2. Lesson 8.1 & 8.2 Solving Problems with Ratio and Proportion Today, we will learn to… …find and simplify ratios...use proportions to solve problems

3. Ratio A ratio is a comparison of two numbers written in simplest form. a a : b a to b b

4. Simplify the ratio. K H D M D C M 1.2 m : 300 cm 200 cm : 300 cm 2 : 3 2. 2 km : 600 m 2000 m : 600 m 10 : 3 3. 10 mm : 5.5 cm 10 mm : 55 mm 2 : 11

5. 4.In the diagram, DE : EF is 1 : 2 and DF = 45. Find DE and EF. D E F 1x + 2x = 45 3x = 45 x = 15 12 DE = EF = xx 15 30

6. 5.In ΔABC, the measures of the angles are in the extended ratio of 3:4:5. Find the measures of the angles. 12x = 180 x = 15 3x + 4x + 5x = 180 °, °, ° What do we know about the angles of a triangle? 45 60 75

7. 6.The perimeter of a rectangle is 70 cm. The ratio of the length to the width is 3 : 2. Find the length and the width of the rectangle. 3x 2x 3x 3x+2x+3x+2x = 70 Length is Width is 10x = 70 x = 7 21 14

8. 7. A triangle has an area of 48 m 2. The ratio of the base to the height is 2 : 3. Find the base and height. A = ½ bh 48 = ½ (2x)(3x) 48 = 3x 2 16 = x 2 4 = x base is height is 8 m 12 m 2x 3x

9. Solve the proportion for x. 8. 2 8 7 x-2 2(x-2) = 56 2x - 4 = 56 2x = 60 x = 30

10. 9.On a map, 2 inch = 180 miles. Two cities are about 2 ¾ inches apart. Estimate the actual distance between them. 2 in 180 mi 2x = 180(2¾) x = 247.5 miles 2 ¾ in x mi

11. 10.In a photograph taken from an airplane, a section of a city street is 3 1 / 2 inches long and 1 / 8 of an inch wide. If the actual street is 30 feet wide, how long is it? 1 / 8 30 x = x = 840 feet 3 1/21/2 x = (3 )(30) 1/81/8 1/21/2

12. 11.AB : AC is 3 : 2. Find x. 3x+3 x+12 = 3(x+1) = 2(x+3) 3x+3 = 2x+6 x + 3 = 6 x = 3

13. 12. Given MN MP find PQ. NO PQ = x 14-x 4 6 x = x = 8.4 4x = 6(14-x) ? ? 4x = 84 - 6x 4 M 6 N OQ P 14

14. 5 A 2 B CE D 7+x x 7 5 7 7 = x = 2.8 Given AB AD find DE. AC AE = 13. 5(7+x) = 49 ? ? 35+5x = 49 5x = 14

15. 14.Standard paper sizes are all over the world. The sizes all have the same width-to-length ratios. Two sizes of paper shown are A4 and A3. Find x. 210 mm x x 420 mm 210x 420 x = x 2 = (210)(420) x ≈ 297 mm

16. 15.The batting average of a baseball player is the ratio of the number of hits to the number of official at-bats. x.308 643 1 = x = (643)(.308) x = 198 hits In 1998, Sammy Sosa of the Chicago Cubs had 643 official at-bats and a batting average of.308. How many hits did Sammy Sosa get?

17. 16.A wheelchair ramp should have a slope of 1 / 12. If a ramp rises 2 feet, what is its run? 1 2 ft x12 = x = (12)(2 ft) x = 24 feet What is its length? length 2 = 2 2 + 24 2 length 2 = 4 + 576 length 2 = 580 length = 24.08 feet 2 ft ?

18. Geometric Mean The geometric mean of two positive numbers (a and b) is …. a x x b

19. Find the geometric mean of the given numbers. 35 and 175 x ≈ 78.3 x x 35 175 = x 2 = 35(175)

21. Lesson 8.3 Similar Polygons Today, we will learn to… …identify similar polygons...use similar polygons

22. Two polygons are similar if all corresponding angles are congruent and corresponding sides are proportional. AB BC AC ΔABC ~ Δ XYZ if  A   B   C  XX YY ZZ XY YZ XZ and

23. B C D A F G H E ABCD ~ EFGH CD GH AD EH AB EF BC FG  A   E,  B   F,  C   G,  D   H Statement of Proportionality

24. Scale Factor The scale factor is the ratio of the lengths of two corresponding sides.

25. 6 8 10 1.Are the triangles similar? If they are, find the scale factor and write a statement of similarity. 912 15 Yes, the scale factor is 2323  XAR ~  __ __ __ MNT

26. 4.5 6 9 2.Are the triangles similar? If they are, find the scale factor and write a statement of similarity. 6 8 12 Yes, the scale factor is 3434  LMN ~  __ __ __ TPO

27. 12 15 x A B C D E F 10 12 15 12 y x 4. Δ ABC ~ Δ DEF 15 10 3 2 = y 10 12 x 12 = 15 10 x = 18 12 y = 15 10 y = 8 Scale Factor?

