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Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

Napoleon's Hat

Three equilateral triangles ABC, AYX and XZB are drawn with the point X a moveable point on AB. The points P, Q and R are the centres of the three triangles. What can you say about triangle PQR?

Plane to See

P is the midpoint of an edge of a cube and Q divides another edge in the ratio 1 to 4. Find the ratio of the volumes of the two pieces of the cube cut by a plane through PQ and a vertex.

Rarity

Age 16 to 18 Challenge Level:

What we were looking for in the problem Euclid's Algorithm and Musical Intervals was, if you like, a 'ratio of ratios' but we were not able to find that exactly. In that problem you are asked to find rational approximations for the 'ratio of ratios' using Euclid's algorithm. If the process terminates then you will have found an exact 'ratio of ratios' but generally the process does not terminate.

Here you are asked to prove that 'ratios of ratios' in this sense are (nearly) always irrational.