### 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.

### Ladder and Cube

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?

# Circle in a Semicircle

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

Ans: ½

Let the radius of the circle be r. This implies that the radius of the semicircle is 2r. The area of the semi circle is $1/2 \times \pi \times(2r)^2$, which is twice the area of the small circle.

This problem is taken from the UKMT Mathematical Challenges.
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