1. **State the problem:** We have a triangle $\triangle DEF$ with vertices $D(1,6)$, $E(2,-5)$, and $F(-4,-2)$. A dilation centered at the origin with scale factor 2 is applied, resulting in $\triangle D'E'F'$. 2. **Formula for dilation centered at the origin:** The dilation rule for a point $(x,y)$ with scale factor $k$ centered at the origin is: $$ (x,y) \to (kx, ky) $$ 3. **Apply the dilation to each vertex:** - For $D(1,6)$: $$ D' = (2 \times 1, 2 \times 6) = (2, 12) $$ - For $E(2,-5)$: $$ E' = (2 \times 2, 2 \times (-5)) = (4, -10) $$ - For $F(-4,-2)$: $$ F' = (2 \times (-4), 2 \times (-2)) = (-8, -4) $$ 4. **Choose the general rule:** From the options, the correct dilation mapping is: $$ (x,y) \to (2x, 2y) $$ because each coordinate is multiplied by 2. **Final answers:** - $D' = (2, 12)$ - $E' = (4, -10)$ - $F' = (-8, -4)$ and the dilation rule is $(x,y) \to (2x, 2y)$.