Symmetry
52. The Mirror Line · Discovering the Line of Symmetry
The line AM always divides the triangle into two identical halves that are mirror images.
Line symmetry (or reflection symmetry) means a figure can be folded along a line so that the two halves match exactly. The fold line is called the line (or axis) of symmetry. An isosceles triangle has one line of symmetry through its apex.
What this lesson covers
Try to break it
Notice that no matter where you move B, point C always matches it perfectly on the other side!
How you build it
Construct an isosceles triangle and its line of symmetry.
- Mark point B — one end of the base.
- Mark point C — the other end of the base.
- Draw the base BC.
- With centre B, open the compass wider than half of BC and draw an arc.
- With the same opening and centre C, draw an arc crossing the first one.
- Mark point A where the two arcs cross above BC — this is the apex.
- Mark point N where the two arcs cross below BC.
- Draw the line through A and N — this is the line of symmetry.
- Join A to B.
- Join A to C. A is the same distance from B and C, so AB = AC — triangle ABC is isosceles.
The proof, step by step
Prove that line AM divides the triangle into two mirror-image halves.
- The line of symmetry is the perpendicular bisector of the segment connecting corresponding points (like B and C).
- Therefore, the distance from any point on one side to the line is equal to the distance from its mirror image on the other side.
- When folded along the line, the two parts of the figure coincide perfectly, proving it is symmetrical.
Worked example
A kite has a line of symmetry. If one wing tip is 5 cm from the line, how far is the other wing tip from the line?
In a symmetrical figure, corresponding points are equidistant from the line of symmetry. Since one wing tip is 5 cm from the line, the other wing tip must also be 5 cm from the line.
- 2 cm
- 5 cm — correct
- 10 cm
- 0 cm