3. Given the similarity transformation shown below; identify the composition:

Transcription

1 Midterm Multiple Choice Practice 1. Based on the construction below, which statement must be true? 1 1) m ABD m CBD 2 2) m ABD m CBD 3) m ABD m ABC 1 4) m CBD m ABD 2 2. Line segment AB is shown in the diagram below. Which two sets of construction marks, labeled I, II, III, and IV, are part of the construction of the perpendicular bisector of line segment AB? 1) I and II 2) I and III 3) II and III 4) II and IV 3. Given the similarity transformation shown below; identify the composition: 1) RC, 80 T D 1 ( ABCD) mn C"', 4 2) T D 1 ( ABCD) mn C"', 4 3) RC, 80 T DC "',4( ABCD) mn 4) RC, 80 D 1 ( ABCD) C"', 4

2 4. The line constructed that connects the vertex of a triangle that is perpendicular to the opposite side is called the: 1) Altitude 2) Median 3) Perpendicular bisector 4) Angle bisector 5. The midpoint of each side of a triangle connects to the opposite vertex to form a: 1) Altitude 2) Median 3) Perpendicular bisector 4) Angle bisector 6. Given the following diagram, which angles are NOT congruent? 1) 1 and 4 2) 4 and 5 3) 2 and 7 4) 3 and 5 7. The perpendicular bisector of a line segment is the endpoints of the line segment. 1) congruent to 2) perpendicular to 3) equidistant from 4) parallel to 8. Two lines perpendicular to the same line are to each other. 1) Perpendicular 2) Parallel 3) Neither 4) Both 9. Two lines parallel to the same line are to each other. 1) Perpendicular 2) Parallel 3) Neither 4) Both

3 10. How many lines of symmetry are there for a regular hexagon? 1) 4 2) 6 3) 0 4) In the diagram below, the vertices of are the midpoints of the sides of equilateral triangle ABC, and the perimeter of is 36 cm. What is the length, in centimeters, of? 1) 6 2) 12 3) 18 4) In the diagram below of, medians,, and intersect at G. If, what is the length of? 1) 8 2) 10 3) 12 4) In the diagram below of, medians,, and intersect at G. The length of is 12 cm. What is the length, in centimeters, of? 1) 24 2) 12 3) 6 4) In the diagram below of and,, and. To prove that and are congruent by SAS, what other information is needed? 1) 2) 3) 4)

4 15. The diagonal is drawn in parallelogram ABCD. Which method cannot be used to prove that? 1) SSS 2) SAS 3) SSA 4) ASA 16. In the diagram below of, side is extended to point D,,, and. What is? 1) 5 2) 20 3) 25 4) Transversal intersects and, as shown in the diagram below. Which statement could always be used to prove? 1) 2 4 2) 7 8 3) 3 and 6 are supplementary 4) 1 and 5 are supplementary 18. In the diagram below, is isosceles with. If and, what is? 1) 27 2) 28 3) 42 4) Let P(2, 4) be a point on a figure, and let P be the corresponding point on the image. The figure is dilated by a scale factor of 4. What are the coordinates of P? 1) (-2, 0) 2) ( ½, 1) 3) (6, 8) 4) (8, 16)

5 20. Which transformation best describes the image of an object viewed through a microscope? 1) dilation 2) reflection 3) rotation 4) translation 21. If the accompanying diagram, DE AC 1) 4.5 2) 3 3) ) 12. Which of the following represents the length of BC? 22. AC is a diagonal of rectangle ABCD and EF joins the midpoints of AB and BC, respectively. If AC = 26, which of the following represents the length of EF? A E B 1) 52 2) 13 3) 10 4) In the diagram below of right triangle ACB, altitude CD is drawn to hypotenuse AB. If AB = 36 and AC = 12, what is the length of AD? 1) 3 2) 4 3) 6 4) B F C

6 AB C is a dilation of ABC from vertex A. CC = 4, AB = 8, AC = 10 and BC = 12. Which of the following represents the length of B C? 1) ) 12 3) ) Which of the following transformations are not distance preserving? 1) Rotation 3) Translation 2) Reflection 4) Dilation 28. In the diagram of quadrilateral ABCD,,, and diagonal is drawn. Which method can be used to prove is congruent to? 1) AAS 2) SSA 3) SAS 4) SSS

