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.

Triangle in a Triangle

Stage: 4 Challenge Level:

Draw a right-angled triangle $ABC$, and mark a point $\frac13$ of the way along $AB$, $\frac13$ of the way along $BC$ and $\frac13$ of the way along $CA$.

Join your three points together to form a new triangle.
Can you work out the fraction of the original triangle that is covered by your new triangle?

You may wish to explore using the interactive diagram below.

Try a few examples. What do you notice?
Can you explain why?

Perhaps it might help to add some extra lines... click below to see.

We started with a right-angled triangle... what about other triangles?

What fraction of the triangle is shaded purple?
Can you prove it?

You may wish to try Areas and Ratios and Another Triangle in a Triangle next.