To determine the side length \( RS \) of the original square RSTU, we need to consider the dilation transformation and its effects on the geometric figure. Let's go through the steps to solve this problem:
1. Understanding Dilation: When a geometric figure is dilated with respect to a point (in this case, the origin) by a factor of \( \frac{1}{2} \), every point on the figure moves closer to the origin by this factor. The side length of the figure gets scaled accordingly.
2. Given Information:
- The side length of the dilated square R'S'TU' is given as \( R'S' = 2 \) units.
- The dilation factor is \( \frac{1}{2} \).
3. Finding the Original Side Length:
- Let the original side length of the square RSTU be \( RS \).
- Under the dilation transformation with a factor of \( \frac{1}{2} \), the new side lengths are scaled to half their original size.
- Therefore, the new side length, \( R'S' \), is given by:
[tex]\[
R'S' = \left(\frac{1}{2}\right) \times RS
\][/tex]
- We are given \( R'S' = 2 \) units, so we can set up the equation:
[tex]\[
2 = \left(\frac{1}{2}\right) \times RS
\][/tex]
4. Solving for RS:
- To solve for \( RS \), we will multiply both sides of the equation by 2 (the reciprocal of the dilation factor):
[tex]\[
2 \times 2 = RS
\][/tex]
This simplifies to:
[tex]\[
RS = 4
\][/tex]
Therefore, the side length \( RS \) of the original square RSTU is 4 units.
So, the correct answer is:
C. 4 units
