Rough diagrams

You are asked to construct a rectangle that can be divided into 3 identical squares. That seems hard, so the book first tries an easier one: a rectangle that divides into 2 identical squares. And before constructing, it plans, by drawing a rough diagram of the finished figure.

In the rough diagram, the rectangle is ACDF. The top side runs A, B, C and the bottom side runs F, E, D, in this order, and a line BE splits it into two squares, ABEF and BCDE. What can we tell from it?

• The squares are identical, so AB = BC and FE = ED.
• In each square all the sides are equal: AF = AB = BE = FE, and BE = BC = CD = ED.

So all the shorter lines are equal! To show equal sides on a diagram, we put the same small mark '|' on each of them. If another set of sides is equal to each other but not to the first, give them two marks, '||'.

Now the construction is easy to plan. The long side AC is made of two short sides, AB and BC. If we choose AF = 4 cm, then AC = 4 + 4 = 8 cm: the long side is twice the short side. For a rectangle that splits into 3 identical squares, the long side is made of three short sides, so it is three times the short side: with a short side of 4 cm, it is 12 cm.

You do not even need a ruler to copy a length. Draw AF, then a line through A perpendicular to AF. Open the compass so its point is on A and its pencil on F. Keep that opening, put the point on A and draw an arc across the perpendicular: that is B, with AB = AF. Put the point on B and mark again: that is C. This is called transferring a length with a compass.