**The **

**Golden Cut**

**of a**

**Segment**

by Teresa Banker

The problem of finding the golden section of a segment is stated and solved in
__Euclid II, 11__. That means it has been of mathematical interest
for over twenty centuries.

In the figure, B cuts segment AC into the golden cut. A demonstration on
__The Geometer's Sketchpad__ has been prepared. If you would like to
see this demonstration after reading the directions which follow,
**click
here.** Once the Sketchpad is open, open a new sketch. (There will be a
script for getting the golden cut for the segment you choose.) On the right
hand side, draw two points by clicking the dot with the mouse. Use the finger
to label these as points A and B. Be sure both points A and B are selected by
putting the mouse pointer on them, one at a time, and holding down SHIFT. Then
click on FAST on the script to find the golden cut for your segment.

Notice the ratio is 1.62 (the precision allowed by Sketchpad). If we label AB
= x and BC =1, we have the following result:

, or
.

The positive solution to the quadratic is

= 1.61803, the value of **Phi**. The other solution, then, is
= - 0.61803 . This can be proved using the quadratic formula, since the
quadratic will not factor over the integers.

and
,

which equal 1.61803 and - 0.61803, respectively.

Back to
**Teresa Banker's Home Page**