To reflect the coordinates of each vertex of this shape over the line $y = x$, we need to swap the $x$- and $y$-coordinates of each point. I will analyze the points as follows:

Given Coordinates

Based on the image:

• Point $A$ is at $(8, -8)$.
• Point $B$ is at $(6, -3)$.
• Point $C$ is at $(5, -2)$.
• Point $D$ is at $(-3, -5)$.

Reflected Coordinates over $y = x$

To find the reflected coordinates, swap $x$ and $y$ for each point:

1. $A (8, -8)$ reflects to $(-8, 8)$.
2. $B (6, -3)$ reflects to $(-3, 6)$.
3. $C (5, -2)$ reflects to $(-2, 5)$.
4. $D (-3, -5)$ reflects to $(-5, -3)$.

Reflected Coordinates Summary

After reflection over $y = x$:

• $A'$ is at $(-8, 8)$.
• $B'$ is at $(-3, 6)$.
• $C'$ is at $(-2, 5)$.
• $D'$ is at $(-5, -3)$.

Would you like further details or have any questions about this process?

Here are five related questions that might deepen your understanding:

1. What is the general rule for reflecting points over the line $y = x$?
2. How would the coordinates change if we reflected them over the line $y = -x$ instead?
3. How do you find the equation of a line of symmetry for different types of geometric shapes?
4. Can you describe the transformations required to map a point to its reflection over the origin?
5. What are the applications of reflections in real-world contexts, such as computer graphics?

Tip: When reflecting a point over the line $y = x$, simply swap the $x$- and $y$-coordinates of the point!