Let me analyze the image you've uploaded and determine the correct set of points for the reflection of triangle XYZ across the line $y = x$.

I'll extract the coordinates of the triangle's vertices and then apply the reflection rule. The rule for reflecting a point $(x, y)$ across the line $y = x$ is to swap the coordinates to $(y, x)$.

Let me proceed with the analysis. PIL import Image

Open the uploaded image to analyze its content

image_path = '/mnt/data/1738965105224ly3b2u3e.jpg' img = Image.open(image_path) img.show() # Display the image to help with analysisI have displayed the image to analyze the coordinates of the triangle. To proceed with identifying the correct reflection across the line $y = x$, I will manually extract the coordinates of the triangle's points based on the image.

Let me now calculate the correct set of points.Upon visual inspection, we can observe that the vertices of triangle XYZ are at the following coordinates:

• $X(−5, −5)$
• $Y(−3, −2)$
• $Z(−3, −5)$

Now, to find the reflection of these points across the line $y = x$, we apply the reflection rule where we swap the coordinates of each point:

• $X(−5, −5)$ reflects to $X'(-5, -5)$
• $Y(−3, −2)$ reflects to $Y'(-2, -3)$
• $Z(−3, −5)$ reflects to $Z'(-5, -3)$

The correct set of points that indicates the reflection of Triangle XYZ across the line $y = x$ is:

X'(-5, -5), Y'(-2, -3), Z'(-5, -3)

Thus, the correct answer is the third option:

X'(-5, -5), Y'(-2, -3), Z'(-5, -3)

Would you like more detailed steps on how the reflection works, or do you have any other questions?

Here are some related questions to explore further:

1. What is the general rule for reflecting a point across the line $y = x$?
2. How do you find the reflection of points across other lines like $y = -x$?
3. Can you describe how a triangle's shape changes under different types of reflections?
4. What happens to the orientation of the triangle after a reflection across the line $y = x$?
5. How do you reflect a triangle across the x-axis or y-axis, and how do the results compare?

Tip: The key concept in reflections is that the distance from the original point to the line of reflection is the same as the distance from the reflected point to the line of reflection.