Symmetry (including Reflection and Rotation)
181. The Mirror Line · folding figures into perfect halves
C is always on the perpendicular bisector of AB, so CA always equals CB — the dashed line is always a line of symmetry.
Linear (line) symmetry means a figure has a line of reflection that maps it exactly onto itself. The line acts like a mirror: each half is the mirror image of the other.
What this lesson covers
Try to break it
Try to make CA unequal to CB by dragging C — you cannot! C is locked to the axis of symmetry, so both halves always mirror each other. This is why the perpendicular bisector of the base is the one and only line of symmetry of any isosceles triangle.
How you build it
Construct an isosceles triangle and its line of symmetry.
- Mark point A — the left end of the base.
- Mark point B — the right end of the base.
- Draw the base AB.
- Construct the perpendicular bisector of AB — this line is the axis of symmetry!
- Place the apex C anywhere on the perpendicular bisector — CA and CB will be equal, making triangle ABC isosceles.
- Draw side CA.
- Draw side CB to complete triangle ABC. The perpendicular bisector you drew is its line of symmetry!
The proof, step by step
Prove that the line through C and the midpoint M of AB is a line of symmetry.
- A line of symmetry divides a figure into two congruent parts.
- Every point on one side has a mirror image at the same distance on the other side.
- For an isosceles triangle, the line of symmetry passes through the apex and the midpoint of the base.
Worked example
How many lines of symmetry does an isosceles triangle have?
An isosceles triangle has exactly one line of symmetry, which passes through the apex and bisects the base.
- 0
- 1 — correct
- 2
- 3