2014-2015 honors geometry b...

HONORS GEOMETRY B Semester Exam Review

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2014–2015 Honors Geometry B

Exam Review

Notes to the student: This review prepares you for the semester B Honors Geometry Exam. The exam will cover units 3, 4, and 5 of the Honors Geometry curriculum.

The exam consists of two parts. Part 1 is 15 selected response items each worth 2 points. The second part is 15 extended response items worth varying numbers of points for a total of 70 points.

A calculator may be used on both parts of the exam.

Many answers will be in terms of or in radical form. You may leave answers in that form or approximate as a decimal to two places past the decimal point. For most accurate approximations use the key on your calculator. Do not use 3.14 for . On the next two pages you will find the formulas that will be available to you in the exam booklet.

FIGURES ARE NOT DRAWN TO SCALE!!!!!


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Area/Circumference

Triangle: 1

2A bh Rectangle: A bh Trapezoid: 1 2

1

2A b b h

Parallelogram: A bh Regular Polygon: 1

apothem perimeter2

A

Circle Area: 2A r Circle Circumference: 2C r d

Volume

Prism/Cylinder: area of base heightV Bh

Pyramid/Cone: 1 1

area of base height3 3

V Bh

Sphere: 34

3V r

Density Formula

Density = Mass Volume

Coordinate Geometry

Slope: 2 1

2 1

y y

x x

Midpoint: 1 2 1 2,2 2

x x y y

Distance: 2 2

2 1 2 1x x y y

Conic Sections – Circle and Parabola

Equation of a circle with center at ,h k and radius r: 2 2 2x h y k r

Equations of a parabola with vertex at ,h k with p the distance from vertex to focus and vertex

to directrix :

2 214 or

4x h p y k y k x h

p ; opens up if 0p , opens down if 0p .

2 214 or

4y k p x h x h y k

p ; opens right if 0p , opens left if 0p .


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Conic Sections – Ellipse and Hyperbola

Equations of an ellipse with center at ,h k , a b :

Major axis horizontal: 2 2

2 21

x h y k

a b

Major axis vertical: 2 2

2 21

x h y k

b a

Equations of a hyperbola with center at ,h k :

Transverse axis horizontal: 2 2

2 21

x h y k

a b

Transverse axis vertical: 2 2

2 21

y k x h

a b

Circles

Arc Length (degrees): 2360

xS r Arc Length (radians): S r

Sector Area (degrees): 2

360

xA r Sector Area (radians): 2

2A r

Angle and Arc Formulas

1

2m A mBC 1

2m GDE mGE mFH 1

2m J mMN mLK

1801 radian = degrees

1 degree = radians

180

B

A

C

E

D

F

G H

L

J K

M

N


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Unit 3

1. Which pairs of figures have the same volume?

a. b. c.

For items 2 through 4, find the volume if each figure is revolved about the dashed line.

2. 3. 4.

5. Name the cross section that is formed in each case.

a. A cone, cut by a plane parallel to the base, not through the apex (top).

b. A cone, cut by a plane perpendicular to the base, through the apex (top).

c. A rectangular pyramid, cut by a plane parallel to the base, not through the apex (top).

d. A square pyramid, cut by a plane perpendicular to the base, not through the apex (top), through one side only.

e. A cylinder, cut by a plane parallel to the base.

f. A cylinder, cut by a plane perpendicular to the base.

4r

5

5

5

5

Cross sections equidistant from the bases are congruent and the heights of the solids are equal.

5

3

7

4

3

6 2

Bases are squares


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6. What is the relationship between the volumes of a cone and a cylinder if the cone and cylinder have the same radii and heights?

7. What is the relationship between the volumes of a pyramid and a prism, if the pyramid and prism have the same base areas and heights?

8. A scoop of ice cream is in the shape of a sphere with radius 3 cm is placed in a cone that has a radius of 2 cm and a height of 9 cm. If the ice cream melts into the cone, will the ice cream overflow the cone? Show how you determined your answer.

9. The scoop of ice cream in item 8 has a mass of 124 grams. What is the density (in g/cm3 ) of the ice cream?

10. A movie theater sells popcorn in pyramid-shaped boxes whose base is a square of side 15 centimeters and a height of 25 centimeters.

a. What is the volume of the popcorn box?

b. The density of popcorn is 0.03 grams per cubic centimeter. What is the mass of the popcorn in the box?

c. When popcorn is popped, it is stored in a cube that is 45 centimeters on each side. How many boxes of popcorn can be filled from this container?


