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Erin Madden

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					                                                                              Erin Madden
                                                                                  10/17/07
                                                                    Christine Von Renesse
                                                Foundations of Math Reasoning MATH-150

                 A Squared Plus B Squared Equals C Squared? But why?

        Why Pythagorean Theorem? Who came up with this nonsense? I will never use

this in my actual day to day life. This was my attitude towards this project. Why this?

Make me teach and write about something that makes sense to my field in elementary

school. No 3rd grader I will have to teach will need to know how to calculate the length

of the hypotenuse of a triangle. But it took someone, my boyfriend none the less, to calm

me down and keep me from crying. He made me realize that no matter how

unproductive and unnecessary I may think this project is, I will learn from it more than I

think or may even want to. I just wanted to get it done and over with. But then I

realized, that it may not be this but, there are going to be things that I will have to teach in

my career that I won’t be able to unless I teach myself first. If I don’t understand

something like the Pythagorean Theorem I will never be able to even pretend I know

what I am talking about. So here is how I began, from the very beginning of course,

which is a very good place to start.

        Who was the person who came up with this way of seeing things? Most people

believe that it was a man named Pythagoras, a Greek man from the island of Samos in

about 570BC. He was the man it was named after and he was the one to make it famous.

Yet, just as we praise Columbus for finding America, even though he was not the first

one here, we give Pythagoras the credit. The first to know the characteristics of a right

triangle was a Chinese man named Tschou-Gun, who lived way back in 1100BC. The

theorem itself was also known to the Caldeans and the Babylonians more than one
thousand years before Pythagoras. Pythagoras gets the credit and the honor of it being

named after him since he was the first to come up with a geometrical demonstration.

This part of his studying was in Egypt where they used a triangle with sides three, four,

and five to make the right angled corners of their buildings. Even if all they knew is that

it worked and did not get an equation out of it, we now know how long ago the idea came

about. In the end, when it all comes down to it, it was Pythagoras that came up with the

actual realization that “the square described upon the hypotenuse of a right-angled

triangle is equal to the sum of the squares described upon the other two sides.”

           The concept of a²+b²=c² is not exactly easy to grasp at first. What do those extra
2’s
      on the side mean? The extra 2’s are called squares, which mean that whatever number

they follow has to be multiplied by itself. In addition, the two legs of a right triangle,

which is what we are working with, are always shorter than the hypotenuse, which is the

side we are trying to find in this situation. Therefore, if the triangle we are working with

has two legs, one with a length of 3 and the other with a length of 4, we want to put these

into the equation as “a” and “b” to find out what “c” would be.


          This makes the equation:


                  3²+4²=c²


          Then we have what (3×3)+(4×4)= 9+19=25


          We now have the two sides 3 and 4 squared to give us 9 and 19. In the case of a

           right triangle we now know that the one mystery side (c²) is equal to 25 but that is

           what the number is squared.
      Now we need to find the square root of 25


       This would look like: 25      This tells us what number squared is 25


      To simply find the square root of a number we can plug it into our calculators

           using the square root button. Which looks like this:


      After doing so we now know that the square root of 25 is 5.


      This means that the three sides of our triangle are 3, 4, and 5.


There are many other ways to figure out. A way to check and make sure that your

process works is to make sure that the areas of your two squares that you made from

finding the squares of the sides add up to the area of the hypotenuse. It can be

represented through blocks in order to fully understand the actual square figures produced

from squaring.




                                      (4×4)


                                                                                   (5×5)




                                                                     (3×3)
In this you can actually see the squared aspect of the problem, as well as count out each

individual square to count out the area of both the squares produced by the legs 3 and 4

and visually realize that they are exactly the same totaled up as the area of the hypotenuse

sides resulting square. This right here proves my original idea that a+b=c is the same as

a²+b²=c². This can not be true because if it were the hypotenuse side would be 7 not 5.

Also 16+9 is 25 not 49. I know finally understand the Pythagorean theorem and why it

works.

				
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