My Math Forum  

Go Back   My Math Forum > High School Math Forum > Geometry

Geometry Geometry Math Forum


Reply
 
LinkBack Thread Tools Display Modes
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.
johnny is offline  
 
September 17th, 2010, 07:19 PM   #2
Global Moderator
 
Joined: Dec 2006

Posts: 20,805
Thanks: 2150

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.
skipjack is offline  
September 17th, 2010, 11:10 PM   #3
Global Moderator
 
The Chaz's Avatar
 
Joined: Nov 2009
From: Northwest Arkansas

Posts: 2,766
Thanks: 4

Re: Pythagorean theorem

"Cut the knot."
There are infinite algebraic and geometric proofs.
The Chaz is offline  
September 18th, 2010, 12:47 PM   #4
Global Moderator
 
Joined: May 2007

Posts: 6,784
Thanks: 707

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.
mathman is offline  
September 18th, 2010, 01:16 PM   #5
Global Moderator
 
Joined: Dec 2006

Posts: 20,805
Thanks: 2150

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.
skipjack is offline  
September 18th, 2010, 09:16 PM   #6
Senior Member
 
Joined: Apr 2007

Posts: 2,140
Thanks: 0

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.
johnny is offline  
September 19th, 2010, 10:10 AM   #7
Global Moderator
 
Joined: Dec 2006

Posts: 20,805
Thanks: 2150

How did you obtain the result for the semicircles without already knowing the Pythagorean theorem?
skipjack is offline  
September 19th, 2010, 06:50 PM   #8
Senior Member
 
Joined: Apr 2007

Posts: 2,140
Thanks: 0

Re: Pythagorean theorem

Pythagorean theorem proofs
johnny is offline  
September 20th, 2010, 07:13 AM   #9
Math Team
 
Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

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]

soroban is offline  
September 20th, 2010, 03:32 PM   #10
Senior Member
 
MarkFL's Avatar
 
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.
MarkFL is offline  
Reply

  My Math Forum > High School Math Forum > Geometry

Tags
pythagorean, theorem



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
How would You get to Pythagorean theorem Skyer New Users 11 August 1st, 2011 05:15 AM
Pythagorean theorem problem xnarutox Geometry 3 October 16th, 2010 08:53 AM
pythagorean theorem moonrains Geometry 2 January 7th, 2009 03:46 PM
new Pythagorean theorem mohanned karkosh Geometry 1 October 22nd, 2007 05:13 AM
Pythagorean Theorem johnny Calculus 2 September 17th, 2007 12:45 PM





Copyright © 2019 My Math Forum. All rights reserved.