1. **State the problem:** We have triangle LMN with vertices L(1, -2), M(3, 0), and N(2, -3). We want to find the coordinates of the dilated triangle L'M'N' after a dilation with scale factor $k=5$ centered at the origin. 2. **Formula for dilation:** If a point $P(x,y)$ is dilated by a scale factor $k$ centered at the origin, the image point $P'(x',y')$ is given by: $$ x' = kx $$ $$ y' = ky $$ 3. **Apply the formula to each vertex:** - For $L(1,-2)$: $$ L' = (5 \times 1, 5 \times -2) = (5, -10) $$ - For $M(3,0)$: $$ M' = (5 \times 3, 5 \times 0) = (15, 0) $$ - For $N(2,-3)$: $$ N' = (5 \times 2, 5 \times -3) = (10, -15) $$ 4. **Final answer:** The coordinates of the dilated triangle are: $$ L'(5, -10),\quad M'(15, 0),\quad N'(10, -15) $$