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TimeStamp: 09/19/97 at 13:15:00
From: Doctor Chita
To: User125310@AOL.com (noelle)
Subject: Re: Geometry
Approved-by: sonya
Asker Age: 13
Date of most recent message in thread: 09/17/97 at 14:44:04

As noelle wrote to Dr. Math
On 07/07/97 at 13:57:09 (Eastern Time),
>
>this_page:
>dr.math_form_submission
>
>Question submitted via WWW:
>I'm having problems with The Pythagorean Theorem! My teacher has 
>explained it but I still don't get it. Some of it is a breeze other 
>parts just stump me!
>                            Thanks,
>                              noelle

Hi Noelle:

More than 2,000 years ago Pythagoras came up with an extraordinary 
discovery about the relationship among the squares on the sides of a right
triangle. You can understand what he discovered by following the next four
steps:  

1. Draw a right triangle. The two sides that form the right angle are called 
   "legs". The third side, opposite the right angle, is called the
   hypotenuse. 

2. Label the vertices of the triangle A, B, and C, with C at the right 
   angle. Then, label the sides opposite each vertex a, b, and c. If you 
   did this correctly, the two legs are a and b, and the hypotenuse is c.

3. Now, draw a square on each side of the triangle. The area of each square 
   is a^2, b^2, and c^2. (The symbol ^ means "raise to a power." In this 
   case square the numbers.)

   I've posted a picture to illustrate what I mean. The address is

     /dr.math/gifs/noelle7.7.97.gif

   You can look at the picture on the computer or draw your own.

4. What Pythagoras demonstrated was that the sum of the areas of the 
   squares on the legs is equal to the area of the largest square. Using
   our labels, you can write: a^2 + b^2 = c^2. This is Pythagoras' theorem.

It's important to remember that we don't have to use A, B, and C (and a, b, 
and c) as our labels. Any labels will do.In general the Theorem says 
that: leg(1)^2 + leg(2)^2 = hypotenuse^2. When using the theorem to solve 
problems it's important to locate the right angle in the triangle and then 
identify the legs and the hypotenuse. Once you have those, you can use the 
formula.

You can solve many problems involving right triangles by knowing any two of 
the three sides. For example, suppose the sides you are given are the two 
legs, and they measure 3 and 4. Then the unknown side is the hypotenuse. 
Substitute 3 and 4 for a and b in the theorem and solve for c.

  3^2 + 4^2 = c^2
     9 + 16 = c^2
         25 = c^2
          5 = c

(Actually, there are two answers to c^2 = 25. One is 5 and the other is -5. 
But in geometry we only use positive lengths so we can disregard -5.)

If the two sides you are given are a leg and the hypotenuse, then to find 
the length of the unknown leg, you have an extra algebra step to complete.
Let's say a = 5 and c = 13, where a is a leg and c is the hypotenuse.
Then, 

  a^2 + b^2 = c^2
  5^2 + b^2 = 13^2
   25 + b^2 = 169

Subtract 25 from both sides of the equation:

        b^2 = 169 - 25 = 144
          b = 12. (Throwing away -12 as a length.)  

Does this help?

-Doctor Chita,  The Math Forum
 Check out our web site!  </dr.math/>

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