To solve this problem, we need to rotate each vertex of triangle $XYZ$ 180° about the origin.

Step-by-Step Solution:

1. 180° Rotation Rule:

   • Rotating a point $(x, y)$ 180° about the origin transforms it to $(-x, -y)$.

2. Apply the Rotation to Each Vertex:

   • For $X(4, -1)$:
     $X' = (-4, 1)$
   • For $Y(8, -2)$:
     $Y' = (-8, 2)$
   • For $Z(1, -8)$:
     $Z' = (-1, 8)$

Final Answers:

• $X'(-4, 1)$
• $Y'(-8, 2)$
• $Z'(-1, 8)$

These values fill in the blanks as follows:

• Blank 1: -4
• Blank 2: 1
• Blank 3: -8
• Blank 4: 2
• Blank 5: -1
• Blank 6: 8

Would you like a more detailed explanation, or do you have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the coordinates change if the triangle were rotated 90° clockwise about the origin instead of 180°?
2. How does a 270° rotation about the origin affect the coordinates of a point?
3. What would be the result of a 360° rotation of point $X(4, -1)$ around the origin?
4. How do transformations like reflections differ from rotations when applied to coordinates?
5. If the origin were moved, how would that change the rotation calculations?

Tip: Remember, a 180° rotation flips both the x and y coordinates to their opposites.