SOLUTION: what is a pythagorean theorem? what is one proof? one example of pythagorean theorem?

The Pythagorean theorem states:

The square of the hypotenuse of any right triangle
equals the sum of the squares of the two legs


Here's a proof:

Start with right triangle ABC

A
|\
| \
|  \
|   \
|   .\D
|·____\
B      C

We are to prove that AC² = AB² + BC²

Draw a perpendicular BD from B to AC.
I can only indicate it on here by the
two dots.

Right triangles ABC, BCD, and ADB are 
all similar because they have 
corresponding angles equal. So 

 AB     AC         AC     BC 
---- = ----  and  ---- = ----
 AD     AB         BC     CD

Cross multiplying these

AB² = AC·AD  and  BC² = AC·CD

Adding equals to equals:

AB² + BC² = AC·AD + AC·CD

Factoring out AC on the right

AB² + BC² = AC(AD + CD)

But AD + CD = AC, so we
can substitute AC for AD + CD

AB² + BC² = AC·AC

AB² + BC² = AC² 

An example of the Pythagorean theorem:

What is the length of the diagonal of 
a rectangle whose width is 3 feet and
whose length is 4 feet:

The diagonal and two adjacent sides of
the rectangle form a right triangle, where
the diagonal is the hypotenuse.

a² + b² = c²
3² + 4² = c²
 9 + 16 = c²
     25 = c²
      5 = c
So the length of the diagonal is 5

Edwin