Cumulative Review ~ Common Core Geometry Name_________________________ 1) Which type of construction is demonstrated in each of the following diagrams?

a) b)

2) Find the measure of each angle in the accompanying picture. Justify each step of your work.

3) Find x:

4) Find the measure of both remote interior angles in the accompanying diagram.

5) a) Find the measure of PR, in the following diagram. b) Find the measure of x:

6) Given: 1 + 2 = 180

Prove: ∥

7) Given the diagram below, prove that <y + <z = <w + <x.

(Hint: You will need to write in a label in the diagram that is not labeled yet for this proof.)

8) Find x, y and z using the accompanying diagram.

9) Triangle ABC has coordinates A(-3,2), B(0,4) and C(-2,8). Reflect ABC over the x-axis followed

by a reflection over the y-axis. Label the image as A”B”C”.

Which type of isometry is this? JUSTIFY your answer.

10) A translation maps X(-3,2) onto X’(0,0). Find the coordinates of the image of Y(5,4) under the same

translation.

11) Which of these symbols have horizontal but not vertical line symmetry.

There may or may not be more than one answer.

12) Which figure has 45 degree rotational symmetry? Justify your answer.

a) square b) equilateral triangle c) regular hexagon d) regular octagon

13) Which statement is illustrated by the following construction?

Justify your choice:

14) In the following diagram, line l is parallel to line m. Line t is a transversal. Use the following diagram to find x:

15) Find the unknown labeled angle in the following diagram.

16) Find the measure of QRS in the following diagram:

17) Use the diagram below to answer the following:

a.) Write the rule of the composition

that maps Figure onto ” ” ”.

b.) List all of the corresponding sides

and angles.

c.) Is ' ' 'ABC A B C ? Justify your

answer.

18) If this figure is rotated clockwise around its center, what is the minimum number of degrees it must

be rotated to carry the figure onto itself?

19) Find the measure of z in the following diagram. Justify each step of your work.

20) Use your compass and straightedge to translate the image using the given vector.

21) A student believes that she can prove these two triangles to be congruent using SSS Congruence Theorem. Which piece of information is she missing in order to use the theorem? Justify your answer.

22) Find the measure of ∠ and ∠ using parallelogram . What do you know about ∠ and ∠ ?

  z 

23) Find the translation that maps the point (3, 7) onto (5,-1). Find the coordinates of (4, 6) under the same translation.

24) What rotational symmetry does a regular hexagon have?

25) Given: ≅

≅

Prove: ∠ ∠

26) Construct a line perpendicular to the given line, through point P:

P 

27) , andAD EB CF are the medians of ABC. If 5DG x and 5 20AG x , find the length of DG

and AG . Explain your answer.

28) In isosceles XYZ, M and N are midpoints of andXY YZ respectively.

a) Find x:

b) Find the perimeter of XYZ.

29) Are the triangles below similar? Justify your answer.

30) Given that ∆ ~∆ , find the measure of all missing angles in BOTH triangles.

31) ΔABC is graphed on the coordinate plane. Its image after a dilation about the origin is ΔDEF.

a) What is the scale factor of the dilation? Explain how you know.

b) How do you figure the scale factor when the origin is not the point of dilation?

32) What are the coordinates of the point (-9,-3) after , ? 33) What are the coordinates of the point (-9,-3) after , ?

34) In the following diagram, ΔABC ~ ΔDEC. Find the measures of sides andDC CE . Show all of your work.

35) In PSR , point P is located at (2,-2), point S at (3,-4) and point R at (-2,1). ' ' 'P S R is the

image of PSR after a reflection across the line .

a) What are the coordinates of each vertex in ' ' 'P S R ?

b) What kind of isometry is ' ' 'P S R ? Explain why.

36) We have learned four transformations: reflection, rotation, dilation and translation. Fill in the table below based on your knowledge of rigid motions and similarity transformations.

Transformation(s)

Identify the transformations that are rigid motions

Identify the transformations that are similarity transformations

What makes rigid motions and similarity transformations different? Explain. 37) Explain what happens when figures are dilated with the following scale factors.

r < 1

r = 1

r > 1

38) Use the accompanying diagram to find x and y:

39) What construction is the following drawing showing?

40) In the diagram below, Circle A has radii AX and AY . Circle B has radii BR and BT . Describe the transformation that will take Circle A to Circle B (you must be as specific as possible!!!).

