https://file.aiquickdraw.com/mathbot/file/1727360006417c2lye3ms.jpg

Let's solve the problem by breaking down the steps needed to find the vertices of the triangle after applying the given transformations.

### Given Information:
- You have triangle \( \Delta XYZ \), and its vertices are provided on a graph.
- You are asked to find the new coordinates of the vertices after applying the composition of two transformations:
  1. \( R_O, 180^\circ \) – A 180° rotation about the origin.
  2. \( T_{-2,4} \) – A translation by \( (-2,4) \).

### Step 1: Determine the Coordinates of \( \Delta XYZ \)
From the graph, we can visually read the coordinates of the vertices of \( \Delta XYZ \).

- \( X = (4, 4) \)
- \( Y = (2, 2) \)
- \( Z = (4, 2) \)

### Step 2: Apply the 180° Rotation \( R_O, 180^\circ \)
A 180° rotation about the origin transforms each point \( (x, y) \) to \( (-x, -y) \).

- \( X(4, 4) \rightarrow X'(-4, -4) \)
- \( Y(2, 2) \rightarrow Y'(-2, -2) \)
- \( Z(4, 2) \rightarrow Z'(-4, -2) \)

### Step 3: Apply the Translation \( T_{-2,4} \)
A translation by \( (-2,4) \) means subtracting 2 from the x-coordinate and adding 4 to the y-coordinate for each point.

- \( X'(-4, -4) \rightarrow X''(-4 - 2, -4 + 4) = X''(-6, 0) \)
- \( Y'(-2, -2) \rightarrow Y''(-2 - 2, -2 + 4) = Y''(-4, 2) \)
- \( Z'(-4, -2) \rightarrow Z''(-4 - 2, -2 + 4) = Z''(-6, 2) \)

### Final Answer:
- \( X'' = (-6, 0) \)
- \( Y'' = (-4, 2) \)
- \( Z'' = (-6, 2) \)

These are the final coordinates of the vertices of \( \Delta XYZ \) after the composition of transformations.

Would you like any further clarification or details on this? Here are some related questions to consider:

1. How do we determine the effect of different types of rotations on a point?
2. What is the formula for a general translation on a point?
3. How does a 180° rotation affect the orientation of a shape?
4. Can the order of transformations change the final result?
5. What is the geometric significance of combining a rotation and a translation?

**Tip:** When performing multiple transformations, always follow the correct order: the first transformation should be applied first, followed by the second, and so on.

What are the vertices for the final image after applying the composition T_{-2,4} ∘ R_O, 180° to ∆XYZ?

Learn how to find the new vertices of triangle XYZ after applying a 180° rotation about the origin followed by a translation by (-2,4). This step-by-step solution demonstrates key transformation concepts such as rotation and translation in coordinate geometry, suitable for students in Grades 9-11.

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