Question

which of the following transformations carry this regular polygon onto itself? reflection across l rotation of 60° counterclockwise rotation of 90° counterclockwise rotation of 90° clockwise

Explanation:

To determine which transformations carry the regular triangle (equilateral triangle) onto itself, we analyze each option:

1. Reflection across \( \ell \):

The line \( \ell \) is a vertical line of symmetry (bisecting the base and the opposite vertex). Reflecting the triangle across \( \ell \) maps each point to a corresponding point on the other side of \( \ell \), preserving the triangle’s shape and position. Thus, this transformation works.

2. Rotation of \( 60^\circ \) counterclockwise:

For a regular \( n \)-sided polygon, the smallest angle of rotation that maps it onto itself is \( \frac{360^\circ}{n} \). For a triangle (\( n = 3 \)), this angle is \( \frac{360^\circ}{3} = 120^\circ \). A \( 60^\circ \) rotation is not a multiple of \( 120^\circ \), so it will not map the triangle onto itself. Thus, this transformation does not work.

3. Rotation of \( 90^\circ \) counterclockwise:

As the smallest rotational symmetry angle is \( 120^\circ \), a \( 90^\circ \) rotation is not a multiple of \( 120^\circ \). It will not align the triangle with its original position. Thus, this transformation does not work.

4. Rotation of \( 90^\circ \) clockwise:

Similar to the counterclockwise \( 90^\circ \) rotation, \( 90^\circ \) is not a multiple of \( 120^\circ \). This rotation will not map the triangle onto itself. Thus, this transformation does not work.

Only the reflection across \( \ell \) is a valid transformation.

Answer:

reflection across \( \ell \)