Subject: Re: Pythagorean Theorem
From: BETH HENTGES (b.hentges@cctc.cc.mn.us)
Date: Wed Mar 22 2000 - 14:41:43 EST
See , The Converse of the Pythagorean Theorem, The Mathematics Teacher,
pp. 692-693, volume 87, number 9, December 1994, Jerome Rosenthal.
Three proofs are given. The third does not depend of the theorem
itself.
I'll try to restate the third proof given.
Start with triangle ABC with a=length of BC, b=length of AC, c=length
of AB, and a^2 + b^2 = c^2.
Extend segment AB through B. On ray AB there are two points that are a
units from B. Lable the point a units from B that is between A and B as
point E. Label the point a units from B, on the other side of B than A,
as point D.
"Construct" segments CD and CE.
Note that length BD = a, length BE = a, and length EA = c-a (since
length AB=c).
a^2 + b^2 = c^2
b^2 = c^2 - a^2
b^2=(c+a)(c-a)
Divide both sides by b and (c-a).
b/(c-a) = (c+a)/b
Look at triangles CAE and DAC. They both have angle A in them.
The above eqaution says AC / AE = AD / AC.
So, triangles CAE and DAC are similar, (in that order) by
side-angle-side similarity.
Then angles ACE and ADC are congruent since corresponding angles of
similar figures are congruent. Let x=the measure of angle ACE=the
measure of angle ADC.
Note that triangle BDC is isosceles, so measure of angle BCD = x,
also.
Then, measure of angle CBD = 180-2x.
So, measure of angle CBE = 2x.
Note that triangle BCE is also isosceles.
Then meaure of angle BCE=measure of angle BEC = (180-2x)/2 =
2(90-x)/2=90-x.
So, measure of angle ACB = x + (90 - x) = 90 which is want we wanted to
prove.
whoo hoo!!!!!!!!!
I like whoo hoo much better than QED.
Beth
>>> Stefan Baratto <sbaratto@earthlink.net> 03/22 11:14 AM >>>
I am trying to find a proof of the converse of the Pythagorean Theorem
that
doesn't require the Pythagorean Theorem as a given. That is, if a
triangle
has sides whose lengths satisfy the relationship a^2 + b^2 = c^2 then
it is
necessarily a right triangle. The only proofs I could find (or deduce)
require one to assume that if a triangle is right, then the sum of the
squares of the lengths of the legs is equal to the square of the length
of
the hypotenuse.
Stefan Baratto
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>>> Stefan Baratto <sbaratto@earthlink.net> 03/22 11:14 AM >>>
I am trying to find a proof of the converse of the Pythagorean Theorem
that
doesn't require the Pythagorean Theorem as a given. That is, if a
triangle
has sides whose lengths satisfy the relationship a^2 + b^2 = c^2 then
it is
necessarily a right triangle. The only proofs I could find (or deduce)
require one to assume that if a triangle is right, then the sum of the
squares of the lengths of the legs is equal to the square of the length
of
the hypotenuse.
Stefan Baratto
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