. Which of these statements is always true? A. Reflecting the quadrilateral across its midsegment carries the quadrilateral onto itself. B. Rotating the quadrilateral by 90º counterclockwise about the midpoint of its midsegment will carry it onto itself. C. Reflecting the quadrilateral across its midsegment carries the quadrilateral onto itself only if the adjacent angles of the quadrilateral are congruent. D. Rotating the quadrilateral by 90º counterclockwise about the midpoint of its midsegment will carry it onto itself only if the adjacent angles of the quadrilateral are congruent. 12. A trapezoid is graphed on the coordinate plane. Which transformation maps the trapezoid onto itself? A. rotation of 90° B. rotation of 180° C. reflection over the xaxis D. reflection over the yaxis Math II Geometry GCO Page 7 of 46 13. Rectangle ABCD is shown on the coordinate grid below. Which sequence of reflections will map rectangle ABCD onto itself? A. a reflection over the xaxis followed by a reflection over the yaxis B. a reflection over the xaxis followed by another reflection over the xaxis C. a reflection over the line followed by a reflection over the line D. a reflection over the line followed by a reflection over the line 14. Which transformation would map a rectangle onto itself? A. reflection over a diagonal B. reflection over the shorter side C. rotation of 40° around its center D. rotation of 180° around its center Math II Geometry GCO Page 8 of 46 15. Mr. Williams drew the image of a windmill with the perpendicular lines of symmetry shown below. He asked two of his students what rotation about the center point, O, will result in an image that looks like the original. Lara said 90° clockwise and Clark said 180°. Which student(s) answered correctly? A. only Lara B. only Clark C. both Lara and Clark D. neither Lara nor Clark 16. A regular hexagon is rotated on a coordinate plane. Which rotation would result in a hexagon with vertices at the same coordinates as the vertices of the original hexagon? A. a 60º clockwise rotation about the center of the hexagon B. a 90° clockwise rotation about the center of the hexagon C. a 60° clockwise rotation about one of the vertices of the hexagon D. a 90° clockwise rotation about one of the vertices of the hexagon Math II Geometry GCO Page 9 of 46 17. Line segment shown below, is reflected over the yaxis to form the line segment Rhoda concludes that the yaxis is a perpendicular bisector of Andy concludes that the lines and and are parallel. Whose conclusion is correct? A. neither Rhoda’s nor Andy’s B. both Rhoda’s and Andy’s C. only Rhoda’s D. only Andy’s 18. Which of these statements is NOT correct? A. A composition of reflections across two intersecting lines is equivalent to a rotation. B. A composition of reflections across three parallel lines is equivalent to one reflection. C. A composition of reflections across two parallel lines is equivalent to a translation. D. A composition of reflections across two intersecting lines is equivalent to a translation. Math II Geometry GCO Page 10 of 46 19. Square ABCD, shown on the graph below, is rotated 90° counterclockwise about point A to form square Which statement best describes line segments and A. and are parallel. B. and are perpendicular. C. and lie on the same line. D. and intersect but are not perpendicular to each other. Math II Geometry GCO Page 11 of 46 20. Line segment n is shown on the coordinate plane below. Which transformation would result in a line segment, perpendicular to n? A. a reflection across the xaxis B. a reflection across the yaxis C. a 90° clockwise rotation about the point D. a 180° counterclockwise rotation about the point that is 21. Clarisse drew line segment and then translated it 3 units to the right and 4 units down to form line segment She then connected and and and to form two more line segments. Which statement(s) about the resulting figure must be true? I. II. III. is a rectangle. A. I only B. III only C. I and II only D. I, II, and III Math II Geometry GCO Page 12 of 46 22. The diagram shows an angle and its image after a transformation. Which definition describes this transformation? A. a transformation that flips a figure across a line B. a transformation that turns a figure around a point C. a transformation that copies a figure directly over the existing figure D. a transformation that slides each point of a figure the same distance in the same direction 23. What is a definition of a translation of a polygon? A. Each point of the polygon moves parallel to, and the same distance as, every other point. B. Each point of the polygon moves perpendicular to, and the same distance as, every other point. C. Each point of the polygon moves the same distance in the same direction along a circle with the same radius. D. Each point of the polygon moves the same distance in the same direction along a perpendicular bisector of a side of the polygon. Math II Geometry GCO Page 13 of 46 24. Triangle with vertices coordinate plane below. is graphed on the Which transformation of would yield so that is parallel to A. a 180° clockwise rotation about the origin B. a 90° counterclockwise rotation about the origin C. a reflection across the yaxis D. a reflection across the xaxis Math II Geometry GCO Page 14 of 46 25. Figure 1 is mapped to Figure 2 under a transformation. Which series of transformations maps Figure 1 to Figure 2? A. reflection over the yaxis and a reflection over the xaxis B. reflection over the yaxis and a translation