reas of Polygons Triangle: 1 = bh Rectangle: bh 1 b1 b h = Trapezoid: = ( + ) Parallelogram: = bh Regular Polygon: 1 1 = ap = apothem perimeter Coordinate Geometry y y1 Slope: Circles 1 1 1 Midpoint: +, y + y Distance: ( ) + ( y y ) 1 1 rea: = π r Circumference: C = πr = πd o m o m o 360 360 rc Length: L ( πr) ( πd) = = Sector rea: = ( π r ) Segment rea: rea of Sector rea of Triangle ngle and rc Formulas o F E N D G H C ( ) o m 360 1 1 = C 1 GDE = GE + FH J = MN LK Right Triangles o J K L M ( ) b C a c c = a + b a opposite sin = = c hypotenuse b adjacent cos = = c hypotenuse a opposite tan = = b adjacent a 60 o 45 o a a 30 o a a 45 o a MCPS 013 014 1
Volume (V) and Surface rea (S) Right Prism V = h = area of base height ( ) ( ) S = + Ph = area of base + perimeter height Right Circular Cylinder V = h = area of base height = π r h Right Pyramid ( ) ( ) π S = + Ch = area of base + circumference height = r + πrh V 1 1 = h = area of base height 3 3 1 1 S = + Pl = area of base + perimeter of base slant height Right Cone Sphere 1 1 1 area of base height 3 3 3 V = h = = π r h S = πr + πrl V 4 = π r 3 3 S = 4π r Other Formulas Diagonal of a Prism: D= L + W + H or D = L + W + H Geometric Mean of Two Numbers a and b: ab Law of Sines: Law of Cosines: sin sin sin C = = a b c c = a + b ab cosc MCPS 013 014
On this review and the eam, answers may be given in radical form, in terms of π, or as a decimal. On the key, decimals are rounded to decimal places. For items 1 through 3, determine the area of the following composite figures. If the figure is shaded, find the area of the shaded portion of the figure. Note: Figures NOT 1.. 3. OR and OS are radii 1 cm 0 cm 16 in 16 in 16 in 8 ft S 8 ft R O 16 in For items 4 and 5, determine the probability that a dart will hit the shaded area of the target. 4. 6 in 3 in Note: Figure NOT 1 4 1 3 C 1 D 3 4 5. 5 in 7 in 10 in Note: Figure NOT 10 in 7 0 3 5 C 13 0 D 7 10 MCPS 013 014 3
6. Which of the following statements are true? a. 7 5 = 35 b. 35 7 = 5 c. 10 3 = 7 d. 5 + 6 = 11 e. 3 = 16 = 4 For items 7 through 9, determine the distance between the following pairs of points. 7. (,6) and ( 5,9 ) 8. ( 3,6) and (,1) 9. ( 1, 1) and (,5 ) 10. Given: Quadrilateral ROTS with vertices as labeled below. Show that: If the midpoints of the sides of this quadrilateral are connected, then the figure formed has opposite sides that are congruent. R(,4) Q S(6,6) P M O(0,0) N T(8,0) MCPS 013 014 4
11. Given: Right triangle C with coordinates as shown. Point M is the midpoint of C. Find the area of triangle M. C(0,6) M (0,0) (6,0) For items 1 through 14, the lengths of three segments are given. State whether the triangle formed by the three segments is acute, right, or obtuse. Give reasons for your answers. 1. 6, 8, 10 13. 3, 8, 9 14. 9, 1, 14 For items 15 through 3, determine the value of and/or y in each figure below. Note: Figures NOT 7 15. 16. 10 4 7 4 17. 18. 19. 1 y 3 15 45o 8 60 o y MCPS 013 014 5
0. 1. 45 o 30 o y. 3. y 1 30 o 45 o 4. regular heagon has an area of 96 3 and an apothem of 4 3. What is the length of each side of the heagon? 4 C 8 D 8 3 For items 5 through 31, complete the information for each solid. Note: Figures NOT 5. 10 Surface rea: r = 6 Volume: 6. regular heagon with perimeter = 36 and area = 1 Surface rea: Volume: MCPS 013 014 6
7. 4 6 Point is at the center of the square base. Slant Height: Surface rea: Volume: 6 8. 4 Slant Height: Surface rea: Volume: r = 3 9. 4 cm Surface rea: Volume: 30. 9 Surface rea: Volume: 10 5 31. 8 6 Surface rea: Volume: MCPS 013 014 7