28. The triangles are similar. Find x and y. 5. A C 8 x 12 Map the triangles to find corresponding sides. B E D F 9 y 18 B A C x 12 8 B A C x 8 9 y 18 x 128 9 x = x = 4 18 8 y 12 = 18 8 y = 27

29. 5 = 3 6. RSTU ~ LMNO. Find the following. 125  m  T = m  S = 55  x 2.4 x x =4

30. 7.You have a 3.5 inch by 5 inch photo that you want to enlarge. You want the enlargement to be 16 inches wide. How long will it be? 3.516 x5 = 3.5x = (16)(5) x = 22.9 ≈ 23 inches

31. A triangular work of art and the frame around it are similar equilateral triangles. 12 in. 16 in. 9. Find the ratio of the perimeters. (artwork : frame) 3434 3434 36 48 8. Find the ratio of the artwork to the frame.

32. The rectangles are similar. 11. Find the ratio of the perimeters. 4545 4545 22 27.5 10. Find the ratio of corresponding sides. 4 7 5 8.75 220 275

33. Theorem 8.1 If 2 polygons are similar, then the ratio of the perimeters is __________ the ratio of corresponding side lengths. equal to

34. 12.The patio around a pool is similar to the pool. The perimeter of the pool is 96 feet. The ratio of the patio to the pool is 3 to 2. Find the perimeter of the patio. 3 x 962 = 2x = (3)(96) x = 144 feet patio pool

35. Turn to page 145 in your workbook!

37. Lesson 8.4 Proving Triangles are Similar Triangles Today, we will learn to… …identify similar triangles...use similar triangles

38. Postulate 25 Angle-Angle (AA) Similarity Two triangles are similar if 2 pairs of corresponding angles are congruent.

39. Determine whether the triangles are similar. If they are, write a similarity statement. 1. R M N L 27˚ L TS 35˚ 80˚ 65˚ 80˚ ΔRTS ~ Δ____ M 35˚ 65˚ LN

40. Determine whether the triangles are similar. If they are, write a similarity statement. 2. G H JK L 27˚ ΔGLH ~ Δ____ GKJ

41. 4.If the triangles are similar, write a similarity statement. 31 ˚ 47 ˚ not similar

42. 5.If the triangles are similar, write a similarity statement. 43 ˚ not similar

43. 6.The triangles are similar, find x. 3 5 7 x 2 = 3x = 10 x ≈ 3.33 yx 2 5 3 y 2 3y = 14 y ≈ 4.67 73

44. m  DEC = m  ECD = m  EBA =  AEB ~ EC =  CED 44 ˚ 20 68 ˚ x 15 = 6x = 120 x = 20 86 68 ˚ 7. Find the following. What kind of trapezoid? What do you know about DE and CE?

45. 8. The triangles are similar. Find x. A B C D E 15 25 18 9 x 15 9 25 x = x = 15

46. Are the triangles similar? If they are, write a similarity statement. Not ~  XZW ~  XTY T Y X X Z W

47. Are the triangles similar? If they are, write a similarity statement. Not ~  ABD ~  BCE 40  75 

48. Lesson 8.5 Proving Triangles are Similar Triangles Today, we will learn to… …use similarity theorems to prove that two triangles are similar

49. Theorem 8.2 Side-Side-Side (SSS) Similarity If all three corresponding sides are proportional, then the triangles are similar.

50. Determine whether the triangles are similar. If they are, write a similarity statement. 1. D E F 8 10 12 A B C 15 12 18 Δ ACB ~ Δ____ by _____ DFE 12 15 18 8 10 12 scale factor? 3:2 SSS

51. Theorem 8.3 Side-Angle-Side (SAS) Similarity If two sides are proportional and the angles between them are congruent, then the triangles are similar.

52. Determine if the triangles are similar. If they are, write a similarity statement. 2. A B C D E F 6 8 8 12 Not similar 8 12 6 8

53. Determine whether the triangles are similar. If they are, write a similarity statement. 3. A B C D E 5 5 3 3 ΔABE ~ Δ____ by _____ ACD SAS 3 5 610 Scale Factor?1:2

54. Separate the triangles if it helps. 3. A C D 10 6 ΔACD ~ ΔABE by B E A 5 3 SA S 3 5 610

55. Find x.  GLH ~  GKJ 4. x = 7.5 G H JK L 6 x 10 8 10 + x 14 8 10 14 10 + x 14 10 + x 8(10 + x) = 140

56. x = 7.5 G H JK L 6 x 10 8 8 10 6 x 8x = 60 What can we conclude? Find x.  GLH ~  GKJ 5.

