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Chapter 8

Chapter 8 . Similarity. 8.1. Ratio and Proportion. Ratios. Ratio- Comparison of 2 quantities in the same units The ratio of a to b can be written as a/b a : b The denominator cannot be zero. Simplifying Ratios. Ratios should be expressed in simplified form 6:8 = 3:4

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Chapter 8

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  1. Chapter 8 Similarity

  2. 8.1 Ratio and Proportion

  3. Ratios • Ratio- Comparison of 2 quantities in the same units • The ratio of a to b can be written as • a/b • a : b • The denominator cannot be zero

  4. Simplifying Ratios • Ratios should be expressed in simplified form • 6:8 = 3:4 • Before reducing, make sure that the units are the same. • 12in : 3 ft 12in : 36 in 1: 3

  5. Examples (page 461) • Simplify each ratio 10. 16 students 24 students 12. 22 feet 52 feet 18. 60 cm 1 m

  6. Examples (page 461) • Simplify each ratio 20. 2 mi 3000 ft 24. 20 oz. 4 lb There are 5280 ft in 1 mi. There are 16 oz in 1 lb.

  7. 16 mm 20 mm 12 in. 2 ft Examples (page 461) • Find the width to length ratio 14. 16.

  8. L N M Using Ratios Example 1 • The perimeter of the isosceles triangle shown is 56 in. The ratio of LM : MN is 5:4. Find the length of the sides and the base of the triangle.

  9. Using Ratios Example 2 • The measures of the angles in a triangle are in the extended ratio 3:4:8. Find the measures of the angles 4x 8x 3x

  10. Q U T V R S Using Ratios Example 3 • The ratios of the side lengths of ΔQRS to the corresponding side lengths of ΔVTU are 3:2. Find the unknown lengths. 2 cm 18 cm

  11. Proportions • Proportion • Ratio = Ratio • Fraction = Fraction Means and Extremes • Extreme: Mean = Mean: Extreme

  12. Solving Proportions • Solving Proportions • Cross multiply • Let the means equal the extremes • Example:

  13. Properties of Proportions • Cross Product Property • Reciprocal Property

  14. Solving Proportions Example 1

  15. Solving Proportions Example 2

  16. Solving Proportions Example 3 A photo of a building has the measurements shown. The actual building is 26 ¼ ft wide. How tall is it? 2.75 in 1 7/8 in

  17. 8.2 Problem solving in Geometry with Proportions

  18. Properties of Proportions

  19. Example 1 • Tell whether the statement is true or false • A. • B.

  20. M 6 N 15 13 Q L 5 P Example 2 • In the diagram Find the length of LQ.

  21. Geometric Mean • Geometric Mean • The geometric mean between two numbers a and b is the positive number x such that ex: 8/4 = 4/2

  22. Example 3 • Find the geometric mean between 4 and 9.

  23. Similar Polygons Polygons are similar if and only if • the corresponding angles are congruent and • the corresponding sides are proportionate.

  24. Similar figures are dilations of each other. (They are reduced or enlarged by a scale factor.) • The symbol for similar is 

  25. Example 1 Determine if the sides of the polygon are proportionate. 8 m 12 m 6 m 6 m 8 m

  26. Example 2 Determine if the sides of the polygon are proportionate. 15 m 5 m 9 m 3 m 4 m 12 m

  27. Example 3 Find the missing measurements. HAPIE  NWYRS 6 A W H 24 18 P Y 4 5 N I E AP = EI = SN = YR = S R 21

  28. Example 4 Find the missing measurements. QUAD  SIML A D S 12 L 8 65º 25 I 95º 125º U Q 20 M QD = MI = mD = mU = mA =

  29. 8.4/8.5 Similar Triangles

  30. Similar Triangles •  To be similar, corresponding sides must be proportional and corresponding angles are congruent.

  31. Similarity Shortcuts AA Similarity Shortcut If two angles in one triangle are congruent to two angles in another triangle, then the triangles are similar.

  32. Similarity Shortcuts SSS Similarity Shortcut If three sides in one triangle are proportional to the three sides in another triangle, then the triangles are similar.

  33. Similarity Shortcuts SAS Similarity Shortcut If two sides of one triangle are proportional to two sides of another triangle and their included angles are congruent, then the triangles are similar.

  34. Similarity Shortcuts We have three shortcuts: AA SAS SSS

  35. 9 g 6 4 7 10.5 Example 1

  36. k 32 h 50 24 30 Example 2

  37. 42 36 m 24 Example 3

  38. 4m 24m 6m 4. A flagpole 4 meters tall casts a 6 meter shadow. At the same time of day, a nearby building casts a 24 meter shadow. How tall is the building?

  39. 5. Five foot tall Melody casts an 84 inch shadow. How tall is her friend if, at the same time of day, his shadow is 1 foot shorter than hers?

  40. 6. A 10 meter rope from the top of a flagpole reaches to the end of the flagpole’s 6 meter shadow. How tall is the nearby football goalpost if, at the same moment, it has a shadow of 4 meters? 10m 6m 4m

  41. 7. Private eye Samantha Diamond places a mirror on the ground between herself and an apartment building and stands so that when she looks into the mirror, she sees into a window. The mirror is 1.22 meters from her feet and 7.32 meters from the base of the building. Sam’s eye is 1.82 meters above the ground. How high is the window? 1.82 1.22 7.32

  42. 8.6 Proportions and Similar Triangles

  43. Proportions • Using similar triangles missing sides can be found by setting up proportions.

  44. Q T R U S Theorem • 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.

  45. Q T R U S Theorem • Converse of the Triangle Proportionality Theorem • If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

  46. U V W X Y Example 1 • In the diagram, segment UY is parallel to segment VX, UV = 3, UW = 18 and XW = 16. What is the length of segment YX?

  47. Q 9.75 P R 9 26 T 24 S Example 2 • Given the diagram, determine whether segment PQ is parallel to segment TR.

  48. Theorem • If three parallel lines intersect two transversals, then they divide the transversals proportionally.

  49. Theorem • If a ray bisects an angle of a triangle, then it divides the opposite side into segments whose lengths are proportional to the lengths of the other two sides.

  50. C 9 3 B 2 6 A 1 8 x F E D Example 3 • In the diagram, 1  2  3, AB =6, BC=9, EF=8. What is x?

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