1. **State the problem:** A rectangle is dilated by a scale factor of $\frac{1}{2}$ to produce a new rectangle. We need to find the relationship between the perimeter of the original rectangle and the perimeter of the new rectangle. 2. **Recall the formula for perimeter of a rectangle:** $$P = 2(l + w)$$ where $l$ is the length and $w$ is the width. 3. **Effect of dilation on dimensions:** When a figure is dilated by a scale factor $k$, all linear dimensions (length and width) are multiplied by $k$. 4. **Calculate the new perimeter:** If the original length is $l$ and width is $w$, the new length is $\frac{1}{2}l$ and the new width is $\frac{1}{2}w$. The new perimeter $P_{new}$ is: $$P_{new} = 2\left(\frac{1}{2}l + \frac{1}{2}w\right) = 2 \times \frac{1}{2}(l + w)$$ 5. **Simplify the expression:** $$P_{new} = \cancel{2} \times \frac{1}{2}(l + w) = \cancel{\frac{2}{2}}(l + w) = 1 \times (l + w)$$ 6. **Compare with original perimeter:** Original perimeter is $P = 2(l + w)$. So, $$\frac{P_{new}}{P} = \frac{1 \times (l + w)}{2(l + w)} = \frac{1}{2}$$ 7. **Conclusion:** The perimeter of the new rectangle is $\frac{1}{2}$ that of the original rectangle.