Question

transformations
identifying transformations that map a regular polygon onto itself
an equilateral triangle is shown below.
line m passes through a vertex and bisects a side.
line n bisects each side it passes through.
point x is the center of the triangle.
which transformation(s) must map the triangle exactly onto itself? choose all that apply.
□ reflection across line n
□ counterclockwise rotation about x by 144°
□ clockwise rotation about x by 360°
□ reflection across line m
□ none of the above

Explanation:

1. Reflection across line \( n \): Line \( n \) bisects sides, but in an equilateral triangle, the lines of reflection are the angle bisectors/medians/altitudes from vertices. Line \( n \) here does not align with a vertex - side bisector (median) that is a line of reflection (since it doesn't pass through a vertex, just bisects sides), so reflection across \( n \) does not map the triangle to itself.
2. Counterclockwise rotation about \( X \) by \( 144^\circ \): For an equilateral triangle, the rotational symmetry angle is \( \frac{360^\circ}{3}=120^\circ \). \( 144^\circ \) is not a multiple of \( 120^\circ \), so this rotation does not map the triangle to itself.
3. Clockwise rotation about \( X \) by \( 360^\circ \): A rotation of \( 360^\circ \) (clockwise or counterclockwise) maps any figure onto itself, as it is a full rotation back to the original position.
4. Reflection across line \( m \): Line \( m \) passes through a vertex and bisects the opposite side (since it's an equilateral triangle, this is a median, angle bisector, and altitude). Reflecting across this line will map the triangle onto itself, as it is a line of symmetry for the equilateral triangle.
5. None of the above: Since options 3 and 4 are valid, this is incorrect.

Answer:

• Clockwise rotation about \( X \) by \( 360^\circ \)
• Reflection across line \( m \)