Count On


Pythagorean Triangles

Pythagoras was a Greek mathematician who lived around 500 BC, but is probably most famous for Pythagoras' Theorem which says that, in any triangle which has a right angle in it, the squares of the two shorter sides always add up to the square of the longest side (the hypotenuse).

Most triangles will have sides with fractional lengths but some triangles have sides which are all integers. In this case we have an equation with two square numbers on one side totalling a single square number on the other side, for instance:

32 + 42 = 9 + 16 = 25 = 52

Such triangles are special and are called Pythagorean Triangles although we are really only referring to the lengths of their sides, which are whole numbers. In the example above, we have sides of lengths 3, 4 and 5.

Are there many other Pythagorean Triangles? Yes, lots! Here is a simple way to generate some of them:

Write down any 2 (whole) numbers, such as 2 and 5. Add them and write the sum down after them. Repeat with the latest two numbers of the three written down - add the latest two up and put their sum as a fourth number in the row. Now we have:

2 5 7 12

To find a Pythagorean Triangle:

Multiply the two middle numbers ... 5 times 7 is 35 ... and double it to get one of the sides in the Pythagorean Triangle: 35 doubled is 70.

Multiply together the two outer numbers in the row. This is the second side of the Pythagorean triangle: 2 times 12 gives 24.

Multiply the second number by itself, multiply the third number by itself and add these two values to get the third (and longest) side of the Pythagorean Triangle:

5 x 5 = 25, 7 x 7 = 49, 25+49 = 74

The square of the first side PLUS the square of the second side EQUALS the square of the third side. Check:

702 + 242 = 4900 + 576 = 5476 = 742 - it works!

More facts about the numbers in every Pythagorean Triangle:

The product of the smaller two sides is always a multiple of 12;
and so, the area of the triangle is a multiple of 6;
The product of all three sides is always a multiple of 60;

How to do Squaring without (much) multiplying!

Suppose we know that 602 is 3600. Can we use this to find 612? Yes: just add on twice times the number squared (60) plus 1 :

612 = 3600 + 2*60 + 1 = 3600 + 120 + 1 = 3721

Also, we can compute 592 just as easily, this time subtracting the double of the original number, but still adding 1:

592 = 3600 - 2*60 + 1 = 3600 - 120 + 1 = 3481

Can you...
write down 112 without using a calculator?
write down 1012 without using a calculator?
write down 992 without using a calculator?
write down 212 without using a calculator?
write down 192 without using a calculator?