39. The Rhombus Rule · four equal sides, one special shape

Try to break it

Try dragging A or B as far as you can. No matter how you stretch it, the sides always stay equal. It's always a rhombus! That's the power of the definition.

How you build it

Construct a rhombus.

• Place point A — one end of a diagonal of the rhombus.
• Place point C, well away from A. AC is one diagonal of the rhombus.
• Set the compass to length AC: click A for the centre, then click exactly on C. An arc swings around A.
• Without changing the compass width, draw an arc centred at C — click once on C. It crosses the arc from A at two points.
• The two arcs cross at two points, one on each side of AC. Mark point B at one of those crossings.
• Mark point D at the other crossing of the two arcs — the one on the opposite side of AC from B. Now AB, CB, AD and CD all equal AC.
• Draw side AB (Segment tool: click A, then B).
• Draw side BC (Segment tool: click B, then C).
• Draw side CD (Segment tool: click C, then D).
• Draw side DA (Segment tool: click D, then A). All four sides equal AC — ABCD is a rhombus.

The proof, step by step

Prove that all four sides of the rhombus are equal.

• We are given a quadrilateral ABCD.
• All four sides are equal: AB = BC = CD = DA.
• By definition, a quadrilateral with all sides equal is a rhombus.
• Therefore, ABCD is a rhombus.

Worked example

A quadrilateral ABCD has AB = BC = CD = DA = 6 cm. Which of the following must be true about ABCD?

A quadrilateral with all four sides equal is defined as a rhombus. While a square is a special type of rhombus, the given information only guarantees it is a rhombus.

• It is a square.
• It is a rhombus. — correct
• It is a rectangle.
• It is a trapezium.