CONSTRUCTIONS (Using ruler and compasses only)

177. The Square's Blueprint · equal sides, perfect corners

All four sides remain equal and all four angles stay exactly 90°.

BCDAB = 220AB = 220BC = 220BC = 220CD = 220CD = 220AD = 220AD = 220∠DAB = 90°∠DAB = 90°∠ABC = 90°∠ABC = 90°∠BCD = 90°∠BCD = 90°∠CDA = 90°∠CDA = 90°A
To construct a square of side a: draw segment AB of length a. At A and B, construct perpendiculars. Mark length a on each perpendicular to find D and C. Connect ABCD — all sides equal a, all angles 90°.

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Selina ICSE: CONSTRUCTIONS (Using ruler and compasses only)

What this lesson covers

Try to break it

Drag A anywhere on the canvas. The square slides with it, but every angle stays at 90° and every side stays the same length. Try to drag A so one corner becomes a non-right angle, or one side becomes shorter than the rest — you can't. The square moves as a rigid block.

How you build it

Construct a square.

  • Draw segment AB — the base side of the square. A and B are already placed; just connect them.
  • With centre A and radius AB, draw an arc swinging below AB into the region where the square will be built.
  • With centre B and radius AB, draw an arc. It crosses the arc from step 2 at two points — P (above AB) and Q (below AB).
  • Draw segment PQ — the perpendicular bisector of AB. It crosses AB at its midpoint M. M is now marked on segment AB.
  • With centre M and radius MA, draw an arc. It meets segment PQ at point R — the centre of the square, halfway between AB and the bottom side.
  • With centre R and radius RA, draw an arc — the circumscribed circle of the square. It meets the arc from step 2 at D (below A) and the arc from step 3 at C (below B).
  • Draw segment AD — the left side. D is where the arc from step 6 meets the arc from step 2.
  • Draw segment DC — the bottom side. C is where the arc from step 6 meets the arc from step 3. Check that DC equals AB.
  • Draw segment BC — the right side. All four sides are now equal and all four angles are 90°. The square is complete!

The proof, step by step

Prove that the constructed figure is a square.

  • We start with side AB and construct a perpendicular at A, ensuring the first corner is exactly 90°.
  • We mark AD = AB on the perpendicular, guaranteeing two adjacent sides are equal and meet at a right angle.
  • Arcs from B and D with radius AB intersect at C, forcing BC = CD = AB and locking all angles at 90°.

Worked example

A square has a perimeter of 36 cm. What is the length of its diagonal? (Round to 1 decimal place)

Perimeter = 4s = 36 cm → s = 9 cm. Diagonal = s√2 = 9 × 1.414 ≈ 12.7 cm.

  • 12.7 cm — correct
  • 18.0 cm
  • 25.5 cm
  • 36.0 cm
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