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11. The three figures below have the same volume. Find the missing height in each case.

a. b.

h = ______ h = ________

For items 12 through 14, find the volume of each solid described.

12. A sphere with radius 6 cm.

13. A cone with radius 5 cm and height 9 cm.

14. A square pyramid with base of side 8 cm and height 3 cm.

12 3

6 h

3 3 2

24

h


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15. A company is producing a special part for a machine. The part consists of a cylinder of tin (white) that is inside of another cylinder made of copper (shaded). The part is shown below.

a. What is the total volume of the entire part? You may give your answer in terms of or to the nearest cubic millimeter.

b. What is the volume of the tin used in the part? You may give your answer in terms of or to the nearest cubic millimeter.

c. What is the volume of copper used in the part? You may give your answer in terms of or to the nearest cubic millimeter.

5 mm

6 mm

4 mm 3 mm

Copper

Tin


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Unit 4

Use the word bank to complete the following definitions. Words may be used more than once.

Word bank: center focus directrix sum

difference radius equidistant foci

16. A circle is the set of points in the plane that are _________________ from a given point, called the _______________. The distance from the given point to every point on the circle is called the ____________.

17. A parabola is a set of points in the plane that are ___________ from a given point, called the _________, and a given line, called the ____________.

18. An ellipse is a set of points in the plane such that the ___________ of the distances from each point on the ellipse to two fixed points, called the ___________, is the same.

19. A hyperbola is a set of points in the plane such that the __________ of the distances from each point on the hyperbola, to two fixed points, called the ________, is the same.

For items 20 and 21, sketch graphs of the following circles.

20. 2 2 4x y 21. 2 22 1 9x y

x

y

O x

y

O


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For 22 and 23, write equations for the following circles and state the circumference and area of each circle.

22. 23.

Equation: _________________________ Equation: _______________________

Circumference: _____________ Circumference: ____________

Area : _______________ Area: _____________

24. Write the equation of a circle whose center is at the point 1,5 with the point 7,20

on the circle.

25. Is the point 5,3 on the circle whose equation is 2 23 2 65x y ? Show how

you determined your answer.

x

y

O x

y

O


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For each equation of a circle, complete the square to put it in the form 2 2 2x h y k r ,

then give the center and the radius.

26. 2 2 6 27x y y

27. 2 22 16 35 0x x y y

For items 28 through 30, sketch graphs of the following parabolas. For each graph, also graph the focus and directrix.

28. 2 6x y 29. 2 8y x 30. 22 4 1x y

x

y

O x

y

O x

y

O


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For items 31 and 32, write the equation for the parabola.

31. 32.

33. Look at the parabola below.

Show that the point 8,8P satisfies the definition of the parabola by showing that the

distance between the focus and the point 8,8P is equal to the distance from the point

8,8P to the directrix.

x

y

O x

y

O

6

‐1

‐2

‐3

‐4

‐5

‐6

x

y

O‐10 ‐ 9 ‐ 8 ‐ 7 ‐ 6 ‐ 5 ‐ 4 ‐ 3 ‐ 2 ‐ 1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

5

4

3

2

1

‐7

‐8

‐9

‐10

F

0, 2F 2.5,0F


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For items 34 and 35, sketch graphs of the following ellipses.

34. 2 2

125 16

x y 35.

2 21 3

19 4

y x

For items 36 and 37, write equations for the ellipses below.

36. 37.

38. Determine whether the point 2,1 is on the ellipse whose equation is 2 2

19 4

x y .

x

y

O x

y

O

x

y

O x

y

O


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For items 39 and 40, sketch the following hyperbolas.

39. 22 2

14 1

yx 40.

2 2

19 9

y x

For items 41 and 42, write equations for the hyperbolas below.

41. 42.

x

y

O x

y

O

x

y

O x

y

O


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43. How many points of intersection do the graphs of the circle 2 2 16x y and the

parabola 2y x have? You may use the grid below to help you.

44. How many points of intersection do the graphs of the circles 2 22 4x y and

2 22 4x y have? You may use the grid below to help you.

x

y

O

x

y

O


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Look at the graph of line AB

on the coordinate plane below.

