Let's analyze the problem in the image.

Problem Context:

We have two polygons, $P$ and $Q$, shown on the coordinate plane. The question asks which rigid transformation allows us to obtain polygon $P$ from polygon $Q$, assuming counterclockwise (anticlockwise) rotations.

Analyzing Transformations:

The options are:

• A) Reflection with respect to the $y$-axis.
• B) 180° rotation about the origin.
• C) Reflection with respect to the $y$-axis, followed by a 180° rotation about the origin.
• D) Reflection with respect to the $x$-axis, followed by a 180° rotation about the origin.
• E) 90° rotation about the origin.

Steps to Solve:

1. Visual Analysis: Polygon $Q$ seems to be a reflection or rotation of polygon $P$. We need to figure out how one can be obtained from the other.
2. Reflection or Rotation: By observing the shapes of $P$ and $Q$, we first look for any symmetries. Reflection across the axes or rotations about the origin may help.

Let's break down each option:

• Option A (Reflection across the $y$-axis): Reflecting across the $y$-axis would flip the polygon horizontally, but this does not produce polygon $P$ from $Q$, since the polygons aren't just mirrored horizontally.

• Option B (180° rotation about the origin): A 180° rotation would flip the shape completely upside down, but this also doesn't seem to map $P$ to $Q$ directly.

• Option C (Reflection across the $y$-axis and 180° rotation): This involves first flipping $Q$ across the $y$-axis, and then rotating it 180° about the origin. This could align the two shapes.

• Option D (Reflection across the $x$-axis and 180° rotation): This sequence involves a reflection across the $x$-axis and a rotation, but it's not as promising since polygon $P$ would end up flipped vertically, which doesn't seem to be the case.

• Option E (90° rotation about the origin): A 90° rotation counterclockwise moves points in a specific way. This doesn't seem to match the transformation needed.

Best Choice:

The correct sequence of transformations appears to be Option C, which involves a reflection across the $y$-axis followed by a 180° rotation about the origin.

Would you like more details on the steps or need clarification? Here are 5 related questions for further exploration:

1. What is the geometric definition of a rigid transformation?
2. How does a reflection across the $y$-axis affect the coordinates of points?
3. How do you perform a 180° rotation around the origin mathematically?
4. What are the differences between rotations and reflections in geometry?
5. How would the transformation differ if the rotation were clockwise instead of counterclockwise?

Tip: When performing transformations, sketching out the steps can often make it easier to visualize the final result!