Honors Geometry B. Exam Review

Transcription

1 Semester Eam Review Honors Geometr B Eam Review Notes to the student: This review prepares ou for the semester B Honors Geometr Eam. The eam will cover units 3, 4, and 5 of the Honors 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

2 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: Densit Formula V V 1 1 Bh area of base height r 3 3 Densit = Mass Volume Coordinate Geometr 1 Slope: Midpoint:, Distance: 1 1 Conic Sections Circle and Parabola Equation of a circle with center at hk, and radius r: h k r Equations of a parabola with verte at hk, with p the distance from verte to focus and verte to directri : 1 ; opens up if p 0, opens down if p 0. 4 p h 4p k or k h 1 ; opens right if p 0, opens left if p 0. 4 p k 4 p h or h k MCPS Page

3 Semester Eam Review Conic Sections Ellipse and Hperbola Equations of an ellipse with center at hk,, a b: Major ais horizontal: Major ais vertical: h k a 1 b h k b 1 a Equations of a hperbola with center at hk, : Transverse ais horizontal: Transverse ais vertical: h k a 1 b k h a 1 b Circles Arc Length (degrees): S r Sector Area (degrees): A r Angle and Arc Formulas Arc Length (radians): S r Sector Area (radians): A r B F E N A D G H C 1 1 ma mbc 1 mgde mge mfh mj mmn mlk J K L M radian = degrees 1 degree = radians 180 MCPS Page 3

4 Semester Eam Review Unit 3 1. Which pairs of figures have the same volume? a. b. c. 5 r Bases are squares Cross sections equidistant from the bases are congruent and the heights of the solids are equal. For items through 4, find the volume if each figure is revolved about the dashed line 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 perpendicular to the base, not through the ape (top), through one side onl. 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

5 Semester Eam Review 6. What is the relationship between the volumes of a cone and a clinder if the cone and clinder have the same radii and heights? 7. 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? 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 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. 9. 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? 10. 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

6 Semester Eam Review 11. The three figures below have the same volume. Find the missing height in each case. a. b h 3 4 h h = h = For items 1 through 14, find the volume of each solid described. 1. A sphere with radius 6 cm. 13. A cone with radius 5 cm and height 9 cm. 14. A square pramid with base of side 8 cm and height 3 cm. MCPS Page 6

7 Semester Eam Review 15. 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 Tin 3 mm 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

8 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 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 ever 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 fied points, called the, is the same. 19. A hperbola is a set of points in the plane such that the of the distances from each point on the hperbola, to two fied points, called the, is the same. For items 0 and 1, sketch graphs of the following circles MCPS Page 8

9 Semester Eam Review For and 3, write equations for the following circles and state the circumference and area of each circle.. 3. Equation: Circumference: Area : Equation: Circumference: Area: 4. Write the equation of a circle whose center is at the point 1, 5 with the point 7,0 on the circle. 5. Is the point 5,3 on the circle whose equation is ou determined our answer. 3 65? Show how MCPS Page 9

10 Semester Eam Review For each equation of a circle, complete the square to put it in the form then give the center and the radius. h k r, For items 8 through 30, sketch graphs of the following parabolas. For each graph, also graph the focus and directri MCPS Page 10

11 Semester Eam Review For items 31 and 3, write the equation for the parabola F 0, F.5,0 33. Look at the parabola below. F 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

12 Semester Eam Review For items 34 and 35, sketch graphs of the following ellipses For items 36 and 37, write equations for the ellipses below Determine whether the point,1 is on the ellipse whose equation is MCPS Page 1

13 Semester Eam Review For items 39 and 40, sketch the following hperbolas For items 41 and 4, write equations for the hperbolas below MCPS Page 13

14 Semester Eam Review 43. 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 44. 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 14

15 Semester Eam Review Look at the graph of line AB on the coordinate plane below. 10, 7 A B 10,8 45. What is the length of AB? 46. Point C is between points A and B and has coordinates,. What is the ratio AC: CB? 47. Determine the coordinates of point D so that the ratio AD : DB 1:4. MCPS Page 15

16 Semester Eam Review 48. 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 16

17 Semester Eam Review Items 49 through 5 (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. 49. 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. 50. 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 17

18 Semester Eam Review 51. 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. 5. 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 18

19 Semester Eam Review For items 53 and 54, determine the area and perimeter of each shaded figure below. Each grid line represents 10 meters Area: Perimeter: Area: Perimeter: Unit 5 For items 55 through 63, find the value of in each figure below C is the center o o 50 o C o 160 o 70 o MCPS Page 19

20 Semester Eam Review 60 o AB is a diameter 60 o o 8 o 85 o B o A o o o 50 o o 0 o 40 o 70 o o 35 o 64. 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 and m B? Eplain our reasoning. MCPS Page 0

21 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 65. mab 66. mef 67. mbc 68. m AFB 69. m FDB 70. m FEA MCPS Page 1

22 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. 71. mprs 7. 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 b arc whose measure is radians? 78. What part of the circumference of a circle is represented b an arc whose measure is 3 radians? MCPS Page

23 Semester Eam Review 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 1 inches. The measure of APC is 40 o. The measure of JPH is 90 o. B D P E A J H C 79. What is the length of AB? 80. What is the radian measure of APB? 81. What is the area of sector EPD? 8. What is the length of AC? 83. What is the area of sector EPH? 84. What is the measure of the circle segment bounded b JH and JH? MCPS Page 3

24 Semester Eam Review 85. A piece of jewelr is a segment of a circle with radius 10 mm, as shown in the shaded region below. 10 mm 90 o 10 mm What is the area, in square mm, of the shaded region? Your answer ma be in terms of or rounded to two decimal places. If ou use in a calculation, use the calculator s ke. MCPS Page 4