What is the difference between the Pythagorean Theorem and Pythagorean Triples?

1 Answer
EZ as pi
Apr 1, 2018

The theorem is a statement of fact about the sides of a right-angled tri9angle, and the triples are set of three exact values which are valid for the theorem.

Explanation:

The theorem of Pythagoras is the statement that there is a specific relationship between the sides of a right-angled triangle.

ie: #a^2 = b^2 + c^2 #

In finding the length of a side, the last step involves finding a square root which is often an irrational number.

For example, if the shorter sides are #6 and 9# cm, then the hypotenuse will be:

#c^2 = 6^2 + 9^2 = 117#

#c = sqrt117 = 10.8166538.........#

This theorem ALWAYS works, but the answers can be rational or irrational.

In some triangles, the sides work out to be exact answers. For example if the shorter sides are #3 and 4# cm, then the hypotenuse is:
#c^2 = 3^2+4^2 = 25#
#c = sqrt25 = 5#

The ratio #3:4:5# is known as a Pythagorean Triple ... meaning a set of three values which works for Pythagoras' Theorem.

Some of the common triples are:

#3:4:5#
#5:12:13#
#7:24:25#
#8:15:17#
#9:40:41#
#11:60:61#

Notice that their multiples also work, so from #3:4:5# we can get:
#6:8:10#
#9:12:15#
#12:16:20#
#15:20:25# ... and so on.

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