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Proportions and Similarity

Proportions and Similarity. § 9.1 Using Ratios and Proportions. § 9.2 Similar Polygons. § 9.3 Similar Triangles. § 9.4 Proportional Parts and Triangles. § 9.5 Triangles and Parallel Lines. § 9.6 Proportional Parts and Parallel Lines. § 9.7 Perimeters and Similarity.

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Proportions and Similarity

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  1. Proportions and Similarity • § 9.1 Using Ratios and Proportions • § 9.2 Similar Polygons • § 9.3 Similar Triangles • § 9.4 Proportional Parts and Triangles • § 9.5 Triangles and Parallel Lines • § 9.6 Proportional Parts and Parallel Lines • § 9.7 Perimeters and Similarity

  2. Vocabulary Using Ratios and Proportions What You'll Learn You will learn to use ratios and proportions to solve problems. 1) ratio 2) proportion 3) cross products 4) extremes 5) means

  3. Using Ratios and Proportions In 2000, about 180 million tons of solid waste was created in the United States. The paper made up about 72 million tons of this waste. The ratio of paper waste to total waste is 72 to 180. This ratio can be written in the following ways. 72 to 180 72:180 72 ÷ 180 a to b a:b a ÷ b where b  0

  4. Using Ratios and Proportions proportion A __________ is an equation that shows two equivalent ratios. Every proportion has two cross products. In the proportion to the right, the terms 20 and 3 are called the extremes, and the terms 30 and 2 are called the means. = 20(3) 30(2) The cross products are 20(3) and 30(2). 60 = 60 equal The cross products are always _____ in a proportion.

  5. Using Ratios and Proportions Likewise,

  6. Using Ratios and Proportions Solve each proportion: = (30 – x)2 = 30(6) 3(x) 15(2x) 3x = 60 – 2x 30x = 180 5x = 60 x = 6 x = 12

  7. Driving gear Driven gear equivalent ratio givenratio = driving gear driving gear = driven gear driven gear x = 14 Using Ratios and Proportions The gear ratio is the number of teeth on the driving gear to the number of teeth on the driven gear. If the gear ratio is 5:2 and the driving gear has 35 teeth, how many teeth does the driven gear have? 35 5 x 2 5x = 70 The driven gear has 14 teeth.

  8. Using Ratios and Proportions End of Section 9.1

  9. Vocabulary Similar Polygons What You'll Learn You will learn to identify similar polygons. 1) polygons 2) sides 3) similar polygons 4) scale drawing

  10. D ΔABC is similar toΔDEF A C B F E Similar Polygons A polygon is a ______ figure in a plane formed by segments called sides. closed It is a general term used to describe a geometric figure with at least three sides. Polygons that are the same shape but not necessarily the same size are called ______________. similar polygons The symbol ~ is used to show that two figures are similar. ΔABC ~ ΔDEF

  11. and D C G H E F A B Similar Polygons proportional Polygon ABCD~ polygon EFGH

  12. 6 4 5 7 5 7 4 6 = Similar Polygons Determine if the polygons are similar. Justify your answer. 6 4 5 7 1) Are corresponding angles are _________. congruent 2) Are corresponding sides ___________. proportional 0.66 = 0.71 The polygons are NOT similar!

  13. R 5 Write the proportion thatcan be solved for y. 4 J S T 6 x = 7 Write the proportion thatcan be solved for x. = K L y + 2 Similar Polygons Find the values of x and y if ΔRST ~ ΔJKL 6 4 y + 2 7 4(y + 2) = 42 4y + 8 = 42 5 4 4y = 34 x 7 4x = 35

  14. 1.25 in. 1 in. .5 in. = = Utility Room Kitchen Dining Room length width .75 in. Living Room 1.25 in. Garage Scale: 1 in. = 16 ft. Similar Polygons Scale drawings are often used to represent something that is too large or too small to be drawn at actual size. Contractors use scale drawings to represent the floorplan of a house. Use proportions to find the actual dimensions of the kitchen. 1.25 in. .75 in. 1 in 1 in L ft. w ft. 16 ft 16 ft (16)(1.25) = w (16)(.75) = L 20 = w 12 = L width is 20 ft. length is 12 ft.

  15. Similar Polygons End of Section 9.2

  16. Vocabulary Similar Triangles What You'll Learn You will learn to use AA, SSS, and SAS similarity tests for triangles. Nothing New!

  17. Similar Triangles Some of the triangles are similar, as shown below. The Bank of China building in Hong Kong is one of the ten tallest buildings in the world. Designed by American architect I.M. Pei, the outside of the 70-story building is sectioned into triangles which are meant to resemble the trunk of a bamboo plant.

  18. Similar Triangles In previous lessons, you learned several basic tests for determining whether two triangles are congruent. Recall that each congruence test involves only three corresponding parts of each triangle. Likewise, there are tests for similarity that will not involve all the parts of each triangle. similar C F D E A B If A ≈ D and B ≈ E, then ΔABC ~ ΔDEF

  19. Similar Triangles Two other tests are used to determine whether two triangles are similar. proportional C 6 F 2 3 1 A E D B 4 8 then the triangles are similar then ΔABC ~ ΔDEF

  20. Similar Triangles proportional C F 2 1 D A E B 4 8 then ΔABC ~ ΔDEF

  21. J 14 G K 9 21 6 10 H M 15 P Similar Triangles Determine whether the triangles are similar. If so, tell which similarity test is used and complete the statement. 6 10 14 , the triangles are similar by SSS similarity. Since = = 15 21 9 JMP Therefore, ΔGHK ~Δ

  22. = Similar Triangles Fransisco needs to know the tree’s height. The tree’s shadow is 18 feet longat the same time that his shadow is 4 feet long. If Fransisco is 6 feet tall, how tall is the tree? 1) The sun’s rays form congruent angles with the ground. 2) Both Fransisco and the tree form right angles with the ground. 6 4 t 18 4t= 108 t= 27 6 ft. The tree is 27 feet tall! 4 ft. 18 ft.

