One of the most interesting and well-known formulas in math is the Pythagorean
Theorem, which only holds true for right triangles. The formula says that
the length of the hypotenuse squared is equal to the sum of the squares of
the lengths of the legs:
c^{2} = a^{2} + b^{2}
If a triangle is right, then this formula holds true. Conversely, if this
formula holds true, then you know that the triangle for which it works is a
right triangle. With this formula, given any two sides of a right triangle, you
can calculate the length of the third side. One way to picture it can be seen below.
In the figure above, the area of the square
with side length c is equal to the sum of the areas of the squares with side
lengths a and b. This is a physical interpretation of the Pythagorean Theorem.
The three sides of a right triangle can be of any length, provided that they
obey the laws of the Pythagorean Theorem. However, only certain groups of three
integers can be the lengths of a right triangle. Such groups of integers
are called Pythagorean triples. Some common
Pythagorean triples include , , , , and . Any multiple of
one of these groups of numbers also can be a Pythagorean triple. For example,
=3. You can check any of them for yourself. All of these
groups of three integers, these Pythagorean triples, satisfy the Pythagorean
Theorem.
The key application of the Pythagorean Theorem is the technique for calculating
the length of the third side given two of the sides' lengths and the fact that
the triangle is right. Knowing the lengths of sides of a triangle is handy in
deciding whether triangles are related.