Pythagorean Theorem

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What was Pythagoras thinking?

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What is the Pythagorean Theorem?
Pythagoras's Diagram
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What is the Pythagorean Theorem?

One of the rules you learn in Geometry is the Pythagorean theorem. This states that the sum of the square of the two legs of a right triangle is equal to the square of the hypotenuse.

What exactly does that mean?

Looking at the diagram on the right, you can see a right triangle. The Pythagorean theorem states that if you add the squares of the two legs equal the square of the hypotenuse (the side of the right triangle opposite the right angle -- this would necessarily be the largest side of the triangle).

Pythagoras's Diagram

What this theorem represents

When Pythagoras developed his theorem, he was talking about the area of squares. The diagram at the right represents what Pythagoras meant. He meant that the area of a square with sides the length of one side of a triangle (side A, in magenta, length 3) added to the area of a square with sides the length of the second side of the triangle (side B, in blue, length 4) would add up to the area of a square with sides the length of the hypotenuse (side C, in purple, length 5).

What this means in our example, the area of the magenta square (with a side length of 3 and an area of 3X3 or 9) added to the area of the blue square (with a side length of 4 and an area of 4X4 or 16) equals the area of the hypotenuse (or purple square -- with a side length of 5 and an area of 5X5 or 25). 9+16=25.

This can help you find the length of the third side of a right triangle when you have the other two.

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Let's see what Wikipedia says about the Pythagorean Theorem:

In mathematics, the Pythagorean theorem (in American English) or 'Pythagoras theorem' (in British English) is a relation in Euclidean geometry among the three sides of a right triangle (right-angled triangle in British English). It states:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

The theorem can be written as an equation:

:a^2 + b^2 = c^2\!\,

where c represents the length of the hypotenuse, and a and b'' represent the lengths of the other two sides.

The Pythagorean theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof,Heath, Vol I, p. 144. although it is often argued that knowledge of the theory predates him. (There is much evidence that Babylonian mathematicians understood the principle, if not the mathematical significance.)

read the rest of the Wikipedia article