In rhombus ABCD the diagonal AC turns out to have exactly the same length as the side AB. What is the ratio of the longer diagonal to the shorter diagonal?

All four sides of a rhombus are equal, so AB = BC = AC makes triangle ABC equilateral, and likewise triangle ACD. Every angle of those triangles is 60°, so the rhombus has angles of 60° at B and D and 120° at A and C, and AC is the SHORTER diagonal. The diagonals bisect each other at right angles, so half of BD is the height of an equilateral triangle of side s, namely s√3/2, making BD = s√3 while AC = s. The ratio is therefore √3 : 1. The trap 1 : √3 turns the ratio upside down by taking AC for the longer diagonal; 2 : 1 would need the rhombus to be flattened much further; and √2 : 1 is the diagonal-to-side ratio of a SQUARE, a different figure.