Overview:
The Pythagorean theorem, involving the relationships between the sides of a right triangle, is one of the most famous geometry theorems. Many proofs have been generated to show the relationship, a2+b2=c2.. The a side and the b side are the legs on either side of the right angle, and the c side is the hypotenuse.
What Are Pythagorean Triples?
For example, suppose a right triangle is constructed with legs measuring 3 units and 4 units. Solving 32 is 9, and 42 is 16. Adding 9 and 16 equals 25 . In turn, 25 equals 52. The numbers 3, 4, and 5 are the first Pythagorean triple. In addition, 52 or 25 plus 122 or 144 equals 169, or132. Pythagorean triples were used in construction very early in history.
How Are the Sides Related?
If the length of the hypotenuse is known and either of the sides, the measurement of the other side can be found. In the above example, suppose the hypotenuse was 5 and one of the sides was 3. The measurement of the other side could be found by subtracting 25 – 9 = 16, and the square root of 16 is 4.
What Are Irrational Numbers?
Not all sides of a right triangle are as neatly rational, however. Suppose the legs of a right triangle were each 1 unit. The hypotenuse would equal √2, because if a2+b2 = c2, c will equal √a2 +b2. That was shown with the Pythagorean triples. But √2 is a decimal that is not very neat, with a nonrepeating decimal value starting with 1.4142. This was not as satisfying to the ancient Greeks, so they called those numbers “immeasurable.”
How Are Other Triangles Related?
Right triangles are the only triangles with the relationship a2 + b2 = c2. However, other types of triangles have similar relationships among the sides of the triangle. If a2 + b2 > c2, the triangle is acute. All its angles measure less than 900. Similarly, if a2 + b2 < c2, the triangle will have one angle that measures more than 900.
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