Quick question: Can a pentagon have an axis of symmetry passing through two (one, none) of its vertices?
I'm given the following definition for axis of symmetry: A figure is said to have an axis of symmetry, $a$, if this figure is symmetric to itself about the line, $a$, i.e. if for any point of the figure the symmetric point also belongs to the figure. Thanks in advance...
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Note that to be a line of symmetry, there should be an equal number of vertices on each side of the line. Consequently, there must be an even number of vertices not lying on the line, so the line must pass through an odd number of vertices. It is impossible to have a pentagon with all 5 vertices lying on the same line. A line passing through one vertex can certainly work (for regular pentagons, the "home plate" shape, etc.). Is it possible that a line passing through 3 vertices of a pentagon to be a line of symmetry?
Hint: Suppose the line of symmetry passes through 1 (or 2, or 3) vertices. Where are all the vertices? Draw some pictures!