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Understanding Similar Polygons and the Golden Ratio

This chapter introduces the concept of similar polygons, emphasizing that two figures can be similar in shape while differing in size. Key criteria for similarity include congruent corresponding angles and proportional corresponding sides, encapsulated in the similarity ratio. Examples demonstrate how to identify similar triangles and determine their similarity ratios. Additionally, the chapter explores the golden rectangle and the golden ratio, approximately 1.618:1, recognized for its aesthetic appeal, as illustrated through an artist's canvas planning scenario. Homework assignments reinforce understanding of these concepts.

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Understanding Similar Polygons and the Golden Ratio

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  1. Chapter 7: Similarity 7.2 Similar Polygons

  2. Similarity • Two figures that have the same shape but not necessarily the same size are similar. • Two polygons are similar if: • (1) corresponding angles are congruent • (2) corresponding sides are proportional • similarity ratio • ratio of the lengths of corresponding sides

  3. Example 1 • ABCD ~ EFGH. Complete the statements:

  4. Example 2 • Determine whether the triangles are similar. If they are, write the similarity statement and give the similarity ratio.

  5. Example 3 • LMNO ~ QRST. Find the value of x.

  6. Golden Ratio • golden rectangle: • a rectangle that can be divided into a square and a rectangle that is similar to the original rectangle • golden ratio: • approximately equal to 1.618:1 • considered pleasing to the human eye • Leonardo DaVinci

  7. Golden Rectangle

  8. Golden Ratio • INSERT PICTURE!!!!

  9. Example 5 • An artist plans to paint a picture. He wants the canvas to be a golden rectangle with its longer horizontal sides 30 cm wide. How high should the canvas be?

  10. Homework • p. 369: 35-42 • p. 375: 13-18, 32, 33

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