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

The length of side \( JK \) in parallelogram \( JKLM \): The \( x -\)coordinate of \( J\) is \(- 2\) and the \( x -\)coordinate of \( K\) is \(8\). Using the distance formula for horizontal line \(d=\vert x_2 - x_1\vert\) (since \(y - \)coordinates are the same, \(y = 4\)), \(JK=\vert8-(-2)\vert = 10\)

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

The length of side \(J'K'\) in parallelogram \(J'K'L'M'\): The \(x -\)coordinate of \(J'\) is \(-1\) and the \(x -\)coordinate of \(K'\) is \(5\). Using the distance formula for horizontal line \(d=\vert x_2 - x_1\vert\) (since \(y - \)coordinates are the same, \(y = 2\)), \(J'K'=\vert5-(-1)\vert = 6\)

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side of image}}{\text{length of side of pre - image}}\). Here, \(k=\frac{J'K'}{JK}\)

Substitute \(J'K' = 6\) and \(JK = 10\) into the formula: \(k=\frac{6}{10}=\frac{3}{5}\)

Answer:

The scale factor of the dilation is \(\frac{3}{5}\)