This question is about identifying the transformations that map a regular triangle (equilateral triangle) onto itself. Regular polygons have specific symmetry properties that allow certain transformations to result in the same shape appearing unchanged. Let's analyze each transformation option:

1. Reflection Across Line ℓ: A regular triangle can be mapped onto itself by reflecting across a line that passes through a vertex and the midpoint of the opposite side. Since line ℓ is positioned correctly in this image, this reflection is valid.

2. Rotation of 120° Clockwise: Rotating an equilateral triangle by 120° (one-third of a full 360° rotation) aligns each vertex with the position of the next vertex. This rotation maps the triangle onto itself, so this option is correct.

3. Rotation of 90° Clockwise: Rotating by 90° does not align the vertices with the original position because 90° is not a divisor of 360° for a three-sided shape. Therefore, a 90° rotation does not carry the triangle onto itself.

4. Rotation of 40° Counterclockwise: Similarly, rotating by 40° will not align the vertices correctly because 40° is also not a multiple that matches the triangle's symmetry. Thus, this option does not work.

Answer

The correct transformations that carry the triangle onto itself are:

• Reflection across line ℓ
• Rotation of 120° clockwise

Would you like more details or have any other questions?

Here are five related questions to expand on this topic:

1. How many lines of symmetry does a regular triangle have?
2. What are all the possible rotational symmetries for a square?
3. Can a 60° rotation map a regular hexagon onto itself? Why or why not?
4. Why does a regular pentagon have different rotational symmetries than a triangle?
5. How do rotational symmetries differ between polygons with odd and even numbers of sides?

Tip: In general, an $n$-sided regular polygon has rotational symmetry for every $\frac{360°}{n}$ increment.