41) Use your compass and ruler to construct a congruent angle to the given angle.

42) In the following diagram of rectangle AFCD, AC DB and ∆ ~∆

a) Describe the similarity transformation that will take ∆ to ∆ .

b) If 70oBAD find ∠ .

43) In ∆ , , , andCE BD AF are medians. Find EG and CG if 3EG y and 5 9CG y

44) Dilate ∆ through point with a scale factor of . Label your new triangle ’ ’.

a) What is the relationship between and ’ ’?

b) Are ∆ and ∆ ’ ’ similar? Explain. 45) Given the following triangle, find the following:

length of side = sin B = sin C =

cos B = cos C =

tan B = tan C =

46) Fill in each of the blanks:

sin 25° = cos _____°

sin 46° = cos _____°

sin 0° = cos _____°

47) Solve for . Round your answer to the nearest hundredth.

15 cm

θ°5 cm

10 cm 50°

x

x

12 60°

48) Solve for x. Round your answer to the nearest tenth.

49) If sin 2 5 ° cos 9 – 3 °, what is the value of to the nearest tenth?

50) For each of the following triangles, solve for x and identify the type of triangle according to its

sides and angles

a) b)

51) If cos , find sin .

52) What is the area of a square whose diagonal has a length of 4?

53) If 3 1 and 4 12, find , , and .

x

87°

70°

54) Given the diagram below, solve for x. Justify each step of your solution.

55) Find the measure of r. Justify each step of your solution.

56) Bill is holding a string for a kite 6 feet off the ground. The kite has an angle of depression of

60 degrees. If the string is 75 feet long, how high off the ground is the kite?

57) Find AC in the picture below:

58) In ABC , AB = 120, ∠ 90, ∠ 43o and ∠ 75o .

a) Find the length of .

b) Find the length of

59) Complete the table:

Figure

Identify the slice

parallel to the base (cross section)

Identify the slice

perpendicular to the base

60) If a 3-4-5 right triangle is revolved about a leg of length 4 to create a solid of revolution. Describe the solid, including its dimensions and its volume to the nearest tenth 61) What is the volume of a sphere to the nearest hundredth that has a diameter of 14.2 units? 62) If the ratio of the side lengths of two rectangular prisms are 3:5. What is the volume of the larger prism to the nearest tenth, if the smaller prism has a volume of 48cm3?

63) What is the volume of the following prism?

64) Find the ratio of the areas of the pair of similar figures.

65) What are the methods of proving triangles similar?

66) What are the methods of proving triangles congruent?

67) A silo is used to store crops on a farm. The height of the silo is 40 feet, the radius of the hemisphere is 8 feet. What is the volume of the silo to the nearest cubic foot?

‐‐‐‐‐5‐‐‐‐‐  ‐‐‐‐‐‐‐‐7‐‐‐‐‐‐‐‐‐‐ 

68) Find x given that .

a) b)

69) In the diagram below of , 70) In the diagram below of ,

, AC = 8, AB = 12, and , DB = 6, AD = 3, and

CM = 6. Find the length of MN. DE = 15. Find AC.

71) If , , , and , what is ?

72) A vertical pole, 15 feet high, casts a shadow 12 feet long. At the same time, a nearby tree casts a

shadow 40 feet long. What is the height of the tree?

ABC ABC

BX 9 BA 15 BY 15 YC

73) In the diagram pictured below, ∥ ∥ , AB = 7, CD = 4, FD = 6 and . If the perimeter of trapezoid ABCD is 31, find AE and EC.

74) Given the similar triangles pictured to the right, find:

a) Ratio of the sides

b) Ratio of the perimeters

c) Ratio of the areas

75) The sides of a triangle are 7, 8 and 10. What is the length of the shortest side of a similar triangle whose perimeter is 75? 76) Caterina’s boat has come untied and floated away on the lake. She is standing atop a cliff that is 35 feet above the water in a lake. If she stands 10 feet from the edge of the cliff, she can visually align the top of the cliff with the water at the back of her boat. Her eye level is 5.5 feet above the ground. How far out from the cliff, to the nearest tenth, is Catarina’s boat?

