Draw and label the final image of $\triangle A B C$ after the given sequence of transformations. 1. Reflect $\triangle A B C$ over the $\mathrm{x}$-axis and rotate 180 degrees around the origin. Transformation rulet \begin{tabular}{|c|c|c|} \hline $\begin{array}{c}\text { Preimage } \\ A B C\end{array}$ & Image A'B'C Image $A^{\prime} B^{\prime} C^{\prime \prime}$ \\ \hline$A(-1,2)$ & & \\ \hline$B(-3,4)$ & & \\ \hline$C(-2,5)$ & \\ \hline \end{tabular}
Solution
Step 1 ：The problem asks to reflect the triangle ABC over the x-axis and then rotate it 180 degrees around the origin. The reflection over the x-axis will change the sign of the y-coordinate of each point. The rotation of 180 degrees around the origin will change the sign of both the x and y coordinates. Therefore, we can combine these two transformations into one operation: changing the sign of the y-coordinate.
Step 2 ：The original points of the triangle ABC are A(-1, 2), B(-3, 4), and C(-2, 5).
Step 3 ：After the transformation, the points become A'(-1, -2), B'(-3, -4), and C'(-2, -5).
Step 4 ：\(\boxed{\text{The final image of } \triangle ABC \text{ after the given sequence of transformations is } \triangle A'B'C' \text{ with } A'(-1, -2), B'(-3, -4), \text{ and } C'(-2, -5).}\)
