Saturday, January 26, 2013

Two picture proofs of the Pythagorean Theorem.

Here are two squares that are the same size. In the left square, we will label the blue square a² and the red square b².  The two white rectangles both have area ab. Together, they show the famous "middle term" representation of

(a + b)² = a² + 2ab + b²

The square on the right has four white right triangles that are the equivalent of the two white rectangles sliced diagonally, where the diagonal is the hypotenuse, which we usually label c. The yellow square with the gap in it has area c².  Since the squares are the same size we get

c² + 4(½ab) = a² + 2ab + b²  Next step: get rid of the parentheses

c² + 2ab = a² + 2ab + b²  Next step: subtract 2ab from both sides

c²  = a² + b²  and we are done.

Second proof just using the yellow square and the gap inside.

The gap in the middle of the yellow square is (a - b)² = a² - 2ab + b². The four yellow triangles are 2ab, exactly the same as the four white triangles. That means this picture tells us

c² = a² - 2ab + b² + 2ab

So we combine like terms to get
c²  = a² + b²  and once again, Q.E.D., the Latin abbreviation for "that which has been demonstrated".

Tomorrow: number theory and Pythagorean Theorem.

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