Midterm Multiple Choice Practice 1. Based on the construction below, which statement must be true?

1) CBDmABDm 2

1

2) CBDmABDm 3) ABCmABDm

4) ABDmCBDm 2

1

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) )(4

1,'"

80, ABCDDTRCmnC

2) )(4

1,'"

ABCDDTCmn

3) )(4,'"80, ABCDDTR CmnC

4) )(4

1,"'

80, ABCDDRC

C

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

10. How many lines of symmetry are there for a regular hexagon? 1) 4 2) 6 3) 0 4) 2

11. 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) 4

12. In the diagram below of , medians , , and intersect at G. If , what is the length of

?

1) 8 2) 10 3) 12 4) 16

13. 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) 4

14. In the diagram below of and , , and .

To prove that and are congruent by SAS, what other information is needed? 1) 2) 3) 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) 55

17. 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) 70

19. 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)

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, ACDE . Which of the following represents the length of BC?

1) 4.5 2) 3 3) 13.5 4) 12

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?

1) 52 2) 13 3) 10 4) 5

23. In the diagram below of right triangle ACB, altitude 𝐶𝐷̅̅ ̅̅ is drawn to hypotenuse 𝐴𝐵̅̅ ̅̅ . If AB = 36 and AC = 12,

what is the length of 𝐴𝐷̅̅ ̅̅ ? 1) 3 2) 4 3) 6 4) 32

24.

C

F

B E A

B

25. 26. ∆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) 11.2 2) 12 3) 16.8 4) 16

27. 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

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.

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 𝐴𝐶̅̅ ̅̅ to point D, and m∠BCD = 120 . Find the value of x. (1) 10 (3) 35 (2) 25 (4) 40

36. In the accompanying figure, ABCD is a parallelogram, mA = 4x + 15, and m C = 9x – 40. Find the measure of angle D.

(1) 11° (3) 59°

(2) 16° (4) 121° 37. Which of the following is not a degree of rotational symmetry for a regular pentagon?

(a) 36o (b) 72o (c) 144o (d) 288o

38. In rectangle ABCD, BCDADB . E is the midpoint of BD. Which of the following rigid motion(s) maps

AD onto CB ?

(a) BD

r (c) oR90

(b) DB

T (d) 180,ER

A B

C D

4x+15

9x - 40

39. Which is an angle that is complementary to ∠BOC ?

(a) ∠BOE (c) ∠DOE

(b) ∠COD (d) ∠BOA 40. If the letter P is rotated 90 degrees, which is the resulting figure?

(a) Df (c)

(b) (d)

41. In the diagram below, RSPQ || and transversal TU intersects PQ and RS at V and W, respectively. If

225 xTVQm and 103 xVWSm , find WVQm .

(a) 16o (c) 24o

(b) 58o (d) 122o

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 3:1.

43. The ratio of the sides of two similar triangles is 4 :1. If the area of the larger triangle is 2144cm , find

the area of the smaller triangle.

(1) 36 (2) 18 (3) 12 (4) 9

P

P

P

44. Given the diagram below, BCDE . ,6,3, ADCExAE and 2DB . What is the value of x?

(1) 7 (2) 9 (3) 4.5 (4) 1

45. 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 = 72o then angle A’ equals A) 72o C) 144o

B) 36o D) Can’t be determined

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

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.

54. Given the diagram below, Drawing 2 is the image of Drawing 1 with a scale factor of r = 2

1centered 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 O3.

b) Find the dilation factor of drawing 1 to drawing 3

55. 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.

A

B

C

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

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.

63. Given: <1 is supplementary to <2 Prove: l1 || l2

64. The triangle on the right has been mapped to the triangle on the left by a 120o 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.

66. Perform the following construction using construction tools. 𝑟𝑚(∆𝐴𝐵𝐶)

67. A student has performed the rotation 𝑅𝑂,50𝑜(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

69. Given: ABCD is a rectangle, M is the midpoint of

AB

Prove: DMC is isosceles

70. Given: Parallelogram FLSH, diagonal , , Prove:

M B

D C

A

71. Given: Quadrilateral ABCD, diagonal , , , , Prove: ABCD is a parallelogram.

72. The diagram below shows rectangle ABCD with points E and F on side . Segments and intersect

at G, and . Prove:

73.

Given: , , , ,

Prove:

74. Given: CFBE

ABCD

rectangle a is

Prove: AFDE

D

A

C

F

E

B

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 ⃡ at A and B, respectively. If m∠RAN = 3x + 24 and m∠RBQ = 7x − 16, find the value of x.

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, 𝐴𝐵̅̅ ̅̅ intersects 𝐶𝐷̅̅ ̅̅ . Solve for x and y.

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 3.

82. Construct the image of ∆ABC after a dilation with center O and a scale factor of ½.

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.

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? 86. Given: NPVR is a parallelogram Prove: SWTNOW ~

160 cm

90 cm 360 cm

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)

90. Given: D is the midpoint of AC

BDAC Prove: ABC is isosceles

91. Write the following composition of transformations in function notation: 92. Complete the chart to the right.

Sequence of rigid motions

Composition in function notation

Triangle congruence statement

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 = _______

97. Given: Isosceles triangle ACE with AEAC FDCBDE Prove: EFDCBD ~

98. Given: AC and BD intersect at E

CDAB

Prove: CEDAEB ~ 99. Solve for x and y y =__________ x = __________

Statements Reasons

A D

E

C B

Statements Reasons

5

23

5y – 2

3x + 2

x

10

100. Perform the following similarity transformation: 𝐷𝑂,

1

2

(𝑇𝐴𝑁 (∆𝑈𝑅𝐵))

101. A similarity transformation for triangle 𝐴𝐵𝐶 is described by (𝐷𝑂,2 (𝑟𝐷𝐺 ⃡ (∆𝐴𝐵𝐶))) . Locate and label

the image of triangle 𝐴𝐵𝐶 under the similarity.

O

102. Dilate circle A by a scale factor of 2 with O being the center of dilation.

103. Dilate circle 𝐶 with radius 𝐶𝐴 from center 𝑂 with a scale factor 𝑟 =1

2.

104. Given: and bisect each other.

Prove:

AB

FD

ADE EFBStatements Reasons

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.

106. 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 0,90. 107. State the coordinates of the image of trapezoid ABCD after a transformation T 3,-4.

108. State the coordinate of the image of triangle XYZ after a transformation r x-axis. 109. Name and draw in a translation vector that takes ∆𝐴𝐵𝐶 to ∆𝐴′𝐵′𝐶′.

110. Translate triangle ABC across vector DE.

111. What are the rigid motions? __________________________________________________ 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 ''' CBA .

a) ''' CBAABC _________________________ b) ''' CBAABC _________________________ c) )''(4 BAAB _________________________

d) '')(4 BAAB _________________________

e) ABBA )''(4

1 _________________________

f) '''' BA

AB

CB

BC _________________________

g) ''

''

AB

CB

BC

BA _________________________

h) ')(4 BB _________________________

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

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.

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.

Do Now Day 3 1. Given: 𝐹𝐸̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ 𝐴𝐵̅̅ ̅̅ ||𝐸𝐷̅̅ ̅̅ 𝐴𝐹̅̅ ̅̅ ||𝐶𝐷̅̅ ̅̅ Prove: 𝐴𝐹̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅

2. Given: <ADC and <BEA are right angles Prove: ∆𝐴𝐷𝐶~∆𝐴𝐸𝐵

Statements Reasons

Statements Reasons

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.

Name: ____________________________________________ Teacher: ______________________________ Geometry Per: ______