The problem involves finding a reflection rule $r_t(x, y)$ that maps the triangle and its image across a specific line, and the equation of the reflection line $t$.

Steps to Solve:

1. Identify the Reflection Line: The red dashed line appears to pass through the origin $(0, 0)$ and has a slope of $1$, based on its angle with the axes. This indicates the line $t$ is: $y = x$

2. Reflection Rule: The general reflection rule over the line $y = x$ is: $r_t(x, y) = (y, x)$ This swaps the $x$- and $y$-coordinates of any point.

Final Answer:

• The reflection rule is: $r_t(x, y) = (y, x)$
• The equation of the line $t$ is: $y = x$

Would you like a detailed explanation of these steps or a related visualization?

Follow-Up Questions:

1. What is the reflection rule for other lines, such as $y = -x$?
2. How can you verify that a point and its image are reflections over $y = x$?
3. What transformations can occur when reflecting over arbitrary lines, such as $y = mx + b$?
4. How do you find the perpendicular distance between a point and the reflection line?
5. How does reflecting over $y = x$ affect the orientation of a shape?

Tip:

Always check the slope and intercept of the line to correctly identify its equation before determining the reflection rule!