1. **Problem Statement:** Draw the preimage and image of triangle PQR under a translation along the vector $\langle -1, 3 \rangle$. The triangle vertices are $P(1, -3)$, $Q(3, -1)$, and $R(4, -3)$. 2. **Translation Formula:** To translate a point $(x, y)$ by vector $\langle a, b \rangle$, the new point $(x', y')$ is given by: $$ x' = x + a $$ $$ y' = y + b $$ 3. **Apply the translation to each vertex:** - For $P(1, -3)$: $$ P' = (1 + (-1), -3 + 3) = (0, 0) $$ - For $Q(3, -1)$: $$ Q' = (3 + (-1), -1 + 3) = (2, 2) $$ - For $R(4, -3)$: $$ R' = (4 + (-1), -3 + 3) = (3, 0) $$ 4. **Result:** The image triangle vertices after translation are $P'(0, 0)$, $Q'(2, 2)$, and $R'(3, 0)$. 5. **Rigid Motion Explanation:** A translation is a rigid motion because it preserves the size and shape of the figure. This means: - All angles remain the same. - All line segments keep their original lengths. - The figure is simply shifted without rotation or resizing. Therefore, translations do not alter the angles or lengths of the figure, confirming they are rigid motions.