105. Folding the Square · Discovering 4 lines of symmetry

Try to break it

Drag point P along the vertical line. No matter where you place it, the square remains symmetric with its 4 lines of symmetry. That's the theorem!

How you build it

Draw the four lines of symmetry of a square.

• Place point A, the first corner of the square.
• Place point B, the second corner of the square.
• Place point C, the third corner of the square.
• Place point D, the fourth corner of the square.
• Draw the side from A to B.
• Draw the side from B to C.
• Draw the side from C to D.
• Draw the side from D to A to close the square.
• Draw diagonal AC.
• Draw diagonal BD.
• Construct the perpendicular bisector through the midpoints of AB and CD.
• Construct the perpendicular bisector through the midpoints of BC and DA.
• Mark the centre O where all lines of symmetry intersect.

The proof, step by step

Prove that each line of symmetry splits the square into two mirror-image halves.

• A line of symmetry divides a figure into two congruent parts that are mirror images of each other.
• The diagonals of a square bisect each other at 90° and divide the square into two congruent triangles.
• The perpendicular bisectors of opposite sides pass through the center and divide the square into two congruent rectangles.
• Since there are 2 diagonals and 2 perpendicular bisectors, a square has exactly 4 lines of symmetry.

Worked example

A regular hexagon has six equal sides and six equal angles. How many lines of symmetry does it have?

A regular polygon with n sides has exactly n lines of symmetry. Since a regular hexagon has n = 6, it has 6 lines of symmetry.

• 3
• 4
• 6 — correct
• 12