Triangle Dissections

 We all know that you can cut (or ‘dissect’) a square into 4 equal, but smaller, squares.  These are ‘self-similar’ to the original square - the same shape, but a different size.

You can do a similar thing to create a diagram with 9 smaller squares. 

Can you do the same thing, dissecting any triangle into 4 smaller self-similar triangles, using a similar approach ?

 Problem Set 1

 Starting with the rather special triangles shown below - the 30°, 60°, 90° triangle that is half an equilateral triangle, and the ‘half-domino’ triangle - there are 2 nice challenges along similar lines ... :


A         Can you show how to dissect the first into just 3 smaller self-similar triangles ?

B         Can you show how to dissect the second into 5 smaller self-similar triangles ?

 Problem Set 2 

Starting with an equilateral triangle, can you show how to dissect it into :

C         12 identical triangles (not necessarily equilateral)

D         13 equilateral triangles (not necessarily all the same size) 

E         14 triangles of the same area (but not necessarily the same shape)           

F          15 equilateral triangles (not necessarily all the same size)

G         16 equilateral triangles.

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Last modified: June 18, 2007