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Areas and Ratios

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

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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?

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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.


Age 16 to 18 Challenge Level:

Why do this problem?

The problem calls for simple geometrical reasoning (nothing more complicated than similar triangles) and leads to an obvious generalisation. Finding the solution is the beginning of the next stage:

how can I explain my method?
is this the best method?
what if...can I generalise this result?

Key questions

What do you know about the tangent to a circle and its radius?

What do you know about the perpendicular from the centre of a circle to a chord?

Can you spot any similar triangles?

If two sides of a right-angled triangle are in the ratio 3:5 what can we say about the third side?