104. The Fold Test · Finding the line that makes both halves match

Try to break it

Oops! Move the line away from the center. Now the two sides are different sizes. They won't match when you fold. A line of symmetry must split the figure into two exactly equal halves.

How you build it

Construct a rectangle and draw its line of symmetry.

• Mark point A — one end of the base.
• Mark point B — the other end of the base.
• Draw the base segment AB.
• Construct the perpendicular to AB at A.
• Construct the perpendicular to AB at B.
• With centre A, draw an arc crossing the perpendicular at A — this sets the rectangle height.
• Mark point D where the arc crosses the perpendicular at A.
• With centre B, draw an arc of the same radius, crossing the perpendicular at B.
• Mark point C where the arc crosses the perpendicular at B.
• Draw segment DC to close the rectangle.
• Construct the perpendicular bisector of AB — the line of symmetry.

The proof, step by step

Prove that a line of symmetry divides the figure into two identical halves.

• A line of symmetry divides a figure into two congruent parts.
• When folded along the line of symmetry, the two parts coincide exactly.
• Every point on one side has a matching point at an equal distance on the other side.
• The line of symmetry is the locus of points equidistant from corresponding vertices.

Worked example

In the given figure, PQRS is a rectangle. Which of the following lines is a line of symmetry for PQRS?

A line of symmetry must divide the rectangle into two identical, coinciding halves. Only a line passing through the midpoints of opposite sides achieves this. Diagonals and sides do not create congruent halves when folded.

• A line passing through P and Q
• A line passing through P and R
• A line passing through the midpoints of opposite sides — correct
• A line passing through P and the midpoint of QR