Question

the rectangle below is the image of a rectangle that was dilated by a scale factor of 3. find the perimeter of the preimage, the original rectangle, before its dilation. figures are not necessarily drawn to scale.

Explanation:

Step1: Calculate the perimeter of the dilated rectangle

The formula for the perimeter of a rectangle is \(P = 2(l + w)\). For the dilated rectangle with length \(l = 30\) and width \(w = 24\), we have:
\(P_{dilated}=2(30 + 24)\)
\(=2\times54\)
\( = 108\)

Step2: Relate the perimeter of the dilated and pre - image rectangles

When a figure is dilated by a scale factor \(k\), the ratio of the perimeters of the dilated figure (\(P_{dilated}\)) and the pre - image figure (\(P_{pre - image}\)) is equal to the scale factor \(k\). Here \(k = 3\), and \(P_{dilated}=k\times P_{pre - image}\). So, \(P_{pre - image}=\frac{P_{dilated}}{k}\)

Substitute \(P_{dilated}=108\) and \(k = 3\) into the formula:
\(P_{pre - image}=\frac{108}{3}=36\)

Answer:

\(36\)