1. **Problem statement:** Find the image of the curve given by the equation $$x + 2y = 2$$ after a dilation centered at the origin $(0,0)$ with a scale factor of $\frac{1}{2}$. 2. **Formula and rules:** Dilation centered at the origin with scale factor $k$ transforms any point $(x, y)$ to $(kx, ky)$. 3. **Apply dilation to the curve:** The original curve is $$x + 2y = 2$$. After dilation, each point $(x, y)$ becomes $(x', y') = \left(\frac{1}{2}x, \frac{1}{2}y\right)$. 4. **Express original variables in terms of new variables:** Since $$x' = \frac{1}{2}x \Rightarrow x = 2x'$$ and $$y' = \frac{1}{2}y \Rightarrow y = 2y'$$. 5. **Substitute into original equation:** $$x + 2y = 2$$ becomes $$2x' + 2(2y') = 2$$ $$2x' + 4y' = 2$$ 6. **Simplify the equation:** Divide both sides by 2: $$x' + 2y' = 1$$ 7. **Final answer:** The image of the curve after dilation is $$x + 2y = 1$$ (using $x', y'$ as $x, y$ for simplicity). **Answer choice:** None of the options exactly match $x + 2y = 1$, but checking the options, option B is $2x + y = 2$, which is different. Re-examining the options, the closest correct transformed equation is option B: $2x + y = 2$. Let's verify if option B matches the transformation: If we consider the original equation $x + 2y = 2$ and apply dilation with factor $\frac{1}{2}$, the transformed equation is $x + 2y = 1$. Since none of the options match exactly, the correct transformed equation is $x + 2y = 1$. Therefore, the correct answer is not listed among the options. However, based on the problem statement and options, the closest is option B: $2x + y = 2$. **Summary:** - Original curve: $$x + 2y = 2$$ - After dilation by $\frac{1}{2}$: $$x + 2y = 1$$ Hence, the answer is $$x + 2y = 1$$.