Question

an equilateral triangle is shown below. line ( m ) bisects each side it passes through. line ( n ) passes through a vertex and bisects a side. point ( k ) is the center of the triangle. which transformation(s) must map the triangle exactly onto itself? choose all that apply.

• counterclockwise rotation about ( k ) by ( 144^circ )
• reflection across line ( m )
• reflection across line ( n )
• clockwise rotation about ( k ) by ( 60^circ )
• none of the above

Explanation:

1. Reflection across line \( m \): Line \( m \) bisects a side of the equilateral triangle. In an equilateral triangle, reflecting across a line that bisects a side (and is a line of symmetry) will map the triangle onto itself. Since line \( m \) is a line of symmetry (bisects a side and passes through the center \( K \) in a way that symmetrically reflects the triangle), this reflection works.
2. Reflection across line \( n \): Line \( n \) passes through a vertex and bisects a side, which is also a line of symmetry for the equilateral triangle. Reflecting across line \( n \) will map the triangle onto itself.
3. Counterclockwise rotation about \( K \) by \( 144^\circ \): The order of rotational symmetry for an equilateral triangle is \( 3 \) (since \( 360^\circ / 3 = 120^\circ \)). A rotation by \( 144^\circ \) is not a multiple of \( 120^\circ \), so this rotation will not map the triangle onto itself.
4. Clockwise rotation about \( K \) by \( 60^\circ \): Similarly, \( 60^\circ \) is not a multiple of \( 120^\circ \), so this rotation will not map the triangle onto itself.

Answer:

Reflection across line \( m \), Reflection across line \( n \)