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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.

Gassy Information

Age 16 to 18 Challenge Level:

You will need to use the Ideal Gas law $PV = nRT$ and proportional reasoning to solve this problem