The diagram above shows a tessellation of a plane of equilateral triangular tiles. A right-angled triangle is of such a shape that it can cover exactly half of any tile. Its initial position is markedtin the diagram.

The operation of reflectingtin its shortest side is written A, and starting with the given positiontthe result is A(t), which is marked with the figure 1.

Reflection in the second side is denoted by B, so that B(t) is 2.

Reflection in the hypotenuse is C so that C(t) is 3.

Position 4 is B(3) = B(t), so that the operation carryingtto 4 is BC.

Questions...

(i) Show that operations AB and BA are the same

(ii) Describe the combined operation BC geometrically.

(iii)Simplify C(CB), (CC)B and BC(CB)...using I to denote the identity operation if necessary.

(iv) Obtain expressions, as combinations of A, B and C for operations which carry

(a) t to x

(b) t to y

(c) x to t

If you can answer some or all of this it is much appreciated.