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A comprehensive approach to the pythagorean triples in mathematics

A comprehensive approach to the pythagorean triples in mathematics

Some visual proofs of Pythagoras' Theorem My favourite proof of the look-and-see variety is on the right. Both squares contain the same four identical right-angled triangles in white so it is white-angled with sides a, b, c.

The left square also has two blue squares with areas a2 and b2 whereas the right hand one replaces them with one red square of area c2. This does not depend on the lengths a, b, c; only that they are the sides of a right-angled triangle. So the two blue squares are equal in area to the red square, for any right-angled triangle: Don't turn them or flip them, just move them to their respective corners.

Click on the image on the right here to see an animation in a new window or to download the active controls version usable with the free Mathematica player. Bill Richardson has a nice animation of Bhaskara's proof The 3-4-5 Triangle In the example above, we chose two whole-number sides and found the longest side, which was not a whole number.

It is perhaps surprising that there are some right-angled triangles where all three sides are whole numbers called Pythagorean Triangles. The three whole number side-lengths are called a Pythagorean triple or triad.

Pythagorean Right-Angled Triangles

We can check it as follows: This triple was known to the Babylonians who lived in the area of present-day Iraq and Iran even as long as 5000 years ago! Perhaps they used it to make a right-angled triangle so they could make true right-angles when constructing buildings - we do not know for certain.

It is very easy to use this to get a right-angle using equally-spaced knots in a piece of rope, with the help of two friends. If you hold the ends of the rope together together and one friend holds the fourth knot and and the other the seventh knot and you all then tug to stretch the rope into a triangle, you will have the 3-4-5 triangle that has a true right-angle in it.

The rope can be as long as you like so you could lay out an accurate right-angle of any size. The sum of the sides of a triangle is called its perimeter. We can also easily draw a 3 4 5 triangle as follows: The is a 3 4 5 triangle But all Pythagorean triangles are even easier to draw on squared paper because all their sides are whole number lengths.

Measure the lengths of the two smaller sides those around the right-angle as lengths along and up from the same point and then join the two end-points together. So Pythagorean triangles also tell us which pairs of points with whole-number coordinates are a whole-number distance apart not in a horizontal or vertical direction. Test a Triangle - is it Pythagorean? Here is a little calculator that, given any two sides of a right-angled triangle will compute the third, or you can give it all three sides to check.

It will check that it is right-angled and, if so, if it is Pythagorean all the sides are integers. Here a and b are the two legs, the sides surrounding the right-angle, and h is the longest side, the hypotenuse: