1. The problem states that a figure on the left was transformed to the figure on the right using rotation and dilation, and we need to find the length of segment $r$. 2. Rotation changes the orientation of a figure but does not change the size or lengths of segments. 3. Dilation changes the size of the figure by a scale factor $k$, multiplying all lengths by $k$. 4. To find the length of $r$ after dilation, we use the formula: $$r_{new} = k \times r_{original}$$ 5. We need the original length of $r$ and the scale factor $k$ from the figure or problem data. 6. Since the problem does not provide numeric values or the scale factor, we cannot calculate a numeric answer. 7. If you provide the original length of $r$ and the scale factor, we can compute the new length using the formula above.