3. If the radius of a sphere is doubled, what will happen to the surface area of the sphere? It will increase by a factor of. It will increase by a factor of 4. C It will increase by a factor of 6. D It will increase by a factor of 8. 33. If each side of a cube is doubled, what will happen to the volume of the cube? It will increase by a factor of. It will increase by a factor of 4. C It will increase by a factor of 6. D It will increase by a factor of 8. 34. Look at the right rectangular prism below. C E D G 7 3 F 5 a. What is the length of segment E? b. What is the relationship between diagonal E and the prism? 35. cube has a surface area of 4 in. What is the volume of the cube? 36. In the figure below, the two cones are similar: 6 4 15 Note: Figures NOT drawn to scale a. What is the value of r? Eplain how you determined your answer. b. What is the ratio of the volumes of the cones? Eplain how you determined your answer. r MCPS 013 014 8
37. n artist makes three-dimensional artwork for museums. There are four basic shapes that the artist provides: cones, cylinders, and hemispheres. Here is the pricing structure that the artist uses to charge for his artwork: ny surface in the shape of a circle: $1.50 per square foot. ny surface shaped like the side of a cone, cylinder, or hemisphere: $15.00 per square foot. For each shape below, determine the total cost the artist will charge. a. cylinder with circular bases. r = 4 ft h = 10 ft b. hemisphere with a circular base. r = 6 ft c. cone with a circular base. Slant height = 13 ft r = 5 ft MCPS 013 014 9
38. If two polygons are similar, what is known about their angles and sides? For items 39 and 40, the polygons are similar. Determine the value of. 39. 40. 0 1 15 5 14 40 For items 41 and 4, determine the value of in each figure below. Note: Figures NOT drawn to scale 41. 4. 5 6 8 1 30 43. In the figure below, // CD E Note: Figure NOT drawn to scale C D True or False? a. E ECD b. E E = EC ED c. E E = C D d. E is isosceles e. 1 = CD MCPS 013 014 10
44. In the figure below, m n p. 3 C m Note: Figure NOT 9 6 D 8 E n F G p What is the perimeter of triangle FG? Eplain how you determined your answer. 45. Look at triangles C and DCE below. C Note: Figure NOT D E In parts (a), (b), and (c) below, determine whether the triangles are similar, based on the given information. Use mathematics to justify your answer. a. DE b. c. C = 14, C = 16 DC = 1, CE = 4 = 10, C = 0 DE = 0, CD = 40 MCPS 013 014 11
46. The ratio of the perimeter of two similar polygons is 5:7. What is the ratio of their areas? 47. The ratio of the volumes of two similar solids is 8:7. What is the ratio of their surface areas? 48. The ratio of the surface areas of two similar solids is 5:36. What is the ratio of their volumes? 5:6 5:36 C 15:16 D 65:196 49. On the coordinate plane below, plot the points ( 1,), ( 0,1), and C (,3 ) y a. Show that the triangle is a right triangle and determine its area. b. Using ( 0,0 ) as the center of dilation and a scale factor of, draw the dilation image and state the coordinates of each verte of the image. c. What is the ratio of the area of the dilation image to the pre-image? Use mathematics to justify your answer. MCPS 013 014 1
50. Regular pentagon PRSTU below has a perimeter of 60. It is dilated with a scale factor of 1 3, with the center of dilation P to produce the shaded pentagon PR S T U. U P R Note: Figure NOT drawn to scale T S What is the length of ST? 51. If 6 is the geometric mean of and, what is the value of? 3 10 C 1 D 18 5. What is the geometric mean of 6 and 1? 7 = 6 C 18 D 7 For items 53 through 55 below, FGH is a right triangle, GK an altitude. F K G H 53. Which of the following are similar to triangle FGH? I II Triangle FKG Triangle GKH I only II only C oth I and II D Neither I nor II 54. If HK = 1 and FK = 3, what is GK? 55. Write three proportions using GK. MCPS 013 014 13
