The Pythagorean Theorem:
Pythagoras and numbers that work.

So many people misstate the theorem as originally discovered because they wish to abbreviate something. There is no shortening of the Pythagorean Theorem, which essentially states that:
"The sum of the areas determined by the squares formed by the leg sides of a right triangle is equal to the area determined by the square formed by the hypotenuse side of the triangle."

A picture may evolve within your mind in which a 3-4-5 triangle, with the hypotenuse side to the right, the 3 at the bottom edge, and the 4 to the left. We have stated the equation:
(3)^2 + (4)^2 = (5)^2; so 9 + 16 = 25 TRUE!Imagine nine blocks forming a three by three matrix under the side of length three, and a fopur by four matrix of blocks forming to the left of the side of length four. Then, when you form the five by five matrix connected to the edge of the side with the length of five, you can see, indeed, that 9 + 16 = 25!

Why this always works remains a mystery to many math teachers, statistic teachers, and other scientists. Yet, we still wish to look at something known as Pythagorean triples.

Pythagorean triples are sets of numbers which will produce a true result within the Pythagorean theorem. A Pythagorean triple, such as (3-4-5) works out as above. Some other famous Pythagorean triples are 5-12-13 , 8-15-17, 10-24-26. and 21-28-35. In all cases:

(5*5) + (12 * 12 ) = 13*13, 25 + 144 = 169

(8*8) + (15*15) = 17*17, 64 + 225 = 289

(10*10) + (24*24) = 26*26, 100 + 576 = 676

and (21*21) + (28*28) = 35*35

so 441 + 784 = 1225

Oh! just a minute, there. 21-28-35 are the same as 3-4-5, just that everyuthing is a multiple of seven!

Right you are, satute observer. Remember in Algebra when your teacher stated one could multiply every term of any equation by the same unit and the equation would remain equivalent. (That's what we were trying to do to add fractions, multiply by a nuber to get equivalent fractions so it would work!) Ah, yes! Now I remember.

So an equation showing (a^2)+ (b^2)= (c^2)

can now become 2(a^2) + 2(b^2) = 2(c^2)

or 7(^2) + 7(b^2) = 7(c^2).

Many students who are about to take any standardized test involving this concept would do well to memorize a few of these Pythagorean triples. Usually, if an equation cannot be reduced to one of these five, it will not come out perfectly, meaniing you don't really have a right triangle in the case you are looking at.

Try for other Pythagorean triples by making a table of squares, seeeing what they add up to. Clue in on prime numbers. Happy hunting.

Learn more about this author, Dallas Brown.
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