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  1. Activator • Solve the proportion: 4/ (x+2) = 16/(x + 5) • Simplify: 5weeks/30days; 85cm/.5m

  2. UNIT E.Q • How can I use all what I am learning on polygon similarity to solve real life problems. • Which careers use similarity most? • Is similarity shown and used only in geometry?

  3. Standards for Mathematical Practice • 1. Make sense of problems and persevere in solving them. • 2.Reason abstractly and quantitatively. • 3. Construct viable arguments and critique the reasoning of others. • 4. Model with mathematics. • 5. Use appropriate tools strategically. • 6.Attend to precision. • 7. Look for and make use of structure. • 8.Look for and express regularity in repeated reasoning.

  4. 8.1 Ratio and Proportion

  5. Essential Question: 1. How do we use proportions to solve problems? 2.How do we use properties of proportions to solve real- life problems, such as using the scale of a map?

  6. Computing Ratios • Ratio of a to b : • Ex: Simplify the ratio of 6 to 8. Simplify the ratio of 12 to 4.

  7. Simplifying Ratios • Simplify the ratios.

  8. The perimeter of rectangle ABCD is 60 cm. The ratio of AB:BC is 3:2. Find the length and width of the rectangle. • Hint: draw a rectangle ABCD.

  9. Using Extended Ratios • The measure of the angles in ∆JKL are in the extended ratio of 1:2:3. Find the measures of the angles. 2x 3x x

  10. Using Ratios • The ratios of the side lengths of ∆DEF to the corresponding side lengths of ∆ABC are 2:1. Find the unknown lengths. A D 3 L B C L F E 8

  11. Properties of proportions • Cross product property then ad = bc. • Reciprocal property

  12. Solve the proportions.

  13. Given that the track team won 8 meets and lost 2, find the ratios. • What is the ratio of wins to loses? • What is the ratio of losses to wins? • What is the ratio of wins to the total number of track meets?

  14. Simplify the ratio. • 3 ft to 12 in • 60 cm to 1 m • 350g to 1 kg • 6 meters to 9 meters

  15. Solve the proportion

  16. Tell whether the statement is true.

  17. 8.2 Problem Solving in Geometry with Proportions

  18. Additional Properties of Proportions • If , then • If , then

  19. Using Properties of Proportions • Tell whether the statement is true.

  20. In the diagram . Find the length of BD. A 30 16 B C x 10 D E

  21. In the diagram • Solve for DE. A 5 2 D B 3 E C

  22. Geometric mean • The geometric mean of two positive numbers a and b is the positive number x such that • Find the geometric mean of 8 and 18.

  23. Geometric Mean • Find the geometric mean of 5 and 20. • The geometric mean of x and 5 is 15. Find the value of x.

  24. Different perspective of Geometric mean • The geometric mean of ‘a’ and ‘b’ is √ab • Therefore geometric mean of 4 and 9 is 6, since √(4)(9) = √36 = 6.

  25. Geometric mean • Find the geometric mean of the two numbers. • 3 and 27 √(3)(27) = √81 = 9 • 4 and 16 √(4)(16) = √64 = 8 • 5 and 15 √(5)(15) = √75 = 5√3

  26. Ex. 3: Using a geometric mean • PAPER SIZES. International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x. Slide #26

  27. Solution: The geometric mean of 210 and 420 is 210√2, or about 297mm. 210 x Write proportion = x 420 X2 = 210 ∙ 420 X = √210 ∙ 420 X = √210 ∙ 210 ∙ 2 X = 210√2 Cross product property Simplify Factor Simplify Slide #27

  28. Using proportions in real life Slide #28 In general when solving word problems that involve proportions, there is more than one correct way to set up the proportion.

  29. Ex. 4: Solving a proportion Width of Titanic Length of Titanic = Width of model Length of model LABELS: Width of Titanic = x Width of model ship = 11.25 in Length of Titanic = 882.75 feet Length of model ship = 107.5 in. Slide #29 MODEL BUILDING. A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it?

  30. Reasoning: Write the proportion. Substitute. Multiply each side by 11.25. Use a calculator. Width of Titanic Length of Titanic = Width of model Length of model x feet 882.75 feet = 11.25 in. 107.5 in. 11.25(882.75) x = 107.5 in. x ≈ 92.4 feet So, the Titanic was about 92.4 feet wide. Slide #30

  31. Note: Slide #31 Notice that the proportion in this Example contains measurements that are not in the same units. When writing a proportion in unlike units, the numerators should have the same units and the denominators should have the same units. The inches (units) cross out when you cross multiply.

  32. A model truck is 13.5 inches long and 7.5 inches wide. The original truck was 12 feet long. How wide was it?

  33. 8.3 Similar Polygons

  34. Activator

  35. Identifying Similar Polygons • When there is a correspondence between two polygons such that their corresponding angles are congruent and the lengths of corresponding sides are proportional the two polygons are called similar polygons.

  36. Similar polygons • If ABCD ~ EFGH, then G H C D F B A E

  37. Similar polygons • Given ABCD ~ EFGH, solve for x. G H C D 6 x B F 2 4 A E 2x = 24 x = 12

  38. Is ABC ~ DEF? Explain. B D 6 E 13 12 5 7 A C F 10 ABC is not similar to DEF since corresponding sides are not proportional. ? ? yes no

  39. Similar polygons Given ABCD ~ EFGH, solve for the variables. G H C D 5 x B F 10 2 y 6 A E

  40. If two polygons are similar, then the ratio of the lengths of two corresponding sides is called the scale factor. Ex: Scale factor of this triangle is 1:2 9 4.5 3 6

  41. Quadrilateral JKLM is similar to PQRS. Find the value of z. R K L S 15 Q z 6 P J M 10 15z = 60 z = 4

  42. Theorem • If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. • If KLMN ~ PQRS, then

  43. Given ABC ~ DEF, find the scale factor of ABC to DEF and find the perimeter of each polygon. E P = 8 + 12 + 20 = 40 B P = 4 + 6 + 10 = 20 12 20 10 6 A C D F 8 4 CORRESPONDING SIDES 4 : 8 1 : 2

  44. 8.4 Similar Triangles

  45. In the diagram, ∆BTW ~ ∆ETC. • Write the statement of proportionality. • Find m<TEC. • Find ET and BE. T 34° E C 3 20 79° B W 12

  46. Postulate 25 • Angle-Angle (AA) Similarity Postulate If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

  47. Similar Triangles • Given the triangles are similar. Find the value of the variable. )) m )) ) 6 8 11m = 48 ) 11

  48. Similar Triangles • Given the triangles are similar. Find the value of the variable. Left side of sm Δ Base of sm Δ Left side of lg Δ Base of lg Δ = 6 5 > 2 6h = 40 > h

  49. ∆ABC ≈ ∆DBE. A 5 D y 9 x B C E 8 4