7 29. In the diagram of below,. Which reasons can be used to prove? 1) reflexive property and addition postulate 2) reflexive property and subtraction postulate 3) transitive property and addition postulate 4) transitive property and subtraction postulate 30. A father who is 6 ft. tall is standing next to his son. The father casts a 9 ft shadow. If the son casts a shadow that is 6 ft long then how tall is he? 1) 3 ft 3) 5 ft 2) 4 ft 4) 6 ft 31. A statue that is 12 ft tall casts a shadow that is 15 ft long. Determine the length of the shadow that a 8 ft statue casts. 1) 11.5 ft 3) 7 ft 2) 4.5 ft 4) 10 ft 32. Quadrilateral MNOP is a trapezoid with. If is the image of MNOP after a reflection over the x-axis, which two sides of quadrilateral are parallel? 1) and 2) and 3) and 4) and 33. Given that ABCD is a parallelogram, a student wrote the proof below to show that a pair of its opposite angles are congruent. What is the reason justifying that? 1) Opposite angles in a quadrilateral are congruent. 2) Parallel lines have congruent corresponding angles. 3) Corresponding parts of congruent triangles are congruent. 4) Alternate interior angles in congruent triangles are congruent.

8 34. Which of the following must be true about the diagonals of a rectangle? A. The diagonals are perpendicular B. The diagonals are congruent C. The diagonals bisect each other (1) A, only (3) B and C, only (2) A and C, only (4) A, B, and C 35. In the diagram, m A = x + 20, m B = 3x, and BCD is an exterior angle formed by extending AC to point D, and m BCD = 120. Find the value of x. (1) 10 (3) 35 (2) 25 (4) In the accompanying figure, ABCD is a parallelogram, m A = 4x + 15, and m C = 9x 40. Find the measure of angle D. D C 9x - 40 (1) 11 (3) 59 (2) 16 (4) Which of the following is not a degree of rotational symmetry for a regular pentagon? (a) 36 o (b) 72 o (c) 144 o (d) 288 o 38. In rectangle ABCD, ADB BCD. E is the midpoint of BD. Which of the following rigid motion(s) maps AD onto CB? (a) r (c) R o BD 90 (b) T (d) R E,180 DB A 4x+15 B

9 39. Which is an angle that is complementary to BOC? (a) BOE (b) COD (c) DOE (d) BOA 40. If the letter P is rotated 90 degrees, which is the resulting figure? (a) Df (b) P P (c) (d) 41. In the diagram below, PQ RS and transversal TU intersects PQ and RS at V and W, respectively. If m TVQ 5x 22 and m VWS 3x 10, find m WVQ. (a) 16 o (b) 58 o P (c) 24 o (d) 122 o 42. Two triangles are similar, and the ratio of each pair of corresponding sides is 3:1. Which statement regarding the two triangles is not true? (1) Their areas have a ratio of 9:1 (2) Their altitudes have a ratio of 3:1 (3) Their perimeters have a ratio of 3:1 (4) Their corresponding angles have a ratio of 3: The ratio of the sides of two similar triangles is 4 :1. If the area of the larger triangle is 144cm, find the area of the smaller triangle. (1) 36 (2) 18 (3) 12 (4) 9

10 44. Given the diagram below, DE BC. AE x, CE 3, AD 6, and DB 2. What is the value of x? (1) 7 (2) 9 (3) 4.5 (4) In the diagram below,. Which statement is not true? (1) (2) (3) (4) 46. If a quadrilateral ABCD is dilated by a scale factor of 2 and angle A = 72 o then angle A equals A) 72 o C) 144 o B) 36 o D) Can t be determined

11 Short Answer Practice 47. Construct an equilateral triangle given the following side length 48. Using a compass and straightedge, determine if ABC is an equilateral triangle. Explain your reasoning. 49. Using a compass and straightedge, construct the bisector of the angle shown below. [Leave all construction marks.] 50. Using a compass and straightedge, construct an equilateral triangle with as a side. Using this triangle, construct a 30 angle with its vertex at A. [Leave all construction marks.] A C B Triangle ABC is not equilateral because C does not fall on the intersection of the arcs drawn from A and B.

12 51. A) Using a compass and straightedge, construct the perpendicular bisector of. B) Construct a 45 angle with its vertex at the midpoint of AB. [Leave all construction marks.] 52. Given the diagram, find the center point of dilation for triangle ABC and triangle A B C 53. Determine the scale factor of the figure below given center O.

