https://file.aiquickdraw.com/mathbot/file/17276503125676sf8yspd.jpg

The problem asks for a rotation of the triangle 180° about the origin. Let’s go through the steps to solve it:

1. **Identify the Coordinates**: 
   - The vertices of the triangle are labeled as \( G \), \( T \), and \( X \). Let's determine their coordinates.
   - From the graph:
     - \( G \) is at approximately \((4, -2)\).
     - \( T \) is at approximately \((2, -4)\).
     - \( X \) is at approximately \((6, 4)\).

2. **Apply the 180° Rotation Rule**:
   - Rotating a point 180° about the origin changes its coordinates according to the rule:
     \[
     (x, y) \to (-x, -y)
     \]
   - Using this rule, let’s determine the new coordinates for each point:
     - \( G(4, -2) \to G'(-4, 2) \)
     - \( T(2, -4) \to T'(-2, 4) \)
     - \( X(6, 4) \to X'(-6, -4) \)

3. **Plot the Rotated Points**:
   - Now plot the new points on the graph:
     - \( G' \) at \((-4, 2)\)
     - \( T' \) at \((-2, 4)\)
     - \( X' \) at \((-6, -4)\)
   - Connect these points to form the rotated triangle.

4. **Final Triangle**:
   - After connecting the new points, you will have a triangle that is the 180° rotated image of the original triangle about the origin.

### Do You Have Any Questions?
Would you like me to clarify any part of the rotation process, or need help with plotting these points?

### Related Questions to Practice
1. What would be the coordinates of the triangle if it were rotated 90° clockwise about the origin?
2. How would the triangle look after a reflection over the y-axis?
3. If you translated the triangle 3 units to the left and 5 units up, what would the new coordinates be?
4. What are the new coordinates of the triangle after a reflection over the x-axis?
5. Can you determine the coordinates if the triangle was rotated 270° counterclockwise about the origin?

### Tip
When performing rotations, always try to visualize or use graph paper to help you keep track of how each point moves, which can make transformations much clearer.

Graph the image of the figure using the transformation of a 180° rotation about the origin.

This problem involves rotating a triangle 180° around the origin on a coordinate plane. Learn how to apply the 180° rotation rule, plot the new points, and graph the rotated triangle. Suitable for students in Grade 8.

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