1. **State the problem:** We have quadrilateral ABCD with vertices A(3,5), B(5,7), C(8,2), and D(4,-1). It is dilated from the origin to quadrilateral A'B'C'D' with vertices A'(12,20), B'(20,28), C'(32,8), and D'(16,-4). We need to find the scale factor of the dilation. 2. **Recall the dilation formula:** When a figure is dilated from the origin by scale factor $k$, each vertex $(x,y)$ maps to $(kx, ky)$. 3. **Apply the formula to vertex A:** Original A is $(3,5)$, image A' is $(12,20)$. Using the formula: $$ (12,20) = (k \times 3, k \times 5) $$ 4. **Solve for $k$ using x-coordinates:** $$ 12 = 3k \implies k = \frac{12}{3} = 4 $$ 5. **Verify $k$ using y-coordinates:** $$ 20 = 5k \implies k = \frac{20}{5} = 4 $$ 6. **Check other vertices for consistency:** - For B: $(5,7) \to (20,28)$, $k = \frac{20}{5} = 4$ and $k = \frac{28}{7} = 4$ - For C: $(8,2) \to (32,8)$, $k = \frac{32}{8} = 4$ and $k = \frac{8}{2} = 4$ - For D: $(4,-1) \to (16,-4)$, $k = \frac{16}{4} = 4$ and $k = \frac{-4}{-1} = 4$ All vertices confirm the scale factor $k=4$. **Final answer:** The scale factor of the dilation is **4**.