The image shows three triangles, labeled $P$, $Q$, and $R$, on a coordinate grid. You are asked to describe the single transformation that maps:

(a) Triangle $P$ onto triangle $Q$

(b) Triangle $Q$ onto triangle $R$

Let's analyze the transformations:

(a) Mapping Triangle $P$ onto Triangle $Q$

The triangles $P$ and $Q$ have the same size and shape, but their orientations and positions differ. Since triangle $Q$ is a reflection of triangle $P$, and both are symmetric about the y-axis, the transformation is a reflection in the y-axis.

Thus, the transformation that maps $P$ to $Q$ is:

• Reflection in the y-axis.

(b) Mapping Triangle $Q$ onto Triangle $R$

Triangles $Q$ and $R$ appear to be congruent, but triangle $R$ is in the third quadrant, rotated and in a different position compared to triangle $Q$. Observing their relative positions, triangle $R$ is a 180-degree rotation of triangle $Q$ around the origin (point $(0, 0)$).

Thus, the transformation that maps $Q$ to $R$ is:

• Rotation of 180 degrees about the origin.

Let me know if you want more details or have any questions. Here are five related questions for further exploration:

1. How can you identify the line of reflection for different transformations?
2. What are the rules for rotating shapes about the origin by different angles?
3. Can you describe a transformation that would map triangle $P$ onto triangle $R$ directly?
4. How would the coordinates of triangle $P$ change under a reflection in the x-axis?
5. What would be the matrix representation of a 180-degree rotation about the origin?

Tip: When working with transformations, always verify the distances and angles to ensure congruency between shapes after the transformation.