Question

which of the following transformations carry this regular polygon onto itself? reflection across ( l ); rotation of ( 36^circ ) counterclockwise; rotation of ( 120^circ ) counterclockwise; rotation of ( 120^circ ) clockwise

Explanation:

Step1: Analyze the regular polygon (equilateral triangle)

A regular triangle (equilateral triangle) has rotational symmetry of order 3. The central angle for each rotation that maps it onto itself is $\frac{360^\circ}{3} = 120^\circ$. Also, it has reflection symmetry over lines that pass through a vertex and the mid - point of the opposite side.

Step2: Evaluate each transformation

• Reflection across \( l \): The line \( l \) in the diagram seems to pass through a vertex and the mid - point of the opposite side (since it bisects a side as seen from the marks on the side). For an equilateral triangle, reflecting across such a line will map the triangle onto itself.
• Rotation of \( 36^\circ \) counterclockwise: Since the rotational symmetry angle is \( 120^\circ \), a \( 36^\circ \) rotation will not map the equilateral triangle onto itself.
• Rotation of \( 120^\circ \) counterclockwise: As we calculated, the rotational symmetry angle for an equilateral triangle is \( 120^\circ \). So a \( 120^\circ \) counterclockwise rotation will map the triangle onto itself.
• Rotation of \( 120^\circ \) clockwise: Rotation in the clockwise direction by \( 120^\circ \) is equivalent to rotation in the counterclockwise direction by \( 360^\circ - 120^\circ=240^\circ \), but also, since the rotational symmetry group includes rotations by \( 120^\circ k \) (where \( k = 0,1,2 \)), a \( 120^\circ \) clockwise rotation (which is a \( 120^\circ \) rotation in the negative direction) will also map the triangle onto itself. And the reflection across \( l \) is also a valid transformation. But let's check the options again. Wait, the original problem's options: the first option "reflection across \( l \)" is correct, the third option "rotation of \( 120^\circ \) counterclockwise" is correct, the fourth option "rotation of \( 120^\circ \) clockwise" is correct. The second option "rotation of \( 36^\circ \) counterclockwise" is incorrect.

Wait, maybe the diagram is an equilateral triangle (regular 3 - gon). The order of rotational symmetry is 3, so the angle of rotational symmetry is \( \frac{360^\circ}{3}=120^\circ \). So rotations by \( 120^\circ \) and \( 240^\circ \) (clockwise or counterclockwise equivalent) will map it onto itself. Also, reflections over the lines of symmetry (which pass through a vertex and the mid - point of the opposite side) will map it onto itself.

So the correct transformations are:

• Reflection across \( l \)
• Rotation of \( 120^\circ \) counterclockwise
• Rotation of \( 120^\circ \) clockwise

Answer:

• reflection across \( l \)
• rotation of \( 120^\circ \) counterclockwise
• rotation of \( 120^\circ \) clockwise