13 54. Given the diagram below, Drawing 2 is the image of Drawing 1 with a scale factor of r = 2 1 centered at O1, and Drawing 3 is the image of Drawing 2 with a scale factor of r = 2 3 centered at O2, find the following: a) Find the center of dilation going from drawing 1 to drawing 3 and label O 3. b) Find the dilation factor of drawing 1 to drawing Find the values of x and y: 56. A) Using a compass and a straight edge, find the midpoints of AB, BC, and AC and label them as D, E, and F respectively. B) Find the centroid of the triangle. B A C

14 57. Construct a square inscribed in the given circle: 58. Explain how you would know if a triangle was isosceles. 59. Answer the following questions using the figures below: a) What transformation was performed? b) Find the center of rotation. 60. Given triangle XYZ, translate it using vector KL, and label its image X Y Z. O If a triangle has two congruent sides it is isosceles Rotation

15 61. Given the diagram below, answer the following questions: a) Find the scale factor of the dilation already drawn. b) Using the triangle A B C create a third triangle dilated by 1.5cm centered at point O. c) Find the scale factor, in centimeters, from triangle ABC to triangle A B C. 62. Answer the following questions using the figures below: a) What transformation was performed? b) Find the line of reflection. Reflection

16 63. Given: <1 is supplementary to <2 Prove: l 1 l The triangle on the right has been mapped to the triangle on the left by a 120 o rotation about point P. a) Identify all six pairs of corresponding parts (vertices and sides). b) Write the transformation in function notation. 65. Rotate the triangle ABC 60 around point F using a compass and straightedge. If two lines are cut by a transversal such that a pair of corresponding angles are congruent then the lines are parallel.

17 66. Perform the following construction using construction tools. r m ( ABC) 67. A student has performed the rotation R O,50 o(bc) = B C. What have they done wrong? 68. In the diagram of below, A is the midpoint of, B is the midpoint of, C is the midpoint of, and and are drawn. If and, what is the length of? m C A B 50 C' B' B C They rotated clockwise instead of counterclockwise

18 A 70. Given: Parallelogram FLSH, diagonal Prove: C D 69. Given: ABCD is a rectangle, M is the midpoint of AB Prove: DMC is isosceles, M B, In a parallelogram, opposite sides are congruent In a parallelogram, opposite sides are parallel. When parallel lines are cut by a transversal, alternate interior angles are congruent

19 71. Given: Quadrilateral ABCD, diagonal,,,, Prove: ABCD is a parallelogram. Given Perpendicular lines form right angles All right angles are congruent Reflexive property Subtraction property ABCD is a parallelogram A quadrilateral with one set of sides parallel and congruent is a parallelogram. Corresponding parts of congruent triangles are congruent If two lines are cut by a transversal such that a pair of alternate interior angles are congruent, then the lines are parallel 72. The diagram below shows rectangle ABCD with points E and F on side. Segments and intersect at G, and. Prove: Given In a rectangle, opposite sides are congruent Each angle in a rectangle is a right angle All right angles are congruent Corresponding parts of congruent triangles are congruent Reflexive property Subtraction property

20 73. Given:,,,, Prove: Given Perpendicular lines form right angles All right angles are congruent If parallel lines are cut by a transversal, alternate interior angles are congruent Corresponding parts of congruent triangles are congruent ABCD is a rectangle 74. Given: BE CF Prove: DE AF D C F Given In a rectangle, opposite sides are congruent Reflexive property Addition property A E B In a rectangle, all angles are right angles All right angles are congruent Corresponding parts of congruent triangles are congruent

21 75. In the diagram, and intersect at E. If m AED = 9x + 10 and m BEC = 2x + 52, find the value of x. 76. In the diagram, transversal intersects parallel lines and PQ m RAN = 3x + 24 and m RBQ = 7x 16, find the value of x. at A and B, respectively. If 77. In the diagram, parallel lines and are intersected by transversal at G and H, respectively. If m AGH = 4x + 30 and m GHD = 7x 9, what is the value of x? 78. In the accompanying figure, AB intersects CD. Solve for x and y.

22 79. In the diagram of shown below, is drawn from vertex B to point A on, such that. In,,, and. In, and. a) Find. b) Find. c) Find the length of. 80. Construct the image of QRS after a dilation with a scale factor of 2 with Q as the center of dilation. 81. Construct the image of ABC after a dilation with center O and a scale factor of Construct the image of ABC after a dilation with center O and a scale factor of ½.

23 83. A B C is the image of ABC under a dilation. Use a straightedge to determine the center of dilation. 84. A student who is 72 inches tall wants to find the height of a flagpole. He measures the length of the flagpole s shadow and the length of his own shadow at the same time of day, as shown in his sketch below. Explain the error in the student s work. 48 and 128 are flipped around in the fraction. 128 is in the triangle with x not 72 in. 85. A girl 160 cm tall, stands 360 cm from a lamp post at night. Her shadow from the light is 90 cm long. How high is the lamp post? 160 cm 90 cm 360 cm 86. Given: NPVR is a parallelogram Prove: NOW ~ SWT Given In a parallelogram opposite sides are parallel When parallel lines are cut by a transversal, alternate interior angles are congruent

