### 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.

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

# Wood Pile Perimeter

##### Stage: 3 and 4 Short Challenge Level:

The centres of the three circles form an equilateral triangle, so each of the sectors marked is $\frac{1}{6}$ of a circle. The shape below is a rectangle, so the sectors are $\frac 14$ of a circle. The curved portion of the perimeters is therefore the same as each of the circles: $2\pi \times 5 = 10\pi \text{cm}$.

The straight part is the same length as two radii, so is $10\text{cm}$ long.

The perimeter of the shaded shapes is therefore $10+10\pi\text{cm}$.

This problem is taken from the UKMT Mathematical Challenges.
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