of 2 units down C. rotation of 90° counterclockwise around the origin and a reflection across the yaxis D. rotation of 90° clockwise around the origin and a translation of 2 units down Math II Geometry GCO Page 15 of 46 26. On the coordinate plane, the location of across the line across the line is found by reflecting is found by reflecting Which ordered pair corresponds with the location of A. B. C. D. 27. The parallelogram graphed on the coordinate plane is rotated 90° counterclockwise about the origin. Math II Geometry GCO Page 16 of 46 Which is the image of the parallelogram after this rotation? A. B. C. Math II Geometry GCO Page 17 of 46 D. 28. The triangle shown below is reflected across line l. It is then rotated 90º counterclockwise about the origin and then translated 2 units to the left to obtain triangle Which figure correctly shows A. Math II Geometry GCO Page 18 of 46 B. C. D. Math II Geometry GCO Page 19 of 46 29. Triangle RST is translated 4 units right and 2 units down. What are the coordinates of point R? A. B. C. D. 30. Triangle ABC is shown on the coordinate plane below. The triangle is reflected across the yaxis and then translated 3 units to the right to form triangle Which coordinate plane shows triangle Math II Geometry GCO Page 20 of 46 A. B. C. Math II Geometry GCO Page 21 of 46 D. 31. Figure 2 is a transformation of Figure 1. Which series of transformations map Figure 1 to Figure 2? A. a reflection across the xaxis and a translation 2 units right B. a reflection across the yaxis and a translation 2 units down C. a rotation of 90° clockwise about the origin and a translation 2 units down D. a rotation of 90° counterclockwise about the origin and a translation 2 units down Math II Geometry GCO Page 22 of 46 32. If is rotated 90 degrees clockwise about the origin and then translated 1 unit up and 2 units to the left, what are the endpoints of the resulting image? A. B. C. D. 33. George designs a computer program that can easily perform any number of transformations. To check his program, he enters the coordinates of the vertices of a quadrilateral as He then types instructions to reflect the quadrilateral across the xaxis 99 times and then across the yaxis 99 times. Which figure shows the transformed quadrilateral generated by the program? A. Math II Geometry GCO Page 23 of 46 B. C. D. Math II Geometry GCO Page 24 of 46 34. Triangle ABC has vertices at It is translated 1 unit left and 2 units up to form triangle vertices of triangle What are the A. B. C. D. 35. Which series of transformations can be used to prove that is congruent to as shown by the graphs below? A. rotate 90° counterclockwise about the origin and then translate it 8 units to the right B. reflect the left across the line and then translate it 4 units to C. rotate 90° clockwise about the origin and then reflect it across that resulted from the rotation D. reflect across the line and then reflect it across that resulted from the first reflection 36. The figure below shows plotted on a coordinate plane. The triangle is first rotated by 90° counterclockwise about the origin and is then reflected about the line Math II Geometry GCO Page 25 of 46 Which graph shows the transformed triangle? A. B. Math II Geometry GCO Page 26 of 46 C. D. 37. Which transformation maps each triangle in the xycoodinate plane to a congruent triangle? A. B. C. D. 38. A rectangle on a coordinate grid has vertices at and The rectangle is rotated 90° clockwise about the origin and then translated 6 units up and 2 units right. Which graph correctly shows the transformed rectangle? Math II Geometry GCO Page 27 of 46 A. B. C. Math II Geometry GCO Page 28 of 46 D. 39. Use the image to answer the question below. Which series of transformations would result in a figure that is congruent to the triangle ABC? A. rotation of clockwise about the origin and then dilation with a scale factor of centered at the origin B. rotation of clockwise about the origin and then dilation with a scale factor of 2, centered at the origin C. reflection across the line centered at the origin D. translation 3 units to right and then dilation with a scale factor of 3, centered at the origin and then dilation with a scale factor of 40. Quadrilateral Math II Geometry GCO shown on the coordinate plane below, is reflected Page 29 of 46 over the yaxis to form quadrilateral Which of these shows the correct location of quadrilateral A. B. Math II Geometry GCO Page 30 of 46 C. D. 41. Triangle is located in the third quadrant of a coordinate plane. If triangle is reflected across the yaxis to obtain triangle, which statement is true? A. lies in quadrant II and is congruent to B. lies in quadrant IV and is congruent to C. lies in quadrant II and is not congruent to D. lies in quadrant IV and is not congruent to Math II Geometry GCO Page 31 of 46 42. Maggie drew triangle ABC and then drew triangle DBE. Triangle ABC is congruent to triangle DBE only if is equal to what value? A. 29° B. 55° C. 62° D. 89° Math II Geometry GCO Page 32 of 46 43. Triangle ABC was reflected over the xaxis and then reflected over the y axis to form the triangle with vertices labeled M, N, and P in the coordinate plane below. Based on these reflections, which statements would prove that the triangles are congruent? A. B. C. D. Math II Geometry GCO Page 33 of 46 44. Triangle JKL and line segment RT are shown on the graph. If triangle JKL is congruent to triangle RST, what is the maximum number of possible coordinates for point S? A. 1 B. 2 C. 3 D. 