2 nd Semester Final Exam Review

Transcription

1 2 nd Semester Final xam Review I. Vocabulary hapter 7 cross products proportion scale factor dilation ratio similar extremes scale similar polygons indirect measurements scale drawing similarity ratio means hapter 8 angle of depression equal vectors sine angle of elevation geometric mean tangent component form magnitude trigonometric ratio cosine parallel vectors vector direction resultant vector hapter 9 apothem circle center of a circle composite figure center of a regular polygon geometric probability central angle of a regular polygon hapter 10 altitude isometric drawing right cone altitude of a cone lateral edge right cylinder altitude of a pyramid lateral face right prism axis of a cone lateral surface slant height of a axis of a cylinder net regular pyramid center of a sphere oblique cone slant height of a right cone cone oblique cylinder space cross section oblique prism sphere cube orthographic drawing surface area cylinder perspective drawing vanishing point edge polyhedron vertex face prism vertex of a cone great circle pyramid vertex of a pyramid hemisphere radius of a sphere volume horizon regular pyramid hapter 11 adjacent arcs exterior of a circle secant segment arc external secant segment sector of a circle arc length inscribed angle segment of a circle central angle intercepted arc semicircle chord interior of a circle subtend common tangent major arc tangent of a circle concentric circles minor arc tangent circles congruent arcs point of tangency tangent segment congruent circles secant hapter 12 center of dilation composition of transformations reduction enlargement glide reflection isometry

2 II. Formulas distance formula midpoint formula geometric mean sine ratio cosine ratio tangent ratio Pythagorean Theorem Law of osines Law of Sines uler s Formula distance formula for three dimensions midpoint formula for three dimensions diagonal of a right rectangular prism area of a parallelogram, triangle, trapezoid, rhombus, kite, circle, and regular polygon circumference of a circle lateral area of a prism, cylinder, pyramid, and cone surface area of a prism, cylinder, pyramid, cone, and sphere volume of a prism, cylinder, pyramid, cone, and sphere measure of a minor arc measure of a semicircle measure of a major arc arc length area of a sector of a circle area of a segment of a circle measure of an angle whose vertex is on the circle measure of an angle whose vertex is inside the circle measure of an angle whose vertex is outside the circle equation of a circle product of the parts of a chord = product of the parts of the intersecting chord product of external part of a secant and the secant = product of external part of another secant and the secant product of external part of a secant and the secant = (tangent segment) 2 reflection across the x-axis reflection across the y-axis reflection across y = x translation in a coordinate plane rotation of 90 about the origin rotation of 180 about the origin dilation in a coordinate plane slope formula point-slope formula slope-intercept form III. Solve for x. 1. x 1 x = x 5 x x 4a x 5a = 7 6. x = Find x & y: 8 x = 56 6 = y 189 Solve. 5. The ratio of the measures of two complementary angles is : 4. Find the measure of each angle. 6. Find the measure of the acute angles of a right triangle if their measures are in the ratio of 7 to Find the measures of two supplementary angles if their measures are in the ratio of 4 to 5.

3 6. If JU ~ K, find the following: Proofs = U = J = 7. Given: F Prove: F ~ K = K = F x 2 2 x 2 + x + 2 x 2 + 4x + K U 7 x + 2 J x + 8. Given: A Prove: A ~ A 9. Given: A ; A Prove: A = A A 10. Given: quad. A is a parallelogram Prove: A A F A H F State whether the triangles are similar. If so, explain why and state the scale factor º º 12 45º 15 50º

4 F with RS F. 15. S = 6; SF = 8; R = 4; R = 16. R = 4; R = 6; F = 14; RS = 17. R = 4x 6; SF = 6x 5; = 2x + 6; F = 8x 2; R = R S F A with an angle bisector of A. 18. A = 6; = 4; A = 8; = 19. A = 8, = 6, = 8, A = A 20. A free-fall ride at an amusement park casts a shadow 4 feet 8 inches long. At the same time, a 6-foot-tall person standing in line casts a shadow 2 feet long. What is the height of the ride? 21. An artist makes a scale drawing of a new lion enclosure at the zoo. The scale is 1 in. : 25 ft. On 1 the drawing the length of the enclosure is 7 inches. What is the actual length of the lion 4 enclosure? Given: rt. A with rt. A and alt. 22. A = 9, = 4, find. 2. =, = 6, find. 24. A = 12, =, find. 25. A = 9, A = 12, find. Use the Law of Sines to find the segment length to the nearest hundredth or the angle measure to the nearest degree GH 29. m J 0. m R 1. m T 2. An escalator from the ground floor to the second floor of a department store is 105 ft. long and rises 5 ft. vertically. What angle (whole degree) does the escalator make with the ground floor?. The angle of elevation to the top of a tree from a point 4 ft from the base of the tree is 72. How tall is the tree? (Round to the nearest hundredth)

5 4. A person at one end of a 260 ft. bridge spots the rivers edge directly below the opposite end of the bridge and finds the angle of depression to be 58. How far below the bridge is the river? (Round to the nearest hundredth) Use the Law of osines to find the segment lengths to the nearest hundredth and the angle measures to the nearest degree. 5. YZ F 8. m I 9. m M 40. m S 41. Find the measure of each number angle: Given: circle O, diameter H, G AH m = 0 ; A ; ; m G = Solve. A and A are tangents segments. 42. A = 12, AG = 4, find G. 4. F = x, F = x + 1, GF = 14, F = 4, find F and F. 44. A = 14, find A. 45. AJ = 12, HJ = 8, AG = 6, find A. 46. A chord 12 units long is 10 units from the center of a circle. Find the length of the radius. 47. Find the length of a diagonal of a rectangle if the length of the rectangle is 12 in. and the area is 72 in Find the area of a triangle if the longest leg has length 12 cm. 49. Find the area of an isosceles right triangle if the hypotenuse has length 18 meters. 50. Find the area of a trapezoid if the height is 6.5 in. and the bases are 12.2 in. and 15.6 in. 51. Find the length of the side of a square that has an area equal to that of a quadrilateral with perpendicular diagonals whose lengths are 10 m and 5 m. 52. Find the area of a sector of a circle with a radius of 8 and an arc length of. 5. Find the area of a segment of a circle with radius of 9 and an arc length of.

