9. The diagram shows a rectangle $A E F J$ inside a regular decagon $A B C D E F G H I J$.
What is the ratio of the area of the rectangle to the area of the decagon?
A $2: 5$
B $1: 4$
C $3: 5$
D 3:10
E $3: 20$
Solution
A

Let $O$ be the centre of the regular decagon. The decagon is divided into ten congruent isosceles triangles by the radii joining $O$ to each of the vertices of the decagon. The triangles $A O J$ and $E O F$ are two of these ten congruent triangles.

Because $O$ is the centre of the rectangle $A E F J$ and the diagonals of a rectangle split its area into four equal areas, the triangles $E O A$ and $F O J$ each have the same area as triangles $A O J$ and $E O F$.

Therefore the area of $A E F J$ is equal to the area of 4 of the 10 congruent triangles that make up the decagon.

Hence the ratio of the area of $A E F J$ to the area of the decagon $A B C D E F G H I$ is $4: 10$ which simplifies to $2: 5$.