57. Theorem 8.4 Triangle Proportionality Theorem If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.

58. Find x. The triangles are similar. 6. 7. x 12 14 39 x = 18 x = 4 12 x 5 10 x 10 5 2 2639 2x

59. Estimate the height of the tree. 8. 4 ft. 6 ft. 16 ft. x ft. x = 24 feet 4 6 = 16 x

60. 3 5.5 = Estimate the height of the tree. 9. 3 ft. 5.5 ft. 12 ft. x ft. x = 27.5 feet 15 x

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62. Lesson 8.6 Proportions and Similar Triangles Today, we will learn to… …use proportionality theorems to calculate segment lengths

63. Find the value of x. 1. 2. x 4 6 12 x = 4.8 x = 2.8 4 x 5 7 x 7 5 2 10 12 2 x

64. Find the value of x. 3. 4. x 5 7 15 x = 6.25 x ≈ 2.67 5 x 6 8 x 8 6 2 2 x 1215

65. Find the value of x. 5. 6. 10 14 x x = 21 x = 11 36 x x 33 x 20 10 2414 1030 36 33

66. Theorem 8.5 Triangle Proportionality Converse If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

67. Use similar triangles to find x. 7. 8. 12 78 x x = 6 x = 20 6 10 8 x x 10 8 6 ?7 ? 8 1612 x

68. Mid-Segment Theorem The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is _____ as long. half

69. Theorem 8.6 If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.

70. 9. Find x and y. y 24 x 10.5 7 8 x = 24 = y x = 12 y = 21 8 7 7 8

71. 10. Find x, y, and z. x 15 13 y z 10 30.4 15 = x 30.4 38 38x = 456 x = 12

72. 10. Find x, y, and z. 12 15 13 y z 10 30.4 15 = 12 y 13 15y = 156 y = 10.4 15 = 12 z 10 15z = 120 z = 8 x = 12

73. Theorem 8.7 An angle bisector of a triangle divides the opposite side into segments whose lengths are proportional to the other two sides.

74. 11. Find x. 12. Find x. 21 = 24 x = 7 24 x8 21 x 8 3 5 2 x 3 = 2 x5 x = 7.5

75. 24 12 8 13. Find x. ? 8 = 12 x 24-x x = 9.6 24 - x

76. 14. Find x and y. 18 16 8.5 What is another way to write y? 8.5 - x 18 = 16 x 8.5-x x = 4.5 8.5 - x =4 y =4

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79. Lesson 8.7 Dilations Today, we will learn to… …identify dilation...use properties of dilations to create a perspective drawing

80. Dilation A dilation is a transformation that results in a reduction or enlargement of a figure.

81. = 1. A circle in a photocopier enlargement has a 6 inch diameter. If the enlargement percentage is 125%, what is the diameter of the preimage circle? 4.8 in. 100 x 125 6 in.

82. C 3 6 Scale Factor = P Q R P´ Q´ R´ Reduction

83. C Scale Factor = 3 8 P Q R P´ Q´ R´ Reduction new image preimage

84. C 2 5 P Q R P´ Q´ R´ Scale Factor = Enlargement

85. C P Q R P´ Q´ R´ 5 15 Scale Factor = Enlargement new image preimage

86. C P Q R P´ Q´ R´ 4 10 Find x. x 6 x = 2.4 4 10 = x 6

87. 25 10 = P C Q R P´ Q´ R´ 10 25 Find x. 10 4 5 y x x = 2 5 x

88. P C Q R P´ Q´ R´ 10 25 Find y. 10 4 5 y x y = 10 25 10 = y 4

89. Dilations in a Coordinate Plane If the origin is the center of the dilation, you can find the image by multiplying each vertex by the scale factor.

90. Rectangle ABCD has vertices A (3,1), B (3, 3), C (2, 3), and D (2, 1). Find the coordinates of the dilation with center (0,0) and scale factor of 2. Graph on next slide…

91. A (3,1), B (3, 3), C (2, 3), D (2, 1) A’(6,2), B’(6, 6), C’(4, 6), D’(4, 2) Scale Factor is 2 x 2x A’ B’ C’ D’ Do you notice a pattern?

92. Rectangle ABCD has vertices A (-3,3), B (3, 6), C (6, -3), and D (-3, -6). Find the coordinates of the dilation with center (0,0) and scale factor of 1 / 3. (-1,1) B’B’ D’D’ C’ C’ A’A’ (2, -1) (1, 2) (-1, -2)

93. A’ (-1,1) B’ (1,2) C’ (2,-1) D’(-1,-2) Scale Factor is 1 / 3 A(-3,3) B(3, 6) C(6, -3) D(-3, -6) A’ B’ C’ D’ A B C D

94. Find x. 5 12 6 = 12x = 30 x = 2.5 x

95. A’ B’ C’ (0, 6) (6, 6) (4.5, 3) A’ B’ C’ ABCABC (0, 4) (4, 4) (3, 2)

96. X’ Y’ Z’ (-0.75,-0.5) (2, 1) (1, -1) XYZXYZ (-1.5, -1) (4, 2) (2, -2) X’ Y’ Z’

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