AE : EC 1: 4

77) Find the value of x:

a) b)

c) d) 78) Simplify the given expressions:

a) 3 72 b) 3 32x2 c) 4 16x3

d) 5 3 7 2 e) 10 6

5 2 f)

2

3

g) 8 18 2 h) 50 98

6

79) Find the area of the rectangle in simplest radical form.

80) Find the width of the rectangle with an area of 4√6 in simplest radical form. 81) Tanya runs diagonally across a rectangular field that has a length of 40 yards and a width of 30 yards. What is the length of the diagonal, in yards, that Tanya runs? 82) The sides of a triangle are 7, 11, and 18. Do these sides form a right triangle? Justify your answer. 83) A cubic box with sides of length s centimeters holds N marbles. The density of the marbles in this b box can

be determined using the expression

(1) 26s

N (2)

3s

N

(3) 2(6 )s

N (4)

3

N

s

84) A solid sphere has a volume of 288 cubic inches. Find the radius of the sphere. 1) 3 inches 2) 6 inches 3) 36 inches 4) 48 inches 85) If you dilate a rectangle by a scale factor of 2, circle the properties of transformations that are preserved under the dilation. If any are not preserved, explain why they are not. Size Orientation Distance angle measure

86) In the following diagram, ∠1 3 8 and ∠2 6 – 10. Find ∠3. 87) Find the measure of CAD. 88) Plane P is parallel to plane Q. If plane P is perpendicular to line l, then plane Q

1) contains line l 2) is parallel to line l 3) is perpendicular to line l 4) intersects, but is not perpendicular to line l

89) Write a formal proof.

Given: BD AE Prove:

90) In the diagram below of ABC, ∥ , 4, 6 8.Find . 91) A cone fits inside a cylinder so that their bases are the same, as shown in the diagram below. Calculate the

volume that is inside the cylinder but outside the cone. Give an exact answer. 92) The coordinates of the vertices of parallelogram SWAN are S(2, -2), W(-2, -4), A(-4, 6) and N(0, 8).

a) State and label the coordinates of parallelogram S’’W’’A’’N’’, the image of SWAN after the composition .

b) Is the composition an isometry? Explain. 93) Max drives 120 miles to get home, at a rate of 55 mph. How many hours will it take Max to get home, to the nearest tenth? 94) Describe the lines represented by 2 4 6 and 2 4 6.

95) Use the following prism to answer the following questions:

a) Name 3 vertices

b) Name 3 edges

c) Name a pair of parallel planes

d) What is the most specific name for this figure. 96) In ABC, is an altitude. If A B , what method can be used to prove ABC BCD ? Explain how you know. 97) A 25 foot tall tree casts a shadow 15 feet long. How long will the shadow be for a person who is 5.9 feet tall? 98) Identify the translation that maps (-5, 6) onto (-3, -8). 99) Write the equation of a line that is parallel to 4x + y = 10, and goes through the point (-2,-4).

100) Given the following segment AB, construct a square whose sides are length AB. 101) In isosceles ∆ , ∠ is the vertex. What do you know about ∠ and ∠ ?

102) In right ∆ , ∠ 32o and AB = 5. What is the length of to the nearest hundredth?

103) Graph the following inequalities:

a) 5 3y x b) 3 – 4 2

104) A cone has a volume of 354 and a radius of 6. Find the slant height of the cone to the nearest tenth. 105) In Quadrilateral ABCD, A(7,8), B(10,5), C(7,2) and D(4,5).

PROVE: ABCD is a rectangle. 106) A quadrilateral is defined by the following inequalities:

y 3

y 2

x 5

x 1

a) Graph the quadrilateral defined by the inequalities.

b) What are the coordinates of the vertices of the region?

c) Is the shape a rectangle? Explain how you know.

107) Find the coordinates of the point C that partitions AB  whose coordinates are A(-3, -5) and B(7, 0)

such that AC:CB = 1:4.

108) Find the value of x:

a. b.

5 x

x 40                                                                                                                                         

8

              c. d. x x 8 4 5 4 5 109) What is the equation of a circle whose center is (0, 3) and whose radius is 5? 110) What is the area of a circle whose diameter has endpoints (4, -5) and (2, -1)? 111) A sector of a circle has a central angle of 54 and a radius of 3.   Find to the nearest tenth the area of the sector and the length of its arc. 112) ABCD is a cyclic quadrilateral. If mA = 67 find mC.