1. **State the problem:** We have a line segment VW with length 26 units. It is dilated by a scale factor of $\frac{1}{4}$ to form a new segment V'W'. We need to find the length of V'W'. 2. **Formula for dilation:** When a segment is dilated by a scale factor $k$, the length of the new segment is given by: $$\text{New length} = k \times \text{Original length}$$ 3. **Apply the formula:** Here, the scale factor $k = \frac{1}{4}$ and the original length is 26. $$\text{V'W'} = \frac{1}{4} \times 26$$ 4. **Calculate:** $$\text{V'W'} = \frac{26}{4} = 6.5$$ 5. **Conclusion:** The length of the dilated segment V'W' is 6.5 units.