The Pythagorean Theorem

Date: 07/07/97 at 13:57:09
From: Noelle
Subject: Geometry

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 but 
other parts just stump me!
																			
Thanks,
Noelle

Date: 09/19/97 at 13:15:00
From: Doctor Chita
Subject: Re: Geometry

Hi Noelle:

More than 2,000 years ago Pythagoras came up with an extraordinary 
discovery about the relation 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 areas of the 
   squares are a^2, b^2, and c^2. (The symbol ^ means "raise to a 
   power." In this case, square the numbers.)

         

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!  http://mathforum.org/dr.math/