5. TRIANGLES – PYTHAGORAS

The standard diagram of Pythagoras’s theorem shows a right-angled triangle with squares attached to each edge. If the side lengths of the right-angles triangle are a, b and c as shown, then according to Pythagoras, a² + b² = c². It is less well known that the theorem can be illustrated with other similar shapes attached to the sides, for example, triangles. This is because the areas of similar figures are proportional to the squares of the side-lengths.

In 1922, this suggested to US professor Harry Bradley the following problem.

Problem T4. Give a 5-piece triangle dissection to illustrate a² + b² = c².

This will have the effect of dissecting a large triangle of side length c into two smaller triangles of side lengths a and b. We demonstrate Bradley’s dissection. Suppose that a b < c.

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Let’s take the c-triangle (at right) and superimpose the green a-triangle. Below the green triangle lies an isosceles trapezium with area b2 and base angles 60°, which is of course a special type of quadrilateral. We observe also that an isosceles triangle with area b2 and base angles 60° is a member of this same class of isosceles trapezia – it is degenerate, with one of the parallel sides of zero length. Hence we can apply the Q-slide to the trapezium to obtain a 4-piece dissection of this b-triangle. Together with the top a-triangle, this gives a 5-piece dissection of the c triangle into an a-triangle and a b-triangle. If we actually want to see the Q-slide at work on a trapezium like this, and the resulting dissection, check out the figure at left.