Semester Eam Review 014 015 Geometr B Eam Review Notes to the student: This review prepares ou for the semester B Geometr Eam. The eam will cover units 3, 4, and 5 of the Geometr curriculum. The eam consists of two parts. Part 1 is 15 selected response items each worth points. The second part is 15 etended response items worth varing numbers of points for a total of 70 points. A calculator ma be used on both parts of the eam. Man answers will be in terms of or in radical form. You ma leave answers in that form or approimate as a decimal to two places past the decimal point. For most accurate approimations use the ke on our calculator. Do not use 3.14 for. n the net two pages ou will find the formulas that will be available to ou in the eam booklet. FIGURES ARE NT DRAWN T SCALE!!!!! MCPS Page 1
Semester Eam Review Area/Circumference Triangle: 1 A bh Rectangle: A bh 1 A b1 b h Trapezoid: Parallelogram: A bh Regular Polgon: 1 A apothem perimeter Circle Area: A r Circle Circumference: C r d Volume Prism/Clinder: V Bh area of base height Pramid/Cone: Sphere: V V 1 1 Bh area of base height 3 3 4 r 3 3 Densit Formula Densit = Mass Volume Coordinate Geometr 1 Slope: 1 1 1 Midpoint:, Distance: 1 1 MCPS Page
Semester Eam Review Conic Sections Equation of a circle with center at hk, and radius r: h k r Equations of a parabola with verte at the origin, with p the distance from the verte to the focus and verte to directri: 1 4p or ; opens up if p 0, opens down if p 0 4 p 1 4p or ; opens right if p 0, opens left if p 0 4 p Circles Arc Length (degrees): S r Sector Area (degrees): A r Arc Length (radians): S r 360 Sector Area (radians): A r 360 Angle and Arc Formulas J B F K E N L A D G H C M 1 1 ma mbc 1 mgde mge mfh mj mmn mlk 180 1 radian = degrees 1 degree = radians 180 MCPS Page 3
Semester Eam Review Unit 3 1. Which pairs of figures have the same volume? a. b. 5 r 4 5 5 5 Bases are squares For items and 3, find the volume if each figure is revolved about the dashed line.. 3. 3 4 5 7 4. Name the cross section that is formed in each case. a. A cone, cut b a plane parallel to the base, not through the ape (top). b. A cone. cut b a plane perpendicular to the base, through the ape (top). c. A rectangular pramid, cut b a plane parallel to the base, not through the ape (top). d. A square pramid, cut b a plane parallel to the base, not through the ape (top). e. A clinder, cut b a plane parallel to the base. f. A clinder, cut b a plane perpendicular to the base. MCPS Page 4
Semester Eam Review 5. What is the relationship between the volumes of a cone and a clinder if the cone and clinder have the same radii and heights? 6. What is the relationship between the volumes of a pramid and a prism, if the pramid and prism have the same base areas and heights? 7. 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 cm and a height of 9 cm. If the ice cream melts into the cone, will the ice cream overflow the cone? Show how ou determined our answer. 8. The scoop of ice cream in item 8 has a mass of 14 grams. What is the densit (in g/cm 3 ) of the ice cream? 9. A movie theater sells popcorn in pramid-shaped boes whose base is a square of side 15 centimeters and a height of 5 centimeters. a. What is the volume of the popcorn bo? b. The densit of popcorn is 0.03 grams per cubic centimeter. What is the mass of the popcorn in the bo? c. When popcorn is popped, it is stored in a cube that is 45 centimeters on each side. How man boes of popcorn can be filled from this container? MCPS Page 5
Semester Eam Review 10. The three figures below have the same volume. Find the missing height in each case. a. b. 6 1 3 3 3 h h 4 h = h = For items 11 through 13, find the volume of each solid described. 11. A sphere with radius 6 cm. 1. A cone with radius 5 cm and height 9 cm. 13. A square pramid with base of side 8 cm and height 3 cm. MCPS Page 6
Semester Eam Review 14. A compan is producing a special part for a machine. The part consists of a clinder of tin (white) that is inside of another clinder made of copper (shaded). The part is shown below. Copper 6 mm 5 mm 4 mm 3 mm Tin a. What is the total volume of the entire part? You ma give our answer in terms of or to the nearest cubic millimeter. b. What is the volume of the tin used in the part? You ma give our answer in terms of or to the nearest cubic millimeter. c. What is the volume of copper used in the part? You ma give our answer in terms of or to the nearest cubic millimeter. MCPS Page 7
Semester Eam Review Unit 4 Use the word bank to complete the following definitions. Words ma be used more than once. Word bank: center focus directri radius equidistant 15. 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 ever point on the circle is called the. 16. A parabola is a set of points in the plane that are from a given point, called the, and a given line, called the. For items 17 and 18, sketch graphs of the following circles. 17. 4 18. 1 9 MCPS Page 8
