### 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?

# Inscribed Hexagon

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

As the sum of the angles in a triangle is 180° and all four angles in a rectangle are 90°, then sum of the two marked angles in the triangle 180° - 90° = 90°.
Each interior angle of a regular hexagon is 120° and the sum of the angles in a quadrilateral is 360°; hence the sum of the two marked angles in the quadrilateral is 360° - 90° - (360° - 120°) = 30°.
Hence the sum of the four marked angles is 90° + 30° = 120°.

This problem is taken from the UKMT Mathematical Challenges.
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