__Ratio on a Line Segment :: Clay
Kitchings :: EMAT 6600 :: Spring 2007__

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Tell Clay what a buffoon he is

Given the
following construction where AB = 2DB, DB = DE, and AF = AE, find AF/BF.

(From http://jwilson.coe.uga.edu/emt725/Ratio/ratio.html)

I spent
several hours doing some algebraic manipulations on the relationships I knew. I
started with the Pythagorean relationship, but went too far at first by
expanding each squared term (such as AB^{2} = (AF + BF)^{2},
etc. After a while of getting
nowhere, I decided to redo the algebraic manipulation behind the Pythagorean
relationship as follows:

What guided
me during this process and what did I learn? I learned that it is helpful to continually monitor what IÕm
doing and check for the end result.
I knew from the hint in the problem that Òsomething goldenÓ would come
out of this somewhere, most likely.
I also learned that it is often helpful to substitute something like ÒxÓ
for some ÒmessyÓ terms to make things look more familiar. I knew the typical
quadratic that can give the golden ratio, so I kept looking for ways to get
something of the form x^{2} + x – 1 = 0. I also highly suspected
it would be nice to keep AF and AB in the manipulations, even if I was
unsuccessful at keeping BF in there. I knew that BF was included in AB and I
suspected I could find BF if I had AB in there the whole time.