From a side and a diagonal

You have built rectangles from their two sides. But what if you are given one side and a diagonal? The book constructs a rectangle ABCD with side DC = 5 cm and diagonal DB = 7 cm, starting with a rough diagram to plan the order.

• Step 1: draw the base DC = 5 cm.
• Step 2: every angle of a rectangle is 90°, so the side CB must be perpendicular to DC. Draw a line through C perpendicular to DC and call it l.

The corner B is somewhere on l. But where?

We also know that B is 7 cm from D, because DB is the diagonal. You could slide a ruler around until you find a point on l that is 7 cm from D, but that is trial and error. There is a better way: think of ALL the points that are 7 cm from D. They form a circle of radius 7 cm with centre D.

• Step 3: draw that circle with the compass. The point where it cuts the line l is both on l and 7 cm from D, so it is B.

In fact you only need the small part of the circle near l, so an arc is enough. Take care with the centre: it is D, the other end of the diagonal, not C, which is already on l.

• Step 4: now three corners, D, C and B, are known. Draw a line perpendicular to DC through D, and a line perpendicular to BC through B. They meet at the fourth corner, A. (You could also use equal sides: mark A so that DA = CB.)

Finally, check that ABCD has opposite sides equal and all angles 90° (R1 and R2).