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.