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?