Pythagoras was an ancient Greek philosopher and mathematician, who is most famous for the theorem which bears his name: The Pythagorean Theorem.

The Pythagorean Theorem is used in right triangle geometry. If you have a triangle which has a right angle, and you label the two legs a and b, with the hypotenuse c, then a² + b² = c². (Which is not quite what the scarecrow says in the movie "The Wizard of Oz," but you can't expect too much of a scarecrow, I suppose.)

Thus, if you know a and b, you can find c. But of course, that's not all; if you know any two sides, you can find the third one.

Example One - If the legs of a right triangle are 3 and 4, what is the length of the hypotenuse?

Answer - 3² + 4² = 9 + 16 = 25. Thus, c = 25.

Example Two - If the hypotenuse of a right triangle is 17, and one of the legs is 15, what is the length of the other leg?

Answer - 17² = 15² + b², so b² = 289 - 225 = 64, and b = 8.

Not all solutions work out to nice natural numbers, though; you might need your calculator to do a square root, as in the next example...

Example Three - If the legs of a right triangle are both 1, what is the length of the hypotenuse?

Answer - c² = 1² + 1² = 2, so c is the square root of 2 (approximately 1.414).

Example Four - The sides of a right triangle are x,x + 7, and x + 8. What is the value of x?

Answer - x² + (x + 7)² = (x + 8)². This simplifies to x² - 2x - 15 = 0, which factors into (x - 5)(x + 3), and so 5 is the answer. 

1.

If the legs of a right triangle are 1 and 2, what is the length of the hypotenuse?

2.

If the length of the hypotenuse of a right triangle is twice one of the legs, and the other leg has length 12, what is the length of the hypotenuse?

3.

(5, 12, 13) is an example of a Pythagorean Triplet; they are sides of a right triangle, and they are all integers. Can you find another triplet (other than the ones in this article)?

4.

The sides of a right triangle are x, x + 1, and x + 2. What is the value of x?

5.

If the hypotenuse of a right triangle has length 10, and the two legs are the same length, what is the perimeter of the triangle?