A rectangle with an area of 32 units \( ^{2} \) is the image of a rectangle that was dilated by a scale factor of \( \frac{5}{4} \). Find the area of the preimage, the original rectangle, before its dilation. Round your answer to the nearest tenth, if necessary.

To find the area of the original rectangle before dilation, we can use the properties of dilation and how it affects area. **Step 1: Understand the Relationship Between Scale Factor and Area** When a figure is dilated by a scale factor \( k \), its area is multiplied by \( k^2 \). Mathematically, this relationship is expressed as: \[ \text{Area}_{\text{image}} = k^2 \times \text{Area}_{\text{original}} \] **Step 2: Plug in the Given Values** You're given: - The area of the image rectangle: \( 32 \) units\(^2\) - The scale factor: \( \frac{5}{4} \) Using the formula: \[ 32 = \left(\frac{5}{4}\right)^2 \times \text{Area}_{\text{original}} \] **Step 3: Calculate the Scale Factor Squared** \[ \left(\frac{5}{4}\right)^2 = \frac{25}{16} \] **Step 4: Solve for the Original Area** \[ 32 = \frac{25}{16} \times \text{Area}_{\text{original}} \] \[ \text{Area}_{\text{original}} = 32 \times \frac{16}{25} = \frac{512}{25} = 20.48 \] **Step 5: Round to the Nearest Tenth** \[ \text{Area}_{\text{original}} \approx 20.5 \text{ units}^2 \] **Answer:** The area of the original rectangle is approximately 20.5 units².