Golden Triangle: What is It?Date: 09/19/1999 at 16:27:01 From: Robin Byerly Subject: Golden Triangle I'm doing a project for math where I have to answer questions using the Internet. I have gone to many different Web sites trying to figure out what the Golden Triangle is, but I cannot find it. It has to do with math but so far all my results have something to do with bicycles. Please help me find out what the Golden Triangle is. Thank you. Robin Byerly Date: 09/20/1999 at 05:34:49 From: Doctor Floor Subject: Re: Golden Triangle Hi, Robin, Thanks for your question. Suppose we have a triangle ABC, such that <A = <B = 72 degrees (< means angle) and <C = 36 degrees. Such a triangle is known as the Golden Triangle. Let D be the point on BC, such that AD is the angle bisector of <A. Then triangle ABD is again a Golden Triangle. When we let lengths AB = AD = CD = x and BD = 1, then we find: AB : BD = BC : AB x : 1 = (x+1) : x This can be rewritten to: x^2 = x + 1 x^2 -x - 1 = 0 The two solutions for x are x = 1/2 +/- sqrt(5)/2. Since AB > BD, in this case we must have x = 1/2 + sqrt(5)/2. The number 1/2 + sqrt(5)/2 is known as the Golden Ratio, or Golden Mean. So BC : AB is this famous ratio; that's why this triangle is called a Golden Triangle. For more about the Golden Ratio, see our FAQ: Golden Ratio, Fibonacci Sequence http://mathforum.org/dr.math/faq/faq.golden.ratio.html As an example of the appearance of Golden Triangles: the outside triangles of a pentagram are Golden Triangles. When we attach to AC and BC two triangles that are congruent to triangle ACD, we find a regular pentagon. I hope this helped. If you need more, just write us back. Best regards, - Doctor Floor, The Math Forum http://mathforum.org/dr.math/ |
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