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This problem presents a geometrical situation where it appears there is not enough information to solve the problem. However, by persevering and making use of the hints given in the task, students will be surprised to discover that an elegant solution can be found.
"Can you find expressions which include P and Q based on what we know about areas and lengths? Does this help us to solve the problem?"
Finally, encourage students to write up their solution formally, making sure that they justify each step of the method clearly.
What do we know?
What can we deduce?
If $O$ lies on $BD$, and $BOC$ has twice the area of $DOC$, what can we say about the lengths $BO$ and $OD$?
What does this tell us about areas $BOA$ and $AOD$?
Students could start by exploring Triangle in a Triangle.
Another Triangle in a Triangle provides a suitable follow-up challenge that uses similar ideas.
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?
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.
A circular plate rolls inside a rectangular tray making five circuits and rotating about its centre seven times. Find the dimensions of the tray.