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The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

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Six circular discs are packed in different-shaped boxes so that the discs touch their neighbours and the sides of the box. Can you put the boxes in order according to the areas of their bases?

Equilateral Areas

ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

Rhombus in Rectangle

Age 14 to 16 Challenge Level:

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.