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

Hint: For solving this question you should know about the Pythagorean theorem for a right angle. In this problem we will describe the Pythagorean theorem and the Pythagorean triples of a right angle. We will define them and will describe the relation between them.

Complete step-by-step solution:
According to our question we have to describe the difference between the Pythagorean theorem and Pythagorean triples.
So, as we know that the Pythagorean theorem is a statement of fact about the sides of a right – angled triangle, and the Pythagorean triples are set of three exact values which are valid for the theorem.
If we see the statement of Pythagorean theorem is defined that there is a specific relation between the side of a right -angled triangle i.e., \[{{a}^{2}}={{b}^{2}}+{{c}^{2}}\] where a, b and c are as:

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 on them the hypotenuse will be:
\[\begin{align}
  & {{c}^{2}}={{6}^{2}}+{{9}^{2}}=36+81=117 \\
 & c=\sqrt{117}=10.816653...... \\
\end{align}\]
This theorem always works but the answer can be rational or irrational.
In the same triangles, the sides work out to be exact answers. For example, if the shorter sides are 3 and 4 can the hypotenuse is:
\[\begin{align}
  & {{c}^{2}}={{3}^{2}}+{{4}^{2}}=9+16=25 \\
 & c=\sqrt{25}=5 \\
\end{align}\]
The ratio of 3 : 4 : 5 is known as a Pythagorean triple meaning a set of three values which works for Pythagorean’s theorem.
Some of common triples are:
3 : 4 : 5
6 : 8 : 10
8 : 15 : 17
9 : 12 :15
So, these were the differences between Pythagorean triple and theorem.

Note: The Pythagorean triple and Pythagorean theorem are related to each other. If the Pythagorean triple is given then it must satisfy the Pythagorean theorem and if any Pythagorean theorem is given then it must be made up with Pythagorean triple.