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 September 17th, 2010, 06:48 PM #1 Senior Member   Joined: Apr 2007 Posts: 2,140 Thanks: 0 Pythagorean theorem Prove that Pythagorean theorem works. Well I know that I need to have three squares on each side of the triangle. September 17th, 2010, 07:19 PM #2 Global Moderator   Joined: Dec 2006 Posts: 21,035 Thanks: 2272 The triangle should have a right angle, and have just one square on each side! I think you already know how to prove the theorem. September 17th, 2010, 11:10 PM #3 Global Moderator   Joined: Nov 2009 From: Northwest Arkansas Posts: 2,767 Thanks: 5 Re: Pythagorean theorem "Cut the knot." There are infinite algebraic and geometric proofs. September 18th, 2010, 12:47 PM #4 Global Moderator   Joined: May 2007 Posts: 6,835 Thanks: 733 Re: Pythagorean theorem Trivial corollary - Place any geometric figure on the hypotenuse, then place geometrically similar figures on each leg. The sum of the areas of the figures on the legs equals the area of the figure on the hypotenuse. September 18th, 2010, 01:16 PM #5 Global Moderator   Joined: Dec 2006 Posts: 21,035 Thanks: 2272 That can be achieved by constructing just one line segment, a perpendicular drawn from the right-angle vertex to the hypotenuse, since the resulting smaller triangles have the same angles as the original triangle. Their areas obviously sum to the area of the original triangle, so the Pythagorean theorem follows immediately from the theorem that the areas of similar figures are proportional to the squares on corresponding sides. September 18th, 2010, 09:16 PM   #6
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Re: Pythagorean theorem

Quote:
 Originally Posted by mathman Trivial corollary - Place any geometric figure on the hypotenuse, then place geometrically similar figures on each leg. The sum of the areas of the figures on the legs equals the area of the figure on the hypotenuse.
Yes, that reminds me of solving this pythagorean theorem place three semicircles on each side of the right triangle. 1/2 pi (1/2 a)^2 + 1/2 pi (1/2 b)^2 = 1/2 pi (1/2 c)^2, therefore a^2 + b^2 = c^2. September 19th, 2010, 10:10 AM #7 Global Moderator   Joined: Dec 2006 Posts: 21,035 Thanks: 2272 How did you obtain the result for the semicircles without already knowing the Pythagorean theorem? September 19th, 2010, 06:50 PM #8 Senior Member   Joined: Apr 2007 Posts: 2,140 Thanks: 0 Re: Pythagorean theorem September 20th, 2010, 07:13 AM   #9
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Re: Pythagorean theorem

Hello, johnny!

Quote:
 Prove that Pythagorean theorem works.

This is my favorite proof . . .

Code:
a        b
*-----*-----------*
|    *    *       | a
|   *       c *   |
b |  * c            *
| *              *|
|*            c * |
*              *  | b
a |   *  c      *   |
|       *    *    |
*-----------*-----*
b        a

.

[color=beige]. . [/color]

[color=beige]. . [/color] September 20th, 2010, 03:32 PM #10 Senior Member   Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: Pythagorean theorem Hello soroban! That is my favorite as well. It is the one I have used over the years that I have had the best success with easily demonstrating the truth of this theorem to students of algebra/geometry.  Tags pythagorean, theorem Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post Skyer New Users 11 August 1st, 2011 05:15 AM xnarutox Geometry 3 October 16th, 2010 08:53 AM moonrains Geometry 2 January 7th, 2009 03:46 PM mohanned karkosh Geometry 1 October 22nd, 2007 05:13 AM johnny Calculus 2 September 17th, 2007 12:45 PM

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