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...

Unusual Polygon

Stage: 4 Short Challenge Level:

The area of square BDFG is $6\times 6 = 36$ square units.
So the total area of the three triangles ABG, BCD and DEF is also $36$ square units.
These three triangles are congruent and so each has an area of $12$ square units.

The area of each triangle is $\frac{1}{2}\times base \times \ height$ and the base is $6$ units and hence we have $\frac{1}{2}\times 6 \times \ height = 12$,
so the height is $4$ units.

Let $X$ be the midpoint of BD. Then CX is perpendicular to the base BD (since BCD is an isosceles triangle).

By Pythagoras' Theorem, $BC = \sqrt{3^2+4^2} = 5$ units.

Therefore the perimeter of ABCDEFG is $6\times 5+ 6 = 36$ units.

This problem is taken from the UKMT Mathematical Challenges.
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