Constructing a square or rectangle

You construct a square PQRS with sides of 6 cm. The book does it in six steps.

• Step 1: draw the side PQ = 6 cm.
• Step 2: draw a line through P perpendicular to PQ (at 90° to it).
• Step 3: on that line, mark S so that PS = 6 cm. You can measure it with a ruler, or set the compass to 6 cm and mark it with an arc.

• Step 4: draw a line through Q perpendicular to PQ.
• Step 5: on it, mark R so that QR = 6 cm. If the compass is still open at 6 cm, this is easy.
• Step 6: join R and S.

Now check: RS is 6 cm, and the angles at R and S are 90°. All the sides are equal and all the angles are 90°, so PQRS is a square.

For the perpendiculars you can use a set square: lay its right-angled corner on the point, with one edge along PQ, and draw along the other edge. It is quicker than a protractor.

A rectangle is built the same way, but with two different lengths. For a rectangle with sides 6 cm and 4 cm: draw PQ = 6 cm, draw perpendiculars at P and at Q, mark 4 cm up each of them, and join the last two corners.

The book asks: can you make a four-sided figure in which all the angles are 90° but the opposite sides are not equal? Try it, and you will find you cannot. When all four angles are 90°, the opposite sides always come out equal. That is why, in the steps above, joining the last two corners gives a side exactly as long as the first.