An Isosceles Right Triangle has _____________ (Line, Point or Both) Symmetry.?

Question

I know that A right triangle has line symmetry, but does It have Point Symmetry?

The Def. of Point Symmetry is Point of rotation (according to The Teacher).

Point of Rotation: A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180 degrees or less.

Rotation: A type of transformation in which a figure is turned around a fixed point, called the center of rotation.

Using these definitions, I know that if you rotate the triangle less than 360 degrees, you don't have symmetry (the figure is not mapped onto itself). Can you however rotate the figure 0 degrees, in which the figure is mapped onto itself? I believe that this follows the defs., and I realize that this gives every figure a point of symmetry (But couldn't this just be a type of reflexive property?). Is there a theorem somewhere that says that a rotation must be greater that 0?

Answer 1

A rotation of 0 degrees doesn't move or transform the figure at all. It's like sliding it by a vector of <0, 0>, or enlarging it by a scale factor of 1.

No movement = no transformation.

Your isosceles right triangle will have line symmetry, only.

Answer 2

Line. But if the isosceles triangle is equilateral (equilateral triangles are still isosceles in this definition) then both.

No, you cannot rotate the figure 0 degrees. Otherwise even Lake Michigan would have point symmetry. The degrees you must rotate it is x | 0 < x <= 180.

In other words, you have to rotate it x degrees, where x is greater than 0, and less or equal to 180.

Answer 3

Your teacher's definition of point symmetry is incorrect. What your teacher is referring to is rotational symmetry about a point. The isosceles right triangle does not have any rotational symmetries. You could argue that a rotation of zero degrees is a rotation of 180 degrees or less, but it isn't really a transformation. Your teacher's definition is a little bit sloppy, but by most definitions, an isosceles right triangle has no rotational symmetry.

By the way, it doesn't have point symmetry either - One definition of point symmetry is 180 degree rotational symmetry... perhaps this is what your teacher was thinking of.

Answer 4

A rotation of 0 degrees is a transformation. However, it is the identity transformation. The identity transformation can by defined by many other types of transformations as well.