The following is a geometric proof of the Pythagorean Theorem. The only assumption it requires is that the area of a rectangle is its height multiplied by its width. I had fun discovering it myself one day while doodling, but I'm sure I was centuries, if not millenia, from the first to note this proof.
At right is a rectangle of height a and width b. Its area is a times b or ab.
|
|
Now draw a diagonal from one corner of the rectangle to the opposite corner. What is left are two identical right triangles. Since their areas are equal, it follows that the area of each is half the area of the rectangle.
|
|
|
|
|
The next drawing is a smaller blue triangle, at right. The triangle is smaller so that there is enough room to draw four of them and make a clever arrangement in which they form a square.
Here is the clever arrangement in which they form a square.
Noting that the length of the square's sides is a+b, it is easy to determine the area of the square to be
 .
|
Henceforth A will denote Area for the sake of brevity.
|
|
The longest side of the triangles is known as the hypotenuse. If we use
c to represent the unknown length of the hypotenuse, then we can also write the area of the large square in terms of the areas of the smaller square and the four triangles.
|
|
If the two formulae for the area of the large square are correct, then they must agree. Setting them equal gives us a formula for the hypotenuse, c.
Subtract 2ab from both sides of the equality and the result is the Pythagorean Theorem.
This is the Pythagorean Theorem. It is true for any right triangle. For more ways to prove it than a reasonable man could require, click
here.
Now use your knowledge of the Pythagorean Theorem to CALCULATE  !
Or use it to make a good estimate of how far you can see off a mountain...
|
