Question

the rectangle efgh is a dilation of the rectangle efgh. 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 side of the original rectangle

The length of \(GH\) (original rectangle). The \(y -\)coordinate of \(G\) is \(8\) and of \(H\) is \(8\) (same \(y -\)coordinate), and the \(x -\)coordinate of \(G\) is \(0\) and of \(H\) is \(4\). Using the distance formula for horizontal line \(d=\vert x_2 - x_1\vert\), \(GH=\vert4 - 0\vert=4\).

Step2: Find the length of the corresponding side of the dilated rectangle

The length of \(G'H'\) (dilated rectangle). The \(y -\)coordinate of \(G'\) is \(2\) and of \(H'\) is \(2\) (same \(y -\)coordinate), and the \(x -\)coordinate of \(G'\) is \(0\) and of \(H'\) is \(1\). Using the distance formula for horizontal line \(d = \vert x_2 - x_1\vert\), \(G'H'=\vert1 - 0\vert = 1\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side of dilated figure}}{\text{length of side of original figure}}\). So \(k=\frac{G'H'}{GH}\). Substituting \(G'H' = 1\) and \(GH = 4\), we get \(k=\frac{1}{4}\).

Answer:

\(\frac{1}{4}\)