  23. 45 m x 8 m 10 m Similar Triangles Slade is a surveyor. To find the distance across Muddy Pond, he forms similar triangles and measures distances as shown. What is the distance across Muddy Pond? 10 8 = It is 36 meters across Muddy Pond! x 45 10x = 360 x = 36

  24. Similar Triangles End of Section 9.3

  25. Vocabulary Proportional Parts and Triangles What You'll Learn You will learn to identify and use the relationships between proportional parts of triangles. Nothing New!

  26. intersects the other two sides of ΔPQR. and P Q R Proportional Parts and Triangles In ΔPQR, Are ΔPQR and ΔPST, similar? corresponding angles PST  PQR P  P ΔPQR ~ΔPST. Why? (What theorem / postulate?) S T AA Similarity (Postulate 9-1)

  27. A C B E D If Proportional Parts and Triangles parallel similar ΔABC ~ ΔADE.

  28. Since , ΔSVW ~ ΔSRT. Complete the proportion: R V S W T SV Proportional Parts and Triangles

  29. A C B D E Proportional Parts and Triangles proportional lengths

  30. A 3 4 H G 5 x + 5 x B C Proportional Parts and Triangles

  31. Brace 4 ft 6 ft 10 ft = 3 x =1 x ft 5 4 ft Proportional Parts and Triangles Jacob is a carpenter. Needing to reinforce this roof rafter, he must find the length of the brace. 4 x 4 10 10x = 16

  32. Proportional Parts and Triangles End of Section 9.4

  33. Vocabulary Triangles and Parallel Lines What You'll Learn You will learn to use proportions to determine whether lines are parallel to sides of triangles. Nothing New!

  34. A 6 4 C B 9 6 D E Triangles and Parallel Lines You know that if a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of proportional lengths (Theorem 9-5). The converse of this theorem is also true.

  35. A x D E 2x C B Triangles and Parallel Lines one-half

  36. 8 5 8 5 Triangles and Parallel Lines Use theorem 9 – 7 to find the length of segment DE. A x 11 D E 22 C B

  37. M 1) MP || ____ A B AC N P C Triangles and Parallel Lines A, B, and C are midpoints of the sides of ΔMNP. Complete each statement. 28 2) If BC = 14, then MN = ____ s 3) If mMNP = s, then mBCP = ___ 4) If MP = 18x, then AC = __ 9x

  38. E 8 A B 5 7 D F C Triangles and Parallel Lines A, B, and C are midpoints of the sides of ΔDEF. 1) Find DE, EF, and FD. 14; 10; 16 2) Find the perimeter of ΔABC 20 3) Find the perimeter of ΔDEF 40 4) Find the ratio of the perimeter of ΔABC to the perimeter of ΔDEF. 1:2 20:40 =

  39. D F G is the midpoint of BA E is the midpoint of AD H is the midpoint of CB C F is the midpoint of DC E H A G B Q1) What can you say about EF and GH ? (Hint:Draw diagonal AC .) Triangles and Parallel Lines ABCD is a quadrilateral. They are parallel Q2) What kind of figure is EFHG ? Parallelogram

  40. Triangles and Parallel Lines End of Section 9.5

  41. Vocabulary Proportional Parts and Parallel Lines What You'll Learn You will learn to identify and use the relationships betweenparallel lines and proportional parts. Nothing New!

  42. D A E B Measure AB, BC, DE, and EF. F C , , DE AB DE AB Calculate each set of ratios: DF BC EF AC Proportional Parts and Parallel Lines Hands-On On your given paper, draw two (transversals) lines intersecting the parallel lines. Label the intersections of the transversals and the parallel lines, as shown here. Do the parallel lines divide the transversals proportionally? Yes

  43. A D l m n B E C F BC DE DE AB EF AB BC AC EF DF DF AC , , = = = and Then Proportional Parts and Parallel Lines If l || m || n

  44. = = G U 12 15 a b c H V 18 x J W 1 x = 22 2 Proportional Parts and Parallel Lines Find the value of x. UV GH VW HJ 15 12 x 18 12x = 18(15) 12x = 270

  45. A D l m n B E C F AB  BC, DE  EF. Proportional Parts and Parallel Lines If l || m || n and Then

  46. Since AB  BC, DE  EF 5 = x Proportional Parts and Parallel Lines Find the value of x. 10 A B 10 Theorem 9 - 9 C (x + 3) = (2x – 2) x + 3 = 2x – 2 (2x – 2) 8 (x +3) 8 F E D

  47. Proportional Parts and Parallel Lines End of Section 9.6

  48. Vocabulary Perimeters and Similarity What You'll Learn You will learn to identify and use proportional relationships of similar triangles. 1)Scale Factor

  49. perimeter of small Δ 6 + 8 + 10 24 2 = = = perimeter of large Δ 9 + 12 + 15 36 3 Perimeters and Similarity These right triangles are similar! Therefore, the measures of their corresponding sides are ___________. proportional Pythagorean Use the ____________ theoremto calculate the length of the hypotenuse. 10 6 15 9 8 12 10 2 8 6 We know that = = = 15 3 12 9 Is there a relationship between the measures of the perimeters of the two triangles?

  50. D A C B F E perimeter of ΔABC CA AB BC = = = FD perimeter of ΔDEF DE EF Perimeters and Similarity the measures of the corresponding perimeters are proportional to the measures of the corresponding sides. If ΔABC ~ ΔDEF, then

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