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Similarity Ratios and Proportions

Similarity Ratios and Proportions. Section 7.1. Ratio. Comparison of two quantities. Ways to write a ratio:. Be sure to SIMPLIFY!. Example. A scale model of a car is 4” long. The actual car is 18’ long. What is the ratio of the length of the model to the length of the car?.

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Similarity Ratios and Proportions

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  1. SimilarityRatios and Proportions Section 7.1

  2. Ratio • Comparison of two quantities Ways to write a ratio: Be sure to SIMPLIFY!

  3. Example • A scale model of a car is 4” long. The actual car is 18’ long. What is the ratio of the length of the model to the length of the car? 4 to 18(12) = 4 to 216 = 1 to 54 4:216 = 1:54 4 = 1 216 54

  4. Proportion An Equation • A statement that two ratios are equal • Extended Proportion: When three or more ratios are equal

  5. Cross-Product Property • The product of the extremes is equal to the product of the means Means Extremes

  6. Examples…solve the following proportions 1) 2 : 5 = n : 35 2) 3) 4) x + 1 : 3 = x : 2

  7. Solving Proportions • Use the cross product property to solve for variables in a proportion

  8. Solving an extended proportion • Choose two ratios to work with first in order to solve for a variable • Use substitution and then solve for the other variable using cross product

  9. Another example!

  10. Properties of proportions: • Given: The following is true: 1) 2) 3) 4) Cross Product Property Reciprocals Exchange the means Adding 1 to each side

  11. Writing equivalent proportions • Given

  12. Write four true statements given:

  13. Similarity Section 7.2

  14. Similar • Two figures with the same shape that have: • Corresponding angles congruent • Corresponding sides proportional

  15. Similarity Ratio • The ratio of the lengths of corresponding sides = 2” 1

  16. To determine if two figures are similar • Check that corresponding angles are congruent and all corresponding sides have an equal similarity ratio

  17. Using Similar Figures • Find the missing angles and side lengths given LMNP~QRST

  18. Solve for x and y given LMNO~QRST

  19. 7.3 Proving Triangles Similar

  20. Proving triangles are similar • Angle Angle Similarity (AA~) Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. ∆LNM ~ ∆RTS

  21. Side-Angle-Side Similarity (SAS~) Theorem • If an angle of one triangle is congruent to an angle of a second triangle and the sides including the two angles are proportional, then the triangles are similar. ∆XYZ ~ ∆ABC

  22. Side-Side-Side Similarity (SSS~) Theorem • If the corresponding sides of two triangles are proportional, then the triangles are similar. ∆QRS ~ ∆TUV

  23. Why don’t we have AAS similarity or ASA Similarity? • They both fall under the umbrella of AA Similarity

  24. Determine if the following triangles are similar.

  25. 7.3 Proof!  Given: Prove:

  26. 7.4 Similar Right Triangles

  27. The Golden Ratio! • A golden rectangle can be divided into a square and rectangle that is similar to the original rectangle. In any golden rectangle the length and width are in the golden ratio, which is about 1.618 to 1. http://www.youtube.com/watch?v=fmaVqkR0ZXg

  28. Similarity in Right Triangles 4 1 2 What do you know about angle D in the two smaller triangles? Is there any relationship between the other angles? 3

  29. Theorem 7-3 • The altitude to the hypotenuse of a right triangle divides the triangle into 2 triangles that are similar to the original triangle and to each other.

  30. Geometric Mean • For 2 positive numbers a and b, x is the number such that • Therefore, Positive only…NOT +/- !

  31. Find the Geometric Mean of:

  32. Similar Right Triangles and Geometric Mean • The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. Geometric Mean of 4 and 5

  33. Find the value of x in simplest radical form:

  34. Try These • Solve for each variable.

  35. Find the value of x in simplest radical form:

  36. Solve for each variable.

  37. Solve for each variable.

  38. Solve for the variables.

  39. Proportions in Triangles Section 7.5

  40. Review:Triangle Proportionality Theorem/Side-Splitter Theorem • If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

  41. Examples: Find the value of x.

  42. Solve for x.

  43. Corollary to Theorem 7.4 • If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

  44. Examples: solve for the variables.

  45. Triangle Angle Bisector Theorem • If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

  46. Examples: solve for x.

  47. Solve for x.

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