A___________C_______B
The ancient Greek mathematicians thought that the most pleasing geometric forms, such as the ratio of the height to the width of a doorway, were created using the golden section. however, they were surprised to learn that the golden section is not a rational number. one way of expressing the golden section is by using a line segment. In the line segment shown, (AB/AC)=(AC/CB). If AC=1 unit, find the ratio (AB/AC).

Question

A___________C_______B 
The ancient Greek mathematicians thought that the most pleasing geometric forms, such as the ratio of the height to the width of a doorway, were created using the golden section. however, they were surprised to learn that the golden section is not a rational number. one way of expressing the golden section is by using a line segment. In the line segment shown, (AB/AC)=(AC/CB). If AC=1 unit, find the ratio (AB/AC).

dumbcow · Answer

AC = 1
AB = 1+CB

Let CB be represented by x:
$$1+x = \frac{1}{x}$$
rewrite as quadratic:
$$x^2 +x - 1 = 0$$
use quadratic formula(only use positive solution):
$$x = \frac{-1 +\sqrt{5}}{2}$$
plug this result back in to find AB
$$AB = 1+CB = 1 + \frac{-1+\sqrt{5}}{2} = \frac{1+\sqrt{5}}{2}$$
This equals the ratio AB/AC, since AC = 1