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

245. Square's Diagonals · equal lengths, perfect right angles

In a square, diagonals are equal in length and intersect at 90°.

OBCDAC = 401AC = 401BD = 401BD = 401∠AOB = 90°∠AOB = 90°A
A square is both a rectangle and a rhombus, so it inherits both properties: its diagonals are equal in length and they bisect each other at 90° (and bisect the corner angles).

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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 around the circle. The square ABCD rotates with it, but its two diagonals always (1) come out equal in length, (2) bisect each other at O, and (3) meet at exactly 90°. Try to make any of those three properties fail — impossible. The square packs all of them in at once.

How you build it

Construct a square with diagonals.

  • 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 B to C.
  • Segment tool: join C to D — the square ABCD is closed.
  • Segment tool: join A to C — diagonal AC.
  • Segment tool: join B to D — diagonal BD. It equals AC and crosses it at 90°.

The proof, step by step

Prove that the diagonals of a square are equal and intersect at right angles.

  • In ΔDAB and ΔCBA: AD = BC (sides of square), ∠DAB = ∠CBA (each 90°), AB = AB (common). ∴ ΔDAB ≅ ΔCBA [SAS].
  • ∴ AC = BD [By CPCTC].
  • In ΔAOB and ΔBOC: AB = BC, BO = BO, AO = CO. ∴ ΔAOB ≅ ΔBOC [SSS].
  • ∴ ∠AOB = ∠BOC. Since ∠AOB + ∠BOC = 180° (linear pair), ∠AOB = ∠BOC = 90°.
  • Hence, diagonals of a square are equal and meet at right angles.

Worked example

In square ABCD, diagonals AC and BD intersect at O. If AC = 12 cm, find the length of BD and the measure of ∠AOB.

Diagonals of a square are always equal, so BD = AC = 12 cm. They also intersect at right angles, making ∠AOB = 90°.

  • BD = 10 cm, ∠AOB = 45°
  • BD = 12 cm, ∠AOB = 90° — correct
  • BD = 12 cm, ∠AOB = 60°
  • BD = 14 cm, ∠AOB = 90°
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