The Golden Ratio and Other Applications of Similarity Lesson 13.7 The Golden Ratio and Other Applications of Similarity pp. 571-575

Objectives: 1. To define the Golden Ratio, golden rectangle, and golden spiral. 2. To identify the Golden Ratio in natural and architectural designs.

Which rectangle do you think is the best looking, has the most pleasing shape? 3 2 1 4 5 5

The Golden Ratio is the ratio of the length to the width of a golden rectangle.

A golden rectangle is a rectangle with the following characteristic: if a square unit is cut from one end of the rectangle, then the resulting rectangle has the same length-to-width ratio as the original rectangle.

1 x

1 x-1 x

x 1 x - 1 = x2 – x = 1 x2 – x – 1 = 0 x = 1± 1 + 4 2 x = 1± 5 2

A Fibonacci Spiral

Explain why rectangle PQRS is not golden. 4 3 P Q R S

Homework pp. 573-575

■ Cumulative Review 21. Which two conditions are theorems for proving triangles similar: SSS, ASA, SSA, SAA, SAS, AAA?

■ Cumulative Review 22. Three of the other conditions in exercise 21 guarantee triangle similarity. Which three? (SSS, ASA, SSA, SAA, SAS, AAA).

■ Cumulative Review 23. Why are the three similarity theorems in exercise 22 not needed?

■ Cumulative Review 24. Which of the six conditions does not guarantee similarity?

■ Cumulative Review 25. Prove (by counterexample) your answer to exercise 24.