The image shows two congruent triangles, $PQR$ and $P'Q'R'$, plotted on a coordinate grid. The problem asks to identify the transformation that maps triangle $PQR$ onto triangle $P'Q'R'$, demonstrating that the triangles are congruent. The possible transformations are:

1. Reflection across the $x$-axis
2. Reflection across the $y$-axis
3. Clockwise rotation by 90° about the origin
4. Counterclockwise rotation by 90° about the origin

Analysis:

• The triangle $PQR$ is located in the first quadrant, and triangle $P'Q'R'$ is located in the fourth quadrant.
• The triangles appear to have the same shape and size, suggesting a reflection across the $x$-axis.

To confirm:

• A reflection across the $x$-axis flips points across the horizontal line $y = 0$. The $y$-coordinates of all points change sign, while the $x$-coordinates remain the same.

For instance:

• Point $P(4, 4)$ would map to $P'(4, -4)$, which matches the image.
• Similarly, the points $Q(2, 2)$ and $R(5, 2)$ would map to $Q'(2, -2)$ and $R'(5, -2)$, respectively, matching the transformation shown.

Conclusion:

The correct transformation is reflection across the $x$-axis.

Would you like more details or have any other questions?

Here are five related questions to explore:

1. What is the difference between reflection and rotation in geometry?
2. How does congruence differ from similarity in triangles?
3. How would a reflection across the $y$-axis affect triangle $PQR$?
4. Can a reflection be combined with a rotation? How would that affect the congruence?
5. How do translations affect the coordinates of a shape?

Tip: When reflecting across the $x$-axis, always remember that only the sign of the $y$-coordinate changes.