Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

Rhombus in Rectangle

Stage: 4 Challenge Level:

Take any rectangle $ABCD$ such that $AB > BC$. The point $P$ is on $AB$ and $Q$ is on $CD$. Show that there is exactly one position of $P$ and $Q$ such that $APCQ$ is a rhombus.

Show that if the rectangle has the proportions of A4 paper ($AB=BC$ $\sqrt 2$) then the ratio of the areas of the rhombus and the rectangle is $3:4$. Show also that, by choosing a suitable rectangle, the ratio of the area of the rhombus to the area of the rectangle can take any value strictly between $\frac{1}{2}$ and $1$.