Question

the rectangle fghi is a dilation of the rectangle fghi. 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 \(FGHI\), let's consider the vertical side \(IF\). The \(y\) - coordinate of \(I\) is \(2\) and the \(y\) - coordinate of \(F\) is \(2\) (same \(x\) - coordinate \(x = 3\)). The length of \(IF=2 - 2= 2\) (using the distance formula \(d=\vert y_2 - y_1\vert\) for vertical lines \(x = x_1=x_2\)).
For rectangle \(F'G'H'I'\), consider the vertical side \(I'F'\). The \(y\) - coordinate of \(I'\) is \(4\) and the \(y\) - coordinate of \(F'\) is \(4\) (same \(x\) - coordinate \(x = 7\)). The length of \(I'F'=8 - 4 = 4\) (using the distance formula \(d=\vert y_2 - y_1\vert\) for vertical lines \(x = x_1=x_2\)).

Step2: Calculate the scale factor

The scale factor \(k\) of a dilation is given by the formula \(k=\frac{\text{length of side in image}}{\text{length of corresponding side in pre - image}}\).
If the pre - image is \(FGHI\) and the image is \(F'G'H'I'\), then \(k=\frac{I'F'}{IF}\).
Substitute \(I'F' = 4\) and \(IF = 2\) into the formula: \(k=\frac{4}{2}\)

Answer:

\(2\)