4 Math II Geometry GCO Page 34 of 46 45. The figure below shows and on a coordinate plane. If and units, which statement is true? A. The triangles are similar because dilations preserve only angle measure, so B. The triangles are similar because dilations preserve only angle measure, so C. The triangles are congruent because they can be mapped onto each other through reflection, so and units. D. The triangles are congruent because they can be mapped onto each other through reflection, so and units. Math II Geometry GCO Page 35 of 46 46. Figure 1 is reflected across a vertical line, translated up, and rotated counterclockwise fewer than 90 degrees. The resulting figure is represented by Figure 2, as shown below. Which angle MUST be congruent to A. B. C. D. 47. Use the given triangles to answer the question. Triangle JKL is reflected across line a to form triangle MNO. Which one of these is true? A. B. C. D. Math II Geometry GCO Page 36 of 46 48. A triangle, PQR, is reflected across the xaxis and translated 3 units up to form triangle MNO. Which statement is true? A. The corresponding sides and angles of the given triangles are congruent. B. Only the corresponding angles of the given triangles are congruent. C. Only the corresponding sides of the given triangles are congruent. D. Only the base lengths of the triangles are congruent. 49. If triangle RST is congruent to triangle WXY, by the AngleSideAngle Theorem, which transformation would not map triangle RST to triangle WXY? A. horizontal shift 4 units left B. reflection across the yaxis C. dilation with a scale factor of 3 D. rotation of 90° around the origin Math II Geometry GCO Page 37 of 46 50. In the figure below, If Amy knows that is rotated and reflected to obtain which statement correctly lists additional information she would need to prove the relationship between these triangles? A. Reflections and rotations preserve angle measures but not side lengths, so showing can prove that the given triangles are similar by AA. B. Reflections and rotations preserve angle measures but not side lengths, so showing can prove that the given triangles are similar by AA. C. Reflections and rotations preserve side lengths and angle measures, so showing and can prove that the given triangles are congruent by SSA. D. Reflections and rotations preserve side lengths and angle measures, so showing and can prove that the given triangles are congruent by SAS. Math II Geometry GCO Page 38 of 46 51. In the figure below, is a reflection of Two students describe the given triangles as shown: Student I: Because reflections preserve side lengths, so and by SSS. Student II: Because reflections preserve both side lengths and angle measures, and so by SAS. Who has proven correctly? A. only student I B. only student II C. both student I and student II D. neither student I nor student II Math II Geometry GCO Page 39 of 46 52. The figure below shows and its reflection Based on this reflection, which of these can be used to prove A. B. C. D. Math II Geometry GCO Page 40 of 46 53. James is practicing proving geometric theorems. He writes the statements needed for the proof below in different boxes and mixes them up so they are not in order. In what order could James arrange the statements to make the proof correct? A. 1, 3, 5, 6, 4, 7, 2, 8 B. 1, 3, 5, 6, 2, 4, 7, 8 C. 3, 5, 6, 7, 1, 4, 2, 8 D. 3, 1, 5, 6, 7, 2, 4, 8 Math II Geometry GCO Page 41 of 46 54. In the diagram, line l is parallel to line m. When proving that the sum of the measures of the angles of a triangle is 180°, it is stated that because Why is this true? A. and because alternate interior angles are supplementary if lines are B. and because alternate interior angles are congruent if lines are parallel. C. and because corresponding angles are congruent if lines are parallel. D. and because sameside interior angles are congruent if lines are parallel. parallel. Math II Geometry GCO Page 42 of 46 55. A proof of the base angle theorem is shown. Which statements correctly complete the proof? A. B. C. D. Math II Geometry GCO Page 43 of 46 56. In triangle and Which of these can be proved using the triangle sum theorem? A. B. C. D. Math II Geometry GCO Page 44 of 46 57. Mike wants to prove that the diagonals of parallelogram JORD bisect each other. To do that, he labels the intersection of the diagonals as point N and composes the proof shown below. Step 1 JORD is a parallelogram. Given 2 Opposite sides of a parallelogram are congruent. ? and 3 4 Justification Definition of a parallelogram ? 5 ? 6 and 7 and Corresponding parts of congruent triangles are congruent. Definition of congruent line segments 8 N is the midpoint of bisects 9 and and Definition of midpoint bisects Definition of segment bisector The statement of step 3, and the justifications for steps 4 and 5, are missing from this copy. Which set of statements correctly completes Mike’s proof? A. B. C. D. Math II Geometry GCO Page 45 of 46 58. John writes the proof below to show that the sum of the angles in a triangle is equal to 180º. Which of these reasons would John NOT need to use in his proof? A. The sum of the angles on one side of a straight line is 180º. B. If a statement about a is true and replacing a with b is also true. C. When two parallel lines are cut by a transversal, the resulting alternate interior angles are congruent. D. When two parallel lines are cut by a transversal, the resulting alternate exterior angles are congruent. the statement formed by Math II Geometry GCO Page 46 of 46