You are currently browsing the tag archive for the ‘pythagoras’ tag.

Screen Shot 2022-05-15 at 10.05.39 AM

Proving Pythagoras Like Einstein?

There are many ways to prove Pythagoras’ theorem – Einstein reputedly used the sketch above to prove this using similar triangles.  To keep in the spirit of discovery I also just took this diagram as a starting point and tried to prove this myself, (though Einstein’s version turns out to be a bit more elegant)!

Step 1: Finding some links between triangles

We can see that our large right angled triangle has sides a,b,c with angles alpha and beta.  Hopefully it should also be clear that the two smaller right angled triangles will also have angles alpha and beta.  Therefore our triangles will all be similar.  It should also be clear that the area of the 2 small triangles will be the same as the area of the large triangle.

Step 2: Drawing a sketch to make things clearer:

Screen Shot 2022-05-15 at 10.07.40 AM

It always helps to clarify the situation with some diagrams.  So, let’s do that first.

Step 3:  Making some equations

As the area of the 2 small triangles will be the same as the area of the large triangle this gives the following equation:

Screen Shot 2022-05-15 at 10.29.13 AM

We also can make the following equation by considering that triangles 2 and 3 are similar

Screen Shot 2022-05-15 at 10.30.17 AM

We can now substitute our previous result for x into this new equation (remember our goal is to have an equation just in terms of a,b,c so we want to eliminate x and y from our equations).

Screen Shot 2022-05-15 at 10.33.43 AM

We can also make the following equation by considering that triangles 1 and 2 are similar:

Screen Shot 2022-05-15 at 10.35.30 AM

And as before, our goal is to remove everything except a,b,c from these equations, so let’s make the substitution for y using our previous result:

Screen Shot 2022-05-15 at 10.36.41 AM

And if by magic, Pythagoras’ theorem appears!  Remember that the original a,b,c  related to any right angled triangle with hypotenuse c, so we have proved that this equation must always be true for right angled triangles.

You can explore some other ways of proving Pythagoras here.  Which is the most elegant?

Website Stats



All content on this site has been written by Andrew Chambers (MSc. Mathematics, IB Mathematics Examiner).

New website for International teachers

I’ve just launched a brand new maths site for international schools – over 2000 pdf pages of resources to support IB teachers.  If you are an IB teacher this could save you 200+ hours of preparation time.

Explore here!

Free HL Paper 3 Questions

P3 investigation questions and fully typed mark scheme.  Packs for both Applications students and Analysis students.

Available to download here

IB Maths Super Exploration Guide

A Super Exploration Guide with 168 pages of essential advice from a current IB examiner to ensure you get great marks on your coursework.

Available to download here.

Recent Posts

Follow IB Maths Resources from Intermathematics on