Question

the parallelogram ( bcde ) is a dilation of the parallelogram ( bcde ). 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 parallelograms

In parallelogram \(BCDE\), if we consider the horizontal side \(BC\). The \(y -\)coordinate of \(B\) and \(C\) is \(4\). The \(x -\)coordinate of \(B=- 3\) and \(x -\)coordinate of \(C = 3\). Using the distance formula for horizontal points \(d=\vert x_2 - x_1\vert\), so \(BC=\vert3-(-3)\vert = 6\).
In parallelogram \(B'C'D'E'\), for the horizontal side \(B'C'\). The \(y -\)coordinate of \(B'\) and \(C'\) is \(1\). The \(x -\)coordinate of \(B'=-1\) and \(x -\)coordinate of \(C' = 1\). Using the distance formula for horizontal points \(d=\vert x_2 - x_1\vert\), so \(B'C'=\vert1-(-1)\vert = 2\).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of side in pre - image}}\).
Here, the image is \(B'C'D'E'\) and the pre - image is \(BCDE\). So \(k=\frac{B'C'}{BC}\).
Substituting the values \(B'C' = 2\) and \(BC = 6\), we get \(k=\frac{2}{6}=\frac{1}{3}\).

Answer:

\(\frac{1}{3}\)