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

Explanation:

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

The original rectangle \(FGHI\). Let's consider the vertical side \(HI\). The \(y\) - coordinate of \(H\) is \(4\) and the \(y\) - coordinate of \(I\) is \(2\). Using the distance formula for a vertical line \(d=\vert y_2 - y_1\vert\), so \(HI=\vert4 - 2\vert=2\).

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

The dilated rectangle \(F'G'H'I'\). Consider the vertical side \(H'I'\). The \(y\) - coordinate of \(H'\) is \(8\) and the \(y\) - coordinate of \(I'\) is \(4\). Using the distance formula for a vertical line \(d = \vert y_2 - y_1\vert\), so \(H'I'=\vert8 - 4\vert = 4\).

Step3: Calculate the scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of side in dilated figure}}{\text{length of side in original figure}}\). Substituting the values we found: \(k=\frac{H'I'}{HI}\). Since \(H'I' = 4\) and \(HI=2\), then \(k=\frac{4}{2}=2\).

Answer:

\(2\)