Golden Triangle: An Isosceles Triangle

Date: 01/23/2001 at 17:09:22
From: Grayson Connors
Subject: The Golden Triangle

I have talked to my Geometry teacher about whether there is such a 
thing as a Golden Triangle, and he says no, but my Algebra Honors 
teacher tells me that there is, and it is a right triangle with the 
ratio 3:4:5, and any of the multiples, like if you multiply by 2, 
6:8:10.  

I want to know if it exists. I am positive it was in my textbook 
from Algebra 1.  Please help solve the issue.

Date: 01/24/2001 at 00:37:12
From: Doctor Schwa
Subject: Re: The Golden Triangle

Hi Grayson,

What I've heard called a golden triangle is more like a golden 
rectangle. A golden rectangle has the property that if you cut off a 
square from it, the remaining rectangle is the same shape as the 
original rectangle, only smaller. Its sides are in the ratio 
1 : (1 + sqrt(5))/2, or about 1 : 1.618033.

For a picture of these rectangles, see the Dr. Math FAQ:

  Golden Ratio, Fibonacci Sequence
  http://mathforum.org/dr.math/faq/faq.golden.ratio.html   

The golden triangle is an isosceles triangle. It has the property 
that, if you bisect one of the base angles, one of the triangles you 
cut off is similar to the original triangle. If its base is 1 unit 
long, the two equal sides are each (1 + sqrt(5))/2 units long, the 
same ratio as the golden rectangle.

Golden triangles can be found in pentagons.  In a regular pentagon
ABCDE, the triangle ACD is a golden triangle:

  

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/