Chapter 3.7 — Exercise 3.6 — Special Quadrilaterals

Why Diagonals Alone Are Enough

In a rhombus, the two diagonals cross at their shared midpoint at exactly 90°. That means once one diagonal is drawn and its perpendicular bisector constructed, the second diagonal has to lie exactly along that perpendicular bisector — its position isn't a separate choice, it's forced by the first diagonal's placement. All four vertices then fall directly out of drawing one diagonal and marking half the second diagonal's length on either side of the perpendicular bisector.

The General Method for a Rhombus

• Draw the first diagonal as a plain line segment, using its full given length.
• Construct its perpendicular bisector, which crosses the first diagonal at its exact midpoint.
• Mark two points on the perpendicular bisector, one on each side of the midpoint, each at a distance equal to half the second diagonal's length.
• Join each end of the first diagonal to both of the newly marked points, completing all four sides of the rhombus.

Worked Example — Rhombus CART

Construct rhombus CART with diagonals CR = 6 cm and AT = 4.8 cm.

Since the diagonals bisect each other, half of AT is 4.8 ÷ 2 = 2.4 cm — this is the radius used to mark A and T on either side of the midpoint.

1. Draw line segment CR of length 6 cm.
2. Construct the perpendicular bisector XY of CR, meeting it at point O.
3. With O as centre, draw two arcs of radius 2.4 cm along XY, on either side of CR, marking A and T.
4. Join C–A, A–R, R–T and T–C to complete the required rhombus.

A Second Rhombus — Confirming the Pattern

Rhombus SOAP, with diagonals SA = 4.3 cm and OP = 5 cm, follows the identical steps: draw SA in full, construct its perpendicular bisector meeting SA at X, then mark O and P on either side of X at a radius of half of OP — 5 ÷ 2 = 2.5 cm. Joining S–O, O–A, A–P and S–P completes the rhombus. The only numbers that change between CART and SOAP are the two diagonal lengths themselves; the sequence of steps is otherwise word-for-word the same.

A Square Needs Only One Diagonal Length

A square is a special rhombus whose diagonals are not just perpendicular bisectors of each other but also equal in length. That extra fact means a square can be constructed from a single diagonal measurement — the second diagonal is automatically the same length, so nothing else needs to be given.

Square JUMP with diagonal JM = 4.2 cm: since both diagonals of a square are equal, UP = 4.2 cm as well, so half of either diagonal is 4.2 ÷ 2 = 2.1 cm.

1. Draw line segment JM of length 4.2 cm.
2. Construct the perpendicular bisector LN of JM, meeting it at point X.
3. With X as centre, draw two arcs of radius 2.1 cm along LN, on either side of JM, marking U and P.
4. Join J–U, U–M, M–P and J–P to complete the required square.

Confirming the Four Sides Really Are Equal

It's worth checking, at least once, that this method genuinely produces a rhombus and not just a shape that looks like one. Every one of the four triangles formed by the two diagonals and the centre point — like triangle COA in rhombus CART — has the same two leg lengths (half of each diagonal) and the same 90° angle between them, since the perpendicular bisector guarantees that angle at every one of the four points around the centre. Four triangles with matching legs and a matching included angle are congruent to each other, and congruent triangles have equal third sides — which is exactly why CA, AR, RT and TC all come out equal, confirming the shape really is a rhombus rather than an approximate one.

Why This Method Doesn't Work for a General Parallelogram

This diagonals-only approach relies entirely on the perpendicular-bisecting property being true — and that property only holds for a rhombus and a square, not for a general parallelogram or rectangle, whose diagonals bisect each other but don't cross at right angles. Trying to apply this method to a rectangle, for instance, would need the angle between the diagonals stated separately, since it isn't automatically 90°.

The Fewest Measurements of Any Method in This Chapter

Every other construction in this chapter needs five separate measurements. A rhombus built this way needs only two — its two diagonal lengths — and a square needs just one, since a square's second diagonal is never independent of the first. This makes diagonals-only construction the most efficient method covered, but only for shapes that guarantee the right diagonal relationship in the first place; the trade-off for needing fewer measurements is that the method only applies to this one narrow family of quadrilaterals, unlike SSSSA or SASAS, which work for any convex shape at all. Recognising that trade-off — fewer measurements in exchange for a narrower range of shapes it applies to — is a useful lens for comparing every method covered across this chapter, not just this last one. A method that needs fewer measurements is always relying on some extra property of the shape to make up the difference — here, that property is the perpendicular-bisecting diagonals themselves — a shortcut that only exists because a rhombus and a square are already special cases of every more general quadrilateral covered earlier in this chapter, inheriting extra structure that a plain parallelogram or trapezium simply doesn't have, which is exactly why this method is the last one covered rather than the first — it only makes sense once the more general methods, and the properties they rely on, are already familiar.

Missteps With Diagonals and Bisectors

• Marking the full diagonal length instead of half. Since the diagonals bisect each other, the arc radius used from the midpoint is always half the second diagonal's length, not the whole thing.
• Constructing an ordinary bisector instead of a perpendicular one. The 90° angle between the diagonals is essential — a bisector that isn't perpendicular produces a shape that isn't actually a rhombus.
• Assuming a square needs two diagonal measurements. Only one is ever needed, since a square's diagonals are always equal to each other by definition.

Five Methods, One Complete Toolkit

This chapter's five construction methods — SSSSA, SSSSD, SSSDD, SASAA and SASAS, plus this diagonal-only approach for rhombuses and squares — together cover every practical way a convex quadrilateral's measurements can be given. The properties they all depend on are explored in far more depth, with formal proofs rather than construction steps, in Quadrilaterals in Class 9. For the side-and-angle methods that precede this one, revisit Exercise 3.5.