1. The problem states that a polygon with vertices V(-5,7), U(3,5), T(1,-4), and S(-6,-7) is dilated by a factor of $\frac{1}{4}$ centered at the origin. 2. The formula for dilation centered at the origin is: $$ (x', y') = (k \cdot x, k \cdot y) $$ where $k$ is the dilation factor. 3. Applying the dilation factor $k=\frac{1}{4}$ to each vertex: - For V(-5,7): $$ (x', y') = \left(\frac{1}{4} \times -5, \frac{1}{4} \times 7\right) = \left(-\frac{5}{4}, \frac{7}{4}\right) $$ - For U(3,5): $$ (x', y') = \left(\frac{1}{4} \times 3, \frac{1}{4} \times 5\right) = \left(\frac{3}{4}, \frac{5}{4}\right) $$ - For T(1,-4): $$ (x', y') = \left(\frac{1}{4} \times 1, \frac{1}{4} \times -4\right) = \left(\frac{1}{4}, -1\right) $$ - For S(-6,-7): $$ (x', y') = \left(\frac{1}{4} \times -6, \frac{1}{4} \times -7\right) = \left(-\frac{3}{2}, -\frac{7}{4}\right) $$ 4. The resulting image is a polygon with vertices at: $$ V'\left(-\frac{5}{4}, \frac{7}{4}\right), U'\left(\frac{3}{4}, \frac{5}{4}\right), T'\left(\frac{1}{4}, -1\right), S'\left(-\frac{3}{2}, -\frac{7}{4}\right) $$ 5. This shows the polygon shrunk by a factor of 4 towards the origin, preserving the shape but reducing the size. Final answer: The dilated polygon vertices are $V'(-1.25,1.75)$, $U'(0.75,1.25)$, $T'(0.25,-1)$, and $S'(-1.5,-1.75)$.