For items 56 and 57, look at circle P below. 36 o P 0 cm Note: Figure NOT drawn to scale 56. What is the length of? 4π 8π C 80π D 160π 57. What is the area of the shaded sector? 4π 8π C 40π D 160π For items 58 through 60, find the measure of the angle marked o 58. 59. Note: Figures NOT o 14 o 74 o 8 o o 60. 36 o o 56 o MCPS 013 014 14
For items 61 through 63, find the measure of the arc marked o. 61. 6. 6 o Note: Figures NOT 157 o 41 o o o 63. 59 o 41 o o 64. In circle E below, is tangent to the circle at. 90 o md = o mdh = 40, mgh = 30 o 1 10 C 8 9 E 3 4 6 F 5 G D Note: Figure NOT drawn to scale Find the measures of the angles numbered 1 10. H 7 J m 1 = m = m 3 = m 4 = m 5 = m 6 = m 7 = m 8 = m 9 = m 10 = MCPS 013 014 15
65. For a game, a spinner is in the shape of a circle with radius 10 cm. The spinner is divided into sectors. One sector intercepts an arc of 7 degrees. What is the area of that sector? For items 66 and 67, find the area of the shaded segments of the circles. 66. 8 cm 90 o Note: Figure NOT 67. 10 o Note: Figure NOT 18 cm M 68. Circle O has a diameter of 0. UT = 16, SM = 4 RS UT, RS SM L S U Note: Figure NOT drawn to scale R W O T a. What is the length of OW? b. What is the length of LM? MCPS 013 014 16
69. Describe and sketch the locus of all points in a plane that are equidistant from each endpoint of. 70. Describe the set of points in space that are equidistant from each endpoint of segment. For items 71 and 7, look at the right triangle below. 40 9 41 Note: Figure NOT 71. Which of the following has a ratio of 40 41? sinθ cosθ C tanθ D None of these 7. Which of the following has a ratio of 40 9? sinθ cosθ C tanθ D None of these MCPS 013 014 17
For items 73 through 77, find the value of. 73. 74. 75. 1 40 14 17 o 40 o 5 o 76. 11 77. 6 8 7 78. jet plane begins a steady climb flies for 3 miles (15840 feet) at an angle of 1 o with the ground. What was its change in altitude in feet? 79. Kyle and Holly are 000 feet apart and both are looking at a balloon. t one time, Holly sees the hot-air balloon with an angle of elevation of 4 o, while at the same time, Kyle sees the balloon with an angle of elevation of 70 o. alloon Note: Figure NOT Kyle 70 o 4 o 000 ft Holly a. How far are Holly and Kyle from the balloon? Eplain how you determined your answer. b. How high is the balloon? Eplain how you determined your answer. MCPS 013 014 18
80. Farreed and Kim begin walking from the intersection of two roads. The angle between the roads is 40 o, as shown in the figure below. Farreed walks 3 miles on one road, and Kim walks 4 miles on the other road. Farreed 3 mi 40 o 4 mi Kim Note: Figure NOT What is the distance between Farreed and Kim? Eplain how you determined your answer. For items 81 and 8, use the unit circle below. y P(1,0) 81. What angle of rotation will transform point P to an image of P ( 0,1)? 8. What is sin 70 o? 83. elow are vectors u and v. Sketch the resultant vector u+ v using both the head-to-tail and the parallelogram methods. MCPS 013 014 19
84. Jill can swim 6 miles per hour in still water. She tries to swim straight across a stream that has a current of 4 miles per hour. The dashed resultant vector is her path, shown in the drawing below. Jill swims 6 mph Current 4 mph Jill s Path a. t what speed is Jill traveling? b. t what angle,θ, is Jill swimming with respect to her intended path? MCPS 013 014 0