Question

the rectangle pqrs is a dilation of the rectangle pqrs. what is the scale factor of the dilation?
simplify your answer and write it as a proper fraction, an improper fraction, or a whole number.

Explanation:

Step1: Find the length of a corresponding side in both rectangles

For rectangle \(PQRS\), let's take side \(PS\). The \(x -\)coordinates of \(P(2,1)\) and \(S(2, - 1)\) (assuming from the grid - like structure, if \(P\) is at \((2,1)\) and \(S\) is at \((2,-1)\)), the length \(PS=\vert1-(-1)\vert = 2\). For rectangle \(P'Q'R'S'\), if \(P'(8, - 4)\) and \(S'(8,4)\), the length \(P'S'=\vert-4 - 4\vert=8\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\). Using the side lengths from Step1, \(k = \frac{P'S'}{PS}\). Substituting the values, \(k=\frac{8}{2}=4\).

Answer:

\(4\)