45. What is the length of AB ?

46. Point C is between points A and B and has coordinates 2, 2 . What is the ratio :AC CB ?

47. Determine the coordinates of point D so that the ratio : 1: 4AD DB .

6

‐1

‐2

‐3

‐4

‐5

‐6

x

y

O‐10 ‐ 9 ‐ 8 ‐ 7 ‐ 6 ‐ 5 ‐ 4 ‐ 3 ‐ 2 ‐ 1 1 2 3 4 5 6 7 8 9 10

10

9

8

7

5

4

3

2

1

‐7

‐8

‐9

‐10

10, 7A

10,8B


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48. Look at line k whose equation is 3 2y x on the coordinate plane below.

Write equations for the following lines.

a. Parallel to line k, passing through the point 0,2 .

b. Perpendicular to line k, passing through 0, 5 .

x

y

O

k


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Items 49 through 52 (this item continues on the next page) uses the quadrilateral below.

Kelly thinks that the figure might be a square. Remembering her geometry she does the following.

49. Kelly first needs to know if the figure is a parallelogram. Show that the figure is a parallelogram by showing that the slopes of opposite sides are equal.

50. Kelly now wants to determine if the figure is a rectangle. There are two ways to do this.

a. Show, by using slopes, that one of the angles is a right angle, and therefore there are four right angles and the figure is a rectangle.

b. Show that the diagonals of the parallelogram are congruent, and the figure is a rectangle.

x

y

O

1,4A

2,0D 5,1B

2, 3C


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51. Now that Kelly knows the figure is a rectangle. If she can show that it is a rhombus then it must be a square. Again there are two ways to do this.

a. Show that all four sides are the same length, and therefore the figure is a rhombus.

b. Show, by using slope computations that the diagonals are perpendicular, and therefore the figure is a rhombus.

52. Kelly also remembers a property of parallelograms that the diagonals bisect each other. Show that this true by determining the midpoint of each diagonal and showing that each midpoint has the same coordinates.


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For items 53 and 54, determine the area and perimeter of each shaded figure below. Each grid line represents 10 meters.

53. 54.

Area: ____________________ Area: _____________________

Perimeter: ________________ Perimeter: _________________

Unit 5

For items 55 through 63, find the value of x in each figure below.

55. 56. 57. C is the center

xo

xo

xo

C

50o

70o

160o

x

y

x

y

0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100

10

20

60

80

10

20

80

50


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58. 59. 60. AB is a diameter

61. 62. 63.

64. Quadrilateral ABCD is inscribed in the circle below.

a. ________m A m C . Explain your reasoning.

b. If AC is a diameter, what is and m D m B ? Explain your reasoning.

xo

xo

A B

28o

60o

60o

85o

xo

xo

xo

xo

50o

20o

290o

70o 35o

40o

A

D

C

B


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Look at the circle O below.

Determine the following:

65. mAB _______ 66. mEF _______

67. mBC _______ 68. m AFB _______

69. m FDB ______ 70. m FEA _______

D

60o

A

B

C E

F

140o

20o

42o

O


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Look at circle P below. Line m is tangent to the circle at point R.

8, 10,PR PS UW TZ

Complete the following.

71. m PRS ________ 72. SR ________

73. TU is congruent to which segment? _______________

74. WTU is congruent to what angle? _______________

75. TW is congruent to which arc? ____________

76. SX _________

77. What part of the circumference of a circle is represented by arc whose measure is radians?

78. What part of the circumference of a circle is represented by an arc whose measure is 3

radians?

P R

S

T

U

W

Z

m

X


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Jerald is creating a new target game. The game involves concentric circles (circles with the same center P) and sectors as shown below.

The radius of the smaller circle is 6 inches, while the radius of the larger circle is 9 inches.

The length of DE is 12 inches.

The measure of APC is 40o. The measure of JPH is 90o.

79. What is the length of AB ? _______

80. What is the radian measure of APB ? __________

81. What is the area of sector EPD ? __________

82. What is the length of AC ? _________

83. What is the area of sector EPH ? _____________

84. What is the measure of the circle segment bounded by JH and JH ? ____________

B

D

P

C

E

H J

A


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85. A piece of jewelry is a segment of a circle with radius 10 mm, as shown in the shaded region below.

What is the area, in square mm, of the shaded region? Your answer may be in terms of or rounded to two decimal places. If you use in a calculation, use the calculator’s key.

10 mm

10 mm

90o

2014-2015 honors geometry b...

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