24 87. XY Z is a dilation of XYZ from vertex X. YY = 2.4, XY = 6, YZ = 10 and XZ = 8. What is the length of Y Z? 88. Given BAT with coordinates B(-3,1), A(1,4) and T(5,-3). Perform a dilation of BAT from center O(0,0) and a scale factor of 2. Graph and determine the coordinates of the images of points B, A and T. 89. Given the diagrams below, state whether or not a dilation that maps A to A and B to B exists. Explain. a) b)

25 90. Given: D is the midpoint of AC AC BD Prove: ABC is isosceles Given A midpoint divides a segment into two congruent parts Perpendicular lines form right angles All right angles are congruent Reflexive property 91. Write the following composition of transformations in function notation: Corresponding parts of congruent triangles are congruent A triangle with two congruent sides is isosceles 92. Complete the chart to the right. Sequence of rigid motions Composition in function notation Triangle congruence statement

26 93. A wall casts that is 4 feet tall casts a shadow that is 6 feet long. How tall is a tree that casts a 24 ft shadow? 94. At noon, Jimmy is standing 22 feet from a flagpole. The sun shines down on Jimmy and casts a shadow 5 feet long. How tall is the flagpole if it casts a 27 foot shadow? 95. Solve for x. 96. Given: j ll k ll m Solve for each of the following if BC = x + 2; BA = 9; EF = x + 3; ED = 12. a) x = b) BC = c) EF =

27 97. Given: Isosceles triangle ACE with AC AE BDE FDC Statements Reasons Prove: CBD ~ EFD Given In a triangle, angles opposite congruent sides are congruent Reflexive property Subtraction property 98. Given: AC and BD intersect at E AB CD Prove: AEB ~ CED Statements Reasons Given A E D When parallel lines are cut by a transversal, alternate interior angles are B C congruent 99. Solve for x and y y = x = y x + 2 x

28 100. Perform the following similarity transformation: D 1 O, (T AN ( URB)) 2 O 101. A similarity transformation for triangle ABC is described by (D O,2 (r DG ( ABC))). Locate and label the image of triangle ABC under the similarity.

29 102. Dilate circle A by a scale factor of 2 with O being the center of dilation Dilate circle C with radius CA from center O with a scale factor r = Given: AB and FD bisect each other. Prove: ADE EFB Statements Reasons Given A bisector divides a segment into two congruent parts Vertical angles are congruent

30 105. For each given pair of triangle, determine if the triangles are similar and provide your reasoning. If the triangles are similar, write a similarity statement relating the triangles. a. b. c The coordinates of triangle ABC are A(5,4), B(3,4) and C(1,1). State the coordinates of triangle A B C after R 0, State the coordinates of the image of trapezoid ABCD after a transformation T 3,-4.

31 108. State the coordinate of the image of triangle XYZ after a transformation r x-axis Name and draw in a translation vector that takes ABC to A B C Translate triangle ABC across vector DE.

32 Translations, rotations, reflections 111. What are the rigid motions? Dilations 112. What transformation is NOT a rigid motion? 113. State 4 degrees of rotational symmetry for each of the following regular polygons. a) Octagon b) Hexagon 114. True or False? ABC is dilated with a scale factor of r =4. The image is A' B' C'. False a) ABC A' B' C' b) ABC A' B' C' True c) AB 4( A' B' ) False d) 4( AB) A' B' True e) 1 ( A' B') AB 4 True f) BC AB B' C' A' B' True B' C' g) False BC B' A' h) 4( B) B' False 115. Alexandra is placing a pond in the backyard of their house. Alexandra would like the pond, the bench, and the swing set, to be equidistant from each other. A sketch of the yard appears below, with the centers of the bench and swing set listed as B and S. Using a compass and a straightedge show all possible location(s) for the pond and mark with an X. Front Yard

33 Do Now Day 1 1. Dilate circle A by a scale factor of 1/2 if A is the center of dilation. 2. Dilate circle A by a scale factor of 2 if O is the center of dilation.

34 Do Now Day 2 1. Determine the number of degrees of rotational symmetry. 2. Solve for x in the following if AB CD. 3. Solve for x in the following if AB CD.

35 1. Given: FE BC AB ED AF CD Prove: AF DC Do Now Day 3 Statements Reasons Given Reflexive property Addition Property If parallel lines are cut by a transversal, alternate interior angles are congruent Corresponding parts of congruent triangles are congruent 2. Given: <ADC and <BEA are right angles Prove: ADC~ AEB Statements Reasons Given All right angles are congruent Reflexive property

36 Do Now Day 4 For each given pair of triangles, determine if the triangles are similar and provide your reasoning. If the triangles are similar, write a similarity statement relating the triangles. a. b. c.

37 Name: Teacher: Geometry Per:

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