Question

the parallelogram ( jklm ) is a dilation of the parallelogram ( jklm ). what is the scale factor of the dilation?

Explanation:

Step1: Find the length of a side of the original parallelogram

For parallelogram \(JKLM\), let's consider the horizontal side \(JK\). The \(x -\)coordinate of \(J\) is \(- 2\) and the \(x -\)coordinate of \(K\) is \(8\). Using the distance formula for a horizontal line \(d=\vert x_2 - x_1\vert\), we have \(JK=\vert8-(-2)\vert = 10\).

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

For parallelogram \(J'K'L'M'\), consider the horizontal side \(J'K'\). The \(x -\)coordinate of \(J'\) is \(0\) and the \(x -\)coordinate of \(K'\) is \(4\). Using the distance formula for a horizontal line \(d = \vert x_2 - x_1\vert\), we have \(J'K'=\vert4 - 0\vert=4\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\). So \(k=\frac{J'K'}{JK}\). Substituting \(J'K' = 4\) and \(JK = 10\), we get \(k=\frac{4}{10}=\frac{2}{5}\).

Answer:

The scale factor of the dilation is \(\frac{2}{5}\)