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Reflectional Symmetry - Concept

Teacher/Instructor Brian McCall
Brian McCall

Univ. of Wisconsin
J.D. Univ. of Wisconsin Law school

Brian was a geometry teacher through the Teach for America program and started the geometry program at his school

Symmetry in a figure exists if there is a reflection, rotation, or translation that can be performed and the image is identical. Reflectional symmetry exists when the figure can be folded over onto itself along a line. This line is called the "line of symmetry". In regular polygons, the number of lines of symmetry equals the number of sides in the polygon.

There are many objects in Geometry and in real life that have symmetry, but how do we define symmetry? Well an object has symmetry if there exist an isometry so reflection a rotation maybe even a translation that maps the figure back onto itself. If that isometry is a reflection then a figure has reflectional symmetry, so another way of thinking about reflectional symmetry is that half the figure is a mirror image of the other half. We also describe reflectional symmetry using the term line symmetry, so those 2 are interchangeable.
Let's apply what we know about reflectional symmetry to some specific examples. Here we're being asked to find the lines of reflectional symmetry which means, what line could I draw on this pentagon so that I could fold the pentagon back onto itself and have an identical image? Well I see that I could draw a line of reflection right there which would enable me to fold my pentagon over onto itself. I could draw a perpendicular through each one of these vertices perpendicular to the opposite side for a total of 5 lines of reflectional symmetry.
If we move onto a square I see that I could draw in a line of symmetry through the half of these 2 sides and I could fold it. I could also draw a horizontal line of symmetry I can also draw in some diagonal lines of symmetry and those would enable me to fold the figure back onto itself, so this regular pentagon had 5, the square has 4 lines of symmetry.
What about an equilateral triangle? Well an equilateral triangle similarly I can draw this line through the vertex perpendicular to the opposite side and I could draw in two more so an equilateral triangle will have 3 lines reflectional symmetry.
We look at an Isosceles triangle, if I label these two sides as being congruent there's only one line of symmetry because if I drew in another line here, it would not be a mirror image of itself.
If we're talking about a scalene triangle, well let's say we gave these sides lengths of 2, 5 and 8 there are no lines of symmetry of reflectional symmetry of the scalene triangle. So to go back over, what exactly reflectional symmetry is? Is it's a more specific definition of symmetry where you can reflect half of the figure back onto itself.

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