two regular hexagons ii solution

Solution to the Two Regular Hexagons (II) Puzzle

Two Regular Hexagons II

Both hexagons are regular. How long is the pink line?

Solution by Lengths in a Regular Hexagon and Similar Triangles

Two regular hexagons ii labelled

Consider the triangles ABCA B C and FHIF H I in the diagram. The length of ABA B is half the length of FHF H. The length of IHI H is the difference of the sides. Using lengths in a regular hexagon, ADA D and CEC E have lengths half the side lengths of their respective hexagons, so the length of CBC B is half the difference of these sides and so has half the length of IHI H. Since both ABCA B C and FHIF H I are right-angled triangles, this establishes them as similar with a scale factor of 22. Therefore the length of FIF I is twice that of ACA C and so is of length 1212.

Solution by Invariance Principle

Two regular hexagons ii same

By drawing the hexagons the same size, the relationship between the two lengths is clear and so the required length is 1212.