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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.

Halving the Triangle

Age 16 to 18
Challenge Level

Imagine cutting off three triangles and finding the area of the remaining triangle, that is triangle ABC. If each side of triangle PQR is divided in the same ratio, say t:1-t, then can you find the areas of all 4 triangles in terms of t?