1. **Problem Statement:** We need to find a sequence of rigid transformations and dilations that maps square ABCD (side length 5) to square EFGH (side length 2). 2. **Understanding the problem:** Square ABCD has side length 5, and square EFGH has side length 2. To map ABCD to EFGH, we need to scale down by a factor of $\frac{2}{5}$, possibly combined with translations and rotations. 3. **Step 1: Translation** Translate square ABCD by the directed line segment $\overrightarrow{AE}$ to move point A to point E. This moves the entire square accordingly. 4. **Step 2: Rotation** Rotate the translated square about point E by angle $\angle B'EF$ to align the sides properly with square EFGH. 5. **Step 3: Dilation** Dilate the rotated square about point E by scale factor $\frac{2}{5}$ to reduce the side length from 5 to 2, matching square EFGH. 6. **Conclusion:** The correct sequence is: translate by $\overrightarrow{AE}$, rotate about E by $\angle B'EF$, then dilate about E by scale factor $\frac{2}{5}$. This corresponds to option B. **Final answer:** Option B.