4. TRIANGLES

We now use the word triangle to signify ‘equilateral triangle’. We also use x-triangle to denote a triangle of side length x.

Many dissections are based on ‘natural’ partitions of the triangle. Here are two examples.

Problem T1. Dissect the adjacent triangle into 3 equal smaller triangles.

Try solving the problem yourself. You can give yourself partial clues by clicking successively the numbers on the right.

Problem T2. Dissect a 2-triangle into a 1-triangle and a 3-triangle.
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Again, try solving the problem yourself. A stepped solution is given at left. We notice that if we draw an altitude to this triangle we obtain a triangle with sides in the ratio 2 : 1 : 3 .

There is another way of stating dissection problems.

Problem T3. Give a 4-piece triangle dissection for 12 + 22 + 22 = 32.

Here we seek to dissect a 3-triangle into two 2-triangles and a 1-triangle. The squares in the equation are proportional to the areas. The solution is quite trivial.

It is also posible to dissect a triangle into smaller triangles of all different sizes. We look at a special case of this next.
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