Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

241. The Quadrilateral Chameleon · Morphing between Rectangle, Rhombus, and Square

Dragging A and B preserves the parallelogram property (diagonals always bisect each other).

OCDRHOMBUS∠AOB = 90°∠AOB = 90°AC = 400AC = 400BD = 300BD = 300AB = 250AB = 250AB
The rectangle, rhombus, and square are all special parallelograms. A rectangle has every angle 90°, a rhombus has all sides equal, and a square has both. Through every change the parallelogram property holds: the diagonals bisect each other.

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Selina ICSE: Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]

What this lesson covers

Try to break it

Drag A and B to change the two diagonals (their lengths and the angle between them). Make them equal length to get a rectangle. Make them perpendicular to get a rhombus. Make them equal AND perpendicular to get a square. The chameleon flips between parallelogram, rectangle, rhombus, and square based purely on the diagonals.

How you build it

Construct a square.

  • Point tool: mark point A — a corner of the square.
  • Point tool: mark point B — AB is the base.
  • Segment tool: join A to B.
  • Perp tool: click A (on AB), then click upward — side AD lies on this perpendicular.
  • Arc tool: centre A, open out to B (radius AB) — the arc cuts the perpendicular at D.
  • Point tool: mark D where the arc meets the perpendicular. AD = AB and AD ⊥ AB.
  • Arc tool: keep the opening and click B — an arc of radius AB from B.
  • Arc tool: keep the opening and click D — an arc of radius AB from D.
  • Point tool: mark C where the two arcs cross (the corner away from A). BC = DC = AB.
  • Segment tool: join A to D.
  • Segment tool: join B to C.
  • Segment tool: join C to D — ABCD is a square.

The proof, step by step

Prove that the rectangle, rhombus, and square are all special parallelograms.

  • In a square, diagonals are equal.
  • In a square, diagonals bisect each other at 90°.
  • In a square, all sides are equal.
  • In a square, each interior angle is 90°.

Worked example

In a square ABCD, the diagonals intersect at O. If AC = 12 cm, find the length of OB.

Diagonals of a square bisect each other. So OB = AC/2 = 12/2 = 6 cm.

  • 3 cm
  • 6 cm — correct
  • 4√2 cm
  • 12 cm
Hold to talk

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