1. **State the problem:** We have a square with vertices A(-9, -9), B(9, -9), C(9, 6), and D(-9, 6). We want to find the coordinates of these vertices after a dilation centered at the origin with a scale factor of $\frac{1}{3}$. 2. **Formula for dilation:** If a point $P(x, y)$ is dilated from the origin by a scale factor $k$, the new point $P'(x', y')$ is given by: $$ x' = kx, \quad y' = ky $$ 3. **Apply the formula to each vertex:** - For $A(-9, -9)$: $$ A' = \left( \frac{1}{3} \times -9, \frac{1}{3} \times -9 \right) = (-3, -3) $$ - For $B(9, -9)$: $$ B' = \left( \frac{1}{3} \times 9, \frac{1}{3} \times -9 \right) = (3, -3) $$ - For $C(9, 6)$: $$ C' = \left( \frac{1}{3} \times 9, \frac{1}{3} \times 6 \right) = (3, 2) $$ - For $D(-9, 6)$: $$ D' = \left( \frac{1}{3} \times -9, \frac{1}{3} \times 6 \right) = (-3, 2) $$ 4. **Final answer:** The coordinates of the vertices after dilation are: $$ A'(-3, -3), \quad B'(3, -3), \quad C'(3, 2), \quad D'(-3, 2) $$