6 54. A sandbox 12 ft. by 14 ft. requires that the sand be spread to a depth of 6 in. How many cubic feet of sand are needed? 55. The base of a right prism is an equilateral triangle with sides of length 8 cm. The height of the prism is 10 cm. Find the volume. 56. A right closed cylindrical tank has a radius of 6 ft. and is 0 ft. tall. To the nearest hundredth of a gallon, find the number of gallons needed to paint the tank if one gallon covers 150 sq. ft. 57. Water is running into a right cylindrical tank at the rate of 4 cubic yards per second. How long will it take to fill the tank that 15 ft. high and 8 ft. in diameter? 58. Suppose the tank being filled in #17 was cone shaped instead of cylindrical. How long would it take to fill the tank? 59. Find the lateral area of a cone that has a base radius of 6 cm. and a volume of 96 cm. 60. What type of triangle is A: A(2, 1); (, 5); (, ) 61. What type of quadrilateral is A? A( 2, 5); (0, ); ( 1, 4); (, 2) 62. Find the slope of the line passing through ( 4, 2) and (, 1). 6. Find the coordinates of the other endpoint of a segment if the midpoint has coordinates (4, ) and one endpoint has coordinates (, 5). 64. Find the slope of the line parallel to and the slope of the line perpendicular to 4x y = Write the equation of the line with slope 2 that passes through ( 2, 6). 66. Write the equation of the line that passes through P( 2, 1) and Q(, 2). 67. Find the center and the radius of a circle having the equation (x + 2) 2 + y 2 = Find the equation of the circle if the endpoints of a diameter (-, -5) and (5, 1). Graph the circle. Solve the following: 69. Find: a. m A b. m c. m 70. Find: a. m A b. m c. mf d. m e. m A f. m AF d. mf e. OG. g. m F h. m F x + 2x + 2x In circle O O A; O F 5x 6 x Find: a. m A b. m A c. m 72. Find: a. m A b. m A d. m A e. m A c. m d. m e. m 181º 160º is a tangent ircle with 8º diameters A and

7 7. Find: m 74. Find: a. m A b. m AF c. m A d. m e. m A 20º 0º 0º 40º 75. Find: a. m A b. m c. m AF d. m A e. m A f. m. A = circle G with diameter A and tangent A m A = 8º tangent raw diagrams when necessary. 76. A balloon is in the shape of a sphere and has a surface area of π 9 ft2. a. What is the length of the radius in inches? b. What is the volume of the sphere in cubic inches? 77. A balloon is formed by a cylinder with two hemispheres one on each end of the cylinder. The radii of the hemispheres are the same as the radius of the cylinder. If the radius is 4 cm., the volume of the balloon is 688π cm. Find the height of the cylinder. 78. A water reservoir is in the form of a right circular cone. The cone is 21 ft. deep and has a radius of 7 ft. Water has collected in the cone to a depth of x ft., and has a radius of ft. Find the volume of the water. 79. A trough is 12 ft. long and its vertical cross sections are inverted isosceles triangles with bases of 6 ft. and heights of 8 ft. Water is being poured into the trough at a rate of 2 cubic feet per minute. If the level of water reached so far is 5 ft., find the volume of the water at this instant. a. Graph, b. shade each region bounded by the following lines, c. identify the shape, d. find the vertices, and e. find the area. 80. y = 0, x = 0, y = 1 x y = 0, y = x, y = 6, x = y = x, y = x, y = 5 8. y = 1, y = 5, y = x + 7, y = x 8

8 a. Graph each region. b. Rotate the region about the given line. c. Name the solid. d. Find the surface area of the solid. e. Find the volume of the solid. 84. y = 0, x = 0, y = 5, x = 85. x = 0, y = x, y = 6 axis of revolution: y axis axis of revolution: y axis 86. y = 0, x = 0, y = 2 x + 4, x = y = 0, x = 0, y = x 6 axis of revolution: y axis axis of revolution: y axis Graph each vector and find its magnitude and direction. 88. HJ = 8, KL = 5, MN = 6, 6 Plot on a coordinate plane the given line and its image under the translation. T : (x, y) (x + 2, y ) 91. T: y = x T: y = 4x Graph each line then reflect the line (a) using M y and state the equation of the image; (b) using M x and state the equation of the image. Write the equation of each line on the graph. y = 2x (a) Using R O, 90 graph the preimage of the line and its image. State the equation of the image on the graph. (b) Using H O graph the preimage of the line and its image. State the equation of the image on the graph. (c) Using R O, 90 graph the preimage of the line and its image. State the equation of the image on the graph y = x + 8 Study: All 2nd Semester Quizzes All 2nd Semester Tests All 2nd Semester Notes