Rectangles in a Regular polygon

In very simple words:

If we divide 12 points of the 12 sided regular polygon into two parts, each will contain 6 points, and we know that a rectangle can be formed by 4 points, which can be selected in 6C4 ways. Thus 15 is the answer.

The diagonals of the rectangle must pass through center of the polygon (circle , since the polygon is definitely cyclic) , we have 6 pairs of points forming diameter , selecting any two out of them will ensure the formation of a rectangle with them as diameter, therefore our answer is $\displaystyle{6\choose{2}}$ $=15$

Note that any diagonal that is not a diameter corresponds to 1 rectangle. However, for each rectangle, there are 4 diagonals; thus, the number of rectangles is the number of non-diameter diagonals divided by 4.

Each of the 12 points connects to 10 non-diameter diagonals. However, each diagonal connects to 2 points; thus, the number of non-diameter diagonals is 12*10/2=60

Thus, the number of rectangles must be 60/4=15.

The number of diagonals of the polygon passing though its center =6 and hence number of rectangles =6C2=15