Name: Cindie

Who is asking: Parent
Level: Elementary

How many lines of symmetry do the following figures contain?





Hi Cindie,

I think there are two levels of this type of problem.

One is the ability to detect at least one mirror. I imagine in the list below, the rhombus, the pentagon, the hexagon (presumably thought of as 'regular') have at least on line. SOME trapezoids have one, some do not, so I am not sure whether there was some additional clue for that example.

Having detected ONE mirror, it makes more sense to look for rotations. If the object has rotational symmetry, then the rotation of a mirror will be a new mirror.

There is a reverse connection - given any PAIR of mirrors, meeting at an angle A, there is a rotation by angel 2A. And given a rotation by angle 2A, and a mirror, there will be a second mirror at an angle A from the first. This larger set of connections provides a context which organizes your pieces.

Finally, there are a couple of 'local' events for mirrors which can focus your search. IF a mirror goes through a vertex, then the mirror will need to take one edge to the other at this vertex. This means the mirror must cut the angle at the vertex - must bisect the angle in the terms of geometry.

Similarly, if the mirror cuts an edge, not at a vertex, then the mirror must take this edge onto itself (one vertex onto the other). That means that the mirror must cut the edge in half (bisect the edge at the midpoint). In fact, it must also bisect the 'straight angle' at the midpoint and therefore be at right angles to the edge - it is a right bisector.

So here are two important geometric techniques to use. Ways of 'seeing' and searching. Locally look for right bisectors of edges and angle bisectors at vertices.

Globally, having found one mirror, look for rotations. Piece together larger connections between a rotation and a mirror, to get a full list.

These are larger principles which recur later in education. I talk about these with students and teachers of Chemistry at University (chiral and achiral molecules - important in biology and medicine). So it is worth assembling the pieces into some bigger picture to work with later.

Walter Whiteley
York University

Go to Math Central