Transformations that carry a polygon onto itself

Key Notes :

Reflecting a regular n-gon across a line of symmetry carries the n-gon onto itself.

  • If n is odd, the lines of symmetry pass through a vertex and the midpoint of the opposite side.
  • If n is even, the lines of symmetry either pass through two opposite vertices, or pass through the midpoints of two opposite sides.

Rotating a regular n-gon by a multiple of 360°/n carries the n-gon onto itself.

Learn with an example

Which of the following transformations carry this regular polygon onto itself?

a ) Rotation of 90° clockwise

b ) Reflection across 𝓁

c ) Rotation of 45° anticlockwise

d ) Rotation of 120° anticlockwise

This regular polygon has 5 sides, so it is a pentagon.

First, see if the reflection answer choice is correct. In other words, is 𝓁 a line of symmetry?

Since n= 5 is odd and 𝓁 passes through a vertex and the midpoint of its opposite side, 𝓁 is a line of symmetry. So, this answer choice is correct:

reflection across 𝓁

Second, see if any of the rotation answer choices are correct. Rotating a regular pentagon by a multiple of 360°/5=72° carries the pentagon onto itself. So, none of the rotation answer choices are correct.

Which of the following transformations carry this regular polygon onto itself?

a ) Rotation of 90° clockwise

b ) Rotation of 120° anticlockwise

c ) Rotation of 90° anticlockwise

d ) Reflection across 𝓁

First, see if the reflection answer choice is correct. In other words, is 𝓁 a line of symmetry?

Since n = 3 is odd and 𝓁 passes through a vertex and the midpoint of its opposite side, 𝓁 is a line of symmetry. So, this answer choice is correct:

reflection across 𝓁

Second, see if any of the rotation answer choices are correct. Rotating an equilateral triangle by a multiple of 360°/3=120°carries the equilateral triangle onto itself. So, this is the only correct rotation answer choice:

rotation of 120° anticlockwise

Which of the following transformations carry this regular polygon onto itself?

a ) Reflection across 𝓁

b ) Rotation of 60° clockwise

c ) Rotation of 72° anticlockwise

d ) Rotation of 60° anticlockwise

This regular polygon has 6 sides, so it is a hexagon.

First, see if the reflection answer choice is correct. In other words, is 𝓁a line of symmetry?

Since n= 6 is even and 𝓁 passes through a vertex and a midpoint, 𝓁 is not a line of symmetry. So, the reflection answer choice is not correct.

Second, see if any of the rotation answer choices are correct. Rotating a regular hexagon by a multiple of 360°/6=60° carries the hexagon onto itself. So, these are the correct rotation answer choices:

  • rotation of 60° clockwise
  • rotation of 60° anticlockwise

let’s practice!