The Pythagorean theorem, named after Pythagoras, states that in a right triangle, the sum of the squares of the two shorter sides equals the square of the hypotenuse. It has been widely proven and can be used to find unknown segments and special forms, such as Pythagorean triples.
The Pythagorean theorem is a mathematical theorem named after Pythagoras, a Greek mathematician who lived around the 5th century BC. Pythagoras is usually credited with inventing the theorem and providing early proofs, although evidence suggests that the theorem actually predates the existence of Pythagoras and that may just have popularized him. Anyone credited with developing the Pythagorean theorem would no doubt be pleased to know that it is taught in geometry classrooms around the world and is used daily for everything from high school math homework to performing complex engineering calculations for the Space Shuttle.
According to the Pythagorean theorem, if the lengths of the sides of a right triangle are squared, the sum of the squares will equal the length of the hypotenuse squared. This theorem is often expressed as a simple formula: a²+b²=c², with a and b representing the sides of the triangle, while c represents the hypotenuse. In a simple example of how this theorem might be used, one might ask how long it would take to cut a rectangular parcel of land, rather than skirting the edges, based on the principle that a rectangle can be divided into two simple right triangles. He or she might measure two adjacent sides, determine their squares, add the squares, and find the square root of the sum to determine the length of the diagonal of the lot.
Like other mathematical theorems, the Pythagorean theorem is based on proofs. Each proof is designed to create more supporting evidence to prove that the theorem is correct, by proving various applications, by showing the forms to which the Pythagorean theorem cannot be applied, and by attempting to disprove the theorem by showing, conversely, that the logic behind the theorem is sound. Because the Pythagorean theorem is one of the oldest mathematical theorems in use today, it is also one of the most widely proven, with hundreds of proofs by mathematicians throughout history adding to the body of evidence showing that the theorem holds.
Some special forms can be described with the Pythagorean theorem. A Pythagorean triple is a right triangle in which the lengths of the sides and the hypotenuse are all integers. The smallest Pythagorean triple is a triangle where a=3, b=4 and c=5. Using the Pythagorean theorem, people can see that 9+16=25. The squares in the theorem can also be literal; if each length of a right triangle were used as a side of a square, the squares of the sides would have the same area as the square created by the length of the hypotenuse.
One can use this theorem to find the length of any unknown segment in a right triangle, making the formula useful for people who want to find the distance between two points. If, for example, we know that one side of a right triangle equals three, and the hypotenuse equals five, we know that the other side is four long, based on the Pythagorean triad discussed above.
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