Sunday Times Teaser 2713 – Very Similar Triangles

by Michael Fletcher

I have cut two mathematically similar triangles of different sizes from an A4 sheet of paper.  All the sides of these triangles are integer numbers of centimetres and two sides of the first are the same as two sides of the second.

How long are the sides of the smaller triangle?

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1. A manual solution of this teaser is quite easy. If we let the sides of the two triangles be $$(a,b,c)$$ and $$(b,c,d)$$, their similarity then requires that: $a/b=b/c=c/d$ So the four different side lengths then become: $(a, b, b^2/a, b^3/a^2)$
We are only interested in the ratios of these four side lengths, so we can now multiply these values by $$a^2$$ to give the four lengths as: $(a^3, a^2 b, a b^2, b^3)$ Since the longest side of an A4 sheet of paper is 29.7 cm, we can see that b is at most 3 and hence that $$(a, b)$$ can only be $$(1,2), (1,3)$$ or $$(2,3)$$.
Of these only $$(2,3)$$ gives valid triangles so the two triangles have the side lengths $$(8, 12, 18)$$ and $$(12,18,27)$$.