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Explanation of the Pythagorean Theorem Formula

Omkar Phatak Oct 12, 2018
The formula of the Pythagorean theorem is one of the most basic relations in Euclidean two-dimensional geometry. Here you will find a simple explanation of the formula.
If you have just taken up a basic geometry course, one of the first relations in Euclidean geometry, that you will come across, is the Pythagorean theorem. It is a fundamental theorem related to right-angled triangles, that is of immense value to geometry and mathematics in general.
This theorem, which details an important property of right-angled triangles, is named after the Greek mathematician and philosopher, Pythagoras, who discovered it. However, there is evidence that this theorem was known to ancient Indian and even Babylonian mathematicians.
An ancient Indian text called the Apastamba Sulba Sutra, dating back to about 600 BC, has a numerical proof of the theorem. Many indirect evidences of knowledge of the theorem have been found in ancient Egyptian scrolls and Babylonian artifacts.

The Theorem Explained

A triangle is a closed three-sided, two-dimensional geometrical object. With three sides, a triangle has three angles and their sum in two-dimensional geometry is always 180 degrees. A right-angled triangle is a special type, that has one of its angles to be exactly 90 degrees.
Traditionally, the side that is exactly opposite to the right angle is called the hypotenuse. The Pythagoras theorem statement is:
For any right-angled triangle, with length of the sides being denoted by a, b, c (c being the length of the hypotenuse), the following relation always holds:

a+ b2 = c2
This is the Pythagorean theorem. In other words, the theorem can be restated as follows: For any right-angled triangle, the square of the hypotenuse is exactly equal to the sum of the squares of the other two sides. (By square of side length, we mean the length multiplied by itself).
There is one more way of simply stating the theorem: 'If you draw squares on all sides of a right-angled triangle (with the same length as the triangle sides), then the area of the square drawn on hypotenuse will be equal to the sum of the areas of squares drawn on the other two sides.' 
There are multiple proofs of this theorem, ranging from the very simple to the complex. You should know at least one proof.

Distance Formula

There is an important application of the formula, in calculating the distance of a point, from the origin of a frame of reference and the distance between two points. The perpendiculars drawn from a point, on the two Cartesian axes, along with the segment joining the point with origin, forms a right-angled triangle.
The Pythagorean distance formula is as follows:

d = √(x2 + y2)

The distance between two points with coordinates (x1, y1) and (x2, y2) is given by:

d = √((x2-x1)2 + (y2-y1)2)

These formulas are very useful in two dimensional (flat) geometry.

Triples

The sets of three numbers that can be the sides of a right-angled triangle and satisfy the theorem, are called Pythagorean triples. The largest among the three is always the length of the hypotenuse. Here are some triples:
  • 3, 4, 5
  • 5, 12, 13
  • 7, 24, 25
  • 8, 15, 17
  • 9, 40, 41
  • 11, 60, 61
The theorem is the first of many more geometrical gems that you will come across, if you continue your study of geometry and mathematics.