The Pythagorean Theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of the legs.
In other words, if c is the length of the hypotenuse, and the two legs have length a and b, then
c² = a² + b²
A visual proof of this theorem follows:
Here you see three squares, with areas equal to the squares of the three
sides of the red triangle. The two smaller squares can be rearranged to
make the larger square, proving the sum of the two smaller squares is the larger
square.
Integer values of a, b, and c that make this
equation true are called Pythagorean
Triples.
Related Pages in this website
Go back to the Number Theory Home
Pythagorean Triples
Triangles
Formulas for Primitive Pythagorean
Triples and their deriviation -- a way to generate all the triples such
that a^2 + b^2 = c^2
Prove that the area
of a right triangle with integer sides is not a perfect square. The
proof is here (with some help from someone from the nrich website)
Puzzle
question -- if m is the product of n distinct primes, how many different
right triangles with integer sides have a leg of length m?
Arithmetic Sequence of Perfect
Squares, page 3 -- If a^2, b^2, c^2 are in arithmetic sequence, why is
their constant difference a multiple of 24? Look at the second answer to
this question for the relationship between Pythagorean Triples and this
arithmetic sequence of squares.
Theorems Involving Perfect Squares
-- answers questions such as "why is the square root of x irrational
unless x is a perfect square?" and other fundamental questions about
perfect squares.
