1. The problem states that triangle $\triangle ABC$ is dilated from point $A$ by a scale factor of $\frac{1}{2}$. We need to find the relationship between segment $D'E'$ (the image of $DE$ after dilation) and the original segment $DE$. 2. The formula for dilation states that when a figure is dilated from a point by a scale factor $k$, the lengths of all segments are multiplied by $k$. That is: $$\text{Length of image segment} = k \times \text{Length of original segment}$$ 3. Here, the scale factor $k = \frac{1}{2}$. Therefore: $$D'E' = \frac{1}{2} DE$$ 4. This means the segment $D'E'$ is half the length of the original segment $DE$ after dilation from point $A$ by scale factor $\frac{1}{2}$. 5. Hence, the correct equation is: $$D'E' = \frac{1}{2} DE$$