Semester Eam Review For 19 and 0, write equations for the following circles and state the circumference and area of each circle. 19. 0. Equation: Circumference: Area : Equation: Circumference: Area: 1. Is the point 5,3 on the circle whose equation is ou determined our answer. 3 65? Show how. For the equation of a circle 6 7, complete the square to put it in the form h k r, then give the center and the radius. MCPS Page 9
Semester Eam Review For items 3 through 5, sketch graphs of the following parabolas. For each graph, also graph the focus and directri. 3. 6 4. 8 5. 4 For items 6 and 7, write the equation for the parabola. 6. 7. F 0, F.5,0 MCPS Page 10
Semester Eam Review 8. Look at the parabola below. F 10 9 8 7 6 5 4 3 1 1 3 4 5 6 7 8 9 10 1 3 4 5 6 7 8 9 10 10 9 8 7 6 5 4 3 1 Show that the point P 8,8 satisfies the definition of the parabola b showing that the distance between the focus and the point P 8,8 is equal to the distance from the point P 8,8 to the directri. MCPS Page 11
Semester Eam Review 9. How man points of intersection do the graphs of the circle parabola have? You ma use the grid below to help ou. 16 and the 30. How man points of intersection do the graphs of the circles 4 have? You ma use the grid below to help ou. 4 and MCPS Page 1
Semester Eam Review Look at the graph of line AB on the coordinate plane below. 10, 7 A 10 9 8 7 6 10 9 8 7 6 5 4 3 1 1 3 4 5 6 7 8 9 10 1 3 4 5 6 5 4 3 1 7 8 9 10 B 10,8 31. What is the length of AB? 3. Point C is between points A and B and has coordinates,. What is the ratio AC: CB? 33. Determine the coordinates of point D so that the ratio AD : DB 1:4. MCPS Page 13
Semester Eam Review 34. Look at line k whose equation is 3 on the coordinate plane below. k Write equations for the following lines. a. Parallel to line k, passing through the point 0,. b. Perpendicular to line k, passing through 0, 5. MCPS Page 14
Semester Eam Review Items 35 through 38 (this item continues on the net page) uses the quadrilateral below. A 1, 4 D,0 B 5,1 C, 3 Kell thinks that the figure might be a square. Remembering her geometr she does the following. 35. Kell first needs to know if the figure is a parallelogram. Show that the figure is a parallelogram b showing that the slopes of opposite sides are equal. 36. Kell now wants to determine if the figure is a rectangle. There are two was to do this. a. Show, b 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. MCPS Page 15
Semester Eam Review 37. Now that Kell 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 was to do this. a. Show that all four sides are the same length, and therefore the figure is a rhombus. b. Show, b using slope computations that the diagonals are perpendicular, and therefore the figure is a rhombus. 38. Kell also remembers a propert of parallelograms that the diagonals bisect each other. Show that this true b determining the midpoint of each diagonal and showing that each midpoint has the same coordinates. MCPS Page 16
Semester Eam Review For items 39 and 40, determine the area and perimeter of each shaded figure below. Each grid line represents 10 meters. 39. 40. 80 80 60 50 0 10 0 10 0 30 40 50 60 70 80 90 100 Area: Perimeter: Unit 5 0 10 0 10 0 30 40 50 60 70 80 90 100 Area: Perimeter: For items 41 through 49, find the value of in each figure below. 41. 4. 43. C is the center o o 50 o C o 160 o 70 o MCPS Page 17
Semester Eam Review 60 o 44. 45. 46. AB is a diameter 60 o o 8 o 85 o B o A o 47. 48. 49. 90 o o 50 o o 0 o 40 o 70 o o 35 o 50. Quadrilateral ABCD is inscribed in the circle below. A B D C a. mamc. Eplain our reasoning. b. If AC is a diameter, what is md or m B? Eplain our reasoning. MCPS Page 18
Semester Eam Review Look at the circle below. 140 o A 0 o 60 o B F E 4 o C Determine the following: D 51. mab 5. mef 53. mbc 54. m AFB 55. m FDB 56. m FEA MCPS Page 19
Semester Eam Review Look at circle P below. Line m is tangent to the circle at point R. PR 8, PS 10, UW TZ T m W X S U P R Z Complete the following. 57. mprs 58. SR 59. TU is congruent to which segment? 60. WTU is congruent to which angle? 61. TW is congruent to which arc? 6. SX 63. What part of the circumference of a circle is represented b arc whose measure is radians? 64. What part of the circumference of a circle is represented b an arc whose measure is 3 radians? MCPS Page 0
Semester Eam Review In items 65 and 66 below, find the radian measure of the central angle and the area of the sector. 65. 66. 4 cm F 6 cm C 8 cm D P 3 cm Q Radian measure of CD Radian measure of FPQ Area of sector CD Area of sector FPQ In items 67 and 68 below, find the length of arc and area of sector. 67. 68. S R T 40 o 9 cm W X 10 o 3 cm Z Length of SU Area of sector STW Length of RZ Area of sector RXZ MCPS Page 1