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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

Why do this problem?

The problem involves simple geometrical reasoning, calculating with ratio and proportion, and solving quadratic equations. It requires a decision to divide the sides of triangle PQR in the same ratio because otherwise there would be too many variables to work with. Discussion of the choices to be made in order to solve this problem provides good experience of how to set about mathematical modelling.

Key questions

How can we express the area of triangle ABC in terms of the area of triangle PQR so that only one variable is involved?

What can we deduce if we know that one are is a certain fraction of the other area?