Investigation 11 Golden Ratio

The Golden Ratio Two numbers display the golden ratio when their sum divided by the larger number is equal to the ratio of the larger number to the smaller number 𝑎+𝑏 𝑎 = 𝑎 𝑏 We can do some algebra to find the value of this ratio so it can be used in calculations

Find the Golden Ratio 𝑎+𝑏 𝑎 = 𝑎 𝑏 𝑎 2 =𝑎𝑏+ 𝑏 2 Now we will need to complete the square for a 𝑎 2 −𝑎𝑏= 𝑏 2 𝑎 2 −𝑏𝑎+?= 𝑏 2 +? 𝑎− 𝑏 2 2 =?

Find the Golden Ratio 𝑎+𝑏 𝑎 = 𝑎 𝑏 𝑎 2 =𝑎𝑏+ 𝑏 2 Now we will need to complete the square for a 𝑎 2 −𝑎𝑏= 𝑏 2 𝑎 2 −𝑏𝑎+ 𝑏 2 4 = 𝑏 2 + 𝑏 2 4 𝑎− 𝑏 2 2 = 5 𝑏 2 4 𝑎− 𝑏 2 =± 5 𝑏 2 4 𝑎= 𝑏 2 ± 𝑏 5 2 Since the number can only be positive, we can eliminate the negative 𝑎= 𝑏+𝑏 5 2 or 𝑏 1+ 5 2 𝑎 𝑏 𝑏 1+ 5 2 𝑏 1+ 5 2

Golden Ratio/The Divine Proportion/Golden Mean/Golden Section 𝜑= 𝑎+𝑏 𝑎 = 𝑎 𝑏 = 1+ 5 2 Since this is an irrational number the Greek letter phi (Φ,φ) is commonly used to represent the ratio The golden ratio is commonly found in nature, spirals, petals of flowers, and is used in art and architecture

If AC=12, find CB to the tenth 𝑎 𝑏 =𝜑 𝐶𝐵 12 = 1+ 5 2 𝐶𝐵=12 1+ 5 2 𝐶𝐵=19.4

If CB=178, find AB to the hundredth 𝑎+𝑏 𝑎 =𝜑 𝐴𝐵 178 = 1+ 5 2 𝐴𝐵=178 1+ 5 2 𝐴𝐵=288.01

If AB=36, find AC to the hundredth 𝑎+𝑏 𝑎 =𝜑 36 𝐶𝐵 = 1+ 5 2 𝐶𝐵(1+ 5 )=72 𝐶𝐵= 72 1+ 5 𝐴𝐶=𝐴𝐵−𝐶𝐵 𝐴𝐶=36− 72 1+ 5 𝐴𝐶=13.75

The Golden Rectangle The golden rectangle’s ratio of length to width is equal to φ 𝑙 𝑤 =φ or 𝑙 𝑤 = 1+ 5 2

This is a golden rectangle, find x 𝑙 𝑤 = 1+ 5 2 34+𝑥 34 = 1+ 5 2 34+𝑥=34 1+ 5 2 𝑥=34 1+ 5 2 −34 𝑥≈21

Questions? Just about every architecture design of the Greek Parthenon comes from the golden ratio From the shape and actual size all the way down to the space between columns and the size of the sculptures used for accents