Matter of Scale

Can you prove Pythagoras' Theorem using enlargements and scale factors?
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Problem



Take any right-angled triangle with side lengths $a, b$ and $c$.  For convenience, label the two acute angles $x^{\circ}$ and $y^{\circ}$.

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Matter of Scale

 

 

 

 

 

 

 



Make two enlargements of the triangle, one by scale factor $a$ and and one by scale factor $b$:

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Matter of Scale

 

 
 
 
 
 
 
 
 
 
 
What are the lengths of these new triangles? 

Draw out a copy of them and indicate what the lengths and the angles are in each.
 
 
We can put these two triangles together to make a larger triangle.
 
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Matter of Scale

 

 

 

 

 

 

 

Find the lengths and angles in this last triangle.

Can you show that this triangle is similar to the original triangle?

What is the scale factor of enlargement between the first and last triangles?

Can you use your results to prove Pythagoras' Theorem?

 

You might like to explore some more proofs of Pythagoras' Theorem, and a proof of The Converse of Pythagoras' Theorem.

 

We are very grateful to the Heilbronn Institute for Mathematical Research for their generous support for the development of this resource.