Similar Shapes and Scale

Congruence, back in the triangle track, asked when two shapes are identical. This track asks the looser question: when is one shape an exact enlargement of another? The answer is strict — matching angles equal and every pair of matching sides in the same ratio — and two rectangles can look thoroughly alike and fail it, which is why a photograph never quite fits a phone screen. The ratio the test produces is the scale factor, one number describing a whole enlargement, and it can perfectly well come out below 1, since every map and model is a reduction. On triangles that single number turns one known pair of sides into every other length, which is how a tree gets measured from its shadow. The track closes on the fact almost everyone gets wrong first: lengths are multiplied by the scale factor, but areas are multiplied by its square, so doubling a shape gives four times as much of it.