### Why do this problem?

This
problem allows students to gain an understanding of a
geometrical situation by using an algebraic representation. It can
be approached numerically at first and then generalised. The
problem can be used to practise expansion of brackets, changing the
subject of a formula, fractions and surds, as well as the
application of Pythagoras' theorem.

### Possible approach

Start with a particular rectangle, for example 6 units by 4
units. If I positioned$Q$ 1 unit from$D$, would it be a rhombus?
Using Pythagoras, show that $AQ$ and $QC$ would not be equal.By
calling the distance $QC$ $x$, students could try to write down an
equation which must be true for the shape to be a rhombus, that is
for $AQ=QC$. By rearranging to make $x$ the subject, students
should be able to justify the statement that there is only one
possible value of $x$.

It may be necessary to try other numerical examples before
generalising, but once a general form is found linking $x$ with the
base and height of the rectangle, the rest of the problem can be
tackled.

### Key questions

What must be true about the lengths $DQ$ and $PB$?

What must be true about the lengths $AQ$ and $QC$?

What happens to the length of $AQ$ as I move $Q$ further from
$D$?

### Possible extension

Consider what would happen if we constructed the shape on the
shorter sides of the rectangle.

### Possible support

Some learners may find it easier to tackle the problem using Trial
and Improvement.