The proof behind one of the most famous theorems in mathematics

Vishaali, Year 10, looks behind the proof of one of the most famous mathematical theorems – that of Pythagoras’ theorem.

 

What is the difference between a theorem and a theory?

A theorem is a mathematical statement that has been proven on the basis of previously established statements. For example, Pythagoras’ theorem uses previously established statements such as all the sides of a square are equal, or that all angles in a square are 90°. The proof of a theorem is often interpreted as justification of the statement that the theorem makes.

On the other hand, a theory is more of an abstract, generalised way of thinking and is not based on absolute facts. Examples of theories include the theory of relativity, theory of evolution and the quantum theory. Take the theory of evolution; this is about the process by which organisms change over time as a result of heritable behavioural or physical traits. This is based on undeniable true facts, but more from experience and from an abstract way of thinking.

It is also important not to confuse mathematical theorems with scientific laws as they are scientific statements based on repeated experiments or observations.

The proof behind Pythagoras’ theorem

You have probably all heard of Pythagoras’ theorem, one of the simplest theorems there is in mathematics, that is relatively easy to remember. Given that it’s so easy to remember and to learn, wouldn’t it be an added bonus to know exactly how this theorem came to be?

The theorem, a²+b²=c², relates the sides of any right-angled triangle enabling you to find the lengths of any side, given you have the lengths of the other two.

This whole theorem is based on a triangle like this:

These four right-angled triangles are exactly the same just rotated slightly differently to create this shape:

Two shapes have been made by putting these triangles in this order. A big square on the outside, and another slightly smaller square in the middle. As all these triangles are the exact same you can label them A, B and C.

You can tell from the labels the triangles have been given, that the bigger square would have the sides (a+b), and the smaller triangle in the middle will have sides of c. Therefore we know the area of the smaller square is c² :

Using the exact same four triangles, we can rotate and translate them to create a slightly different shape:

Now two more squares have been added to this shape. We can call them  a² and b².

Thinking back to the shape we made before, we can also see that the length of this shape is also (a+b). As we know we used the same four right-angled triangles for the shape before and now, we can infer that the two squares  a² and b² are exactly the same as the square from the first shape, c². Hence we get Pythagoras’ theorem, a²+b²=c²:


References:

https://www.livescience.com/474-controversy-evolution-works.html
https://www.askdifference.com/theorem-vs-theory/