1. **State the problem:** We have a pentagon with vertices A, B, C, D, E and a center of dilation P. The scale factor is $k=3$. We want to find the image distances from P to each vertex after dilation. 2. **Formula for dilation:** The distance from P to each image point is given by $$\text{Image Distance} = k \times \text{Pre-image Distance}$$ 3. **Given:** - $k=3$ - $PA=0.5$ and image distance $PA' = 1.5$ (already calculated) 4. **Calculate image distances for other points:** Let $PB = x$, then image distance $PB' = 3x$ Similarly for $PC, PD, PE$: - $PC' = 3 \times PC$ - $PD' = 3 \times PD$ - $PE' = 3 \times PE$ 5. **Explanation:** Since dilation scales distances from the center P by the factor $k=3$, each image distance is three times the corresponding pre-image distance. 6. **Summary:** | Point | Pre-image Distance | Image Distance | |-------|--------------------|----------------| | PA | 0.5 | 1.5 | | PB | $PB$ | $3 \times PB$ | | PC | $PC$ | $3 \times PC$ | | PD | $PD$ | $3 \times PD$ | | PE | $PE$ | $3 \times PE$ | This completes the calculation for the dilation image distances.