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?

Days and Dates

Investigate how you can work out what day of the week your birthday will be on next year, and the year after...

Circled Corners

Stage: 4 Short Challenge Level:

The three angles of the triangle add to $180^{\circ}$, so the combined area of the three sectors of the circles that are inside the triangle add up to half a circle with area:

$\frac{1}{2} \times \pi \times 2^2 = \frac {4\pi}{2}= 2\pi$.

So the grey area is $(80- 2\pi) cm^2$.

This problem is taken from the UKMT Mathematical Challenges.
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