54. Folding Triangles · Finding lines of symmetry in triangles

Try to break it

Try to break the symmetry — drag the apex C as far up and as far down as it will go. C only slides along the centre line, so AC and BC always stay equal and the two halves always match. This single centre line is the one line of symmetry of an isosceles triangle; a scalene triangle, with all three sides unequal, has none.

How you build it

Construct the perpendicular bisector of a base.

• Mark point A — one end of the base.
• Mark point B — the other end of the base.
• Draw the base AB.
• Construct the perpendicular bisector of AB.
• Place the apex C anywhere on the perpendicular bisector — AC and BC come out equal, so ABC is isosceles.
• Draw the side BC — join B and C.
• Draw the side CA — join C and A. Triangle ABC is now complete, with BC = CA.

The proof, step by step

Prove that the line through C and the midpoint of AB is a line of symmetry.

• C lies on the perpendicular bisector of AB.
• Therefore, CA = CB (any point on the bisector is equidistant from endpoints).
• Folding along the bisector maps A onto B, proving it is a line of symmetry.

Worked example

In an isosceles triangle, the line of symmetry is the bisector of the angle between the two equal sides. How many lines of symmetry does an equilateral triangle have?

An equilateral triangle has three lines of symmetry, each passing through a vertex and the midpoint of the opposite side.

• 1
• 2
• 3 — correct
• 0