The Mathematics Education Department
EMT 725, Lisa F. Garey
Click here for the Problem statement
Let BD = a = DE. Then AB = 2a. 
Using the Pythagorean formula, 
Then, AE=AD-DE and BF = AB-AF, so
which is the Golden Ratio.
Click here for a GSP script constructing the Golden Ratio on a line segment.
The Sublime triangle is a special triangle constructed from a line segment divided into the Golden ratio. A GSP script provides the step-by-step construction. Point G divides segment AB into the Golden ratio. Observe that the Sublime triangle is an isosceles triangle inside another isosceles triangle.
This triangle can be used to construct a pentagon and a decagon. A decagon is made up of 10 Sublime triangles and a pentagon can be constructed from the decagon.
Click here for a GSP script for a decagon or a pentagon.
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