A right circular cone is filled with liquid to a depth of half its
vertical height. The cone is inverted. How high up the vertical
height of the cone will the liquid rise?

Using a ruler, pencil and compasses only, it is possible to
construct a square inside any triangle so that all four vertices
touch the sides of the triangle.

Similar Rectangles

Age 14 to 16 Challenge Level

The smaller of two similar rectangles has height $2$ units; the larger rectangle has length $6$ units.

If one rectangle has twice the area of the other, find the length of the smaller rectangle.

This problem is taken from Tony Gardiner's 'Extension Mathematics Gamma' book.