### Consecutive Numbers

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

### 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.

The diagram below shows isosceles triangles $T$ and $U$. The perpendicular from the top vertex to the base divides an isosceles triangle into two congruent right-angled triangles as shown in both $T$ and $U$. Evidently, by Pythagoras' Theorem, $h = 4$ and $k = 3$. So both triangles $T$ and $U$ consist of two $3$, $4$, $5$ triangles and therefore have equal areas.