Question

1. a figure and its dilation are shown. maria thinks the scale factor is $\frac{1}{3}$. marsha thinks the scale factor is $\frac{5}{6}$. who is correct and why?

Explanation:

Step1: Recall the formula for scale factor

The scale factor \(k\) of a dilation is given by \(k=\frac{\text{length of a side in the image}}{\text{length of the corresponding side in the pre - image}}\)

Step2: Calculate the scale factor using side \(GD\) and \(D'G'\)

The length of \(GD = 12\space\text{cm}\) and the length of \(D'G'=10\space\text{cm}\). Then \(k_1=\frac{D'G'}{GD}=\frac{10}{12}=\frac{5}{6}\)

Step3: Calculate the scale factor using side \(DE\) and \(D'E'\)

The length of \(DE = 30\space\text{cm}\) and the length of \(D'E' = 25\space\text{cm}\). Then \(k_2=\frac{D'E'}{DE}=\frac{25}{30}=\frac{5}{6}\)

Answer:

Marsha is correct. The scale factor \(k=\frac{\text{length of a side in the image}}{\text{length of the corresponding side in the pre - image}}\). Using \(GD = 12\space\text{cm}\) (pre - image) and \(D'G'=10\space\text{cm}\) (image), \(k=\frac{10}{12}=\frac{5}{6}\); using \(DE = 30\space\text{cm}\) (pre - image) and \(D'E'=25\space\text{cm}\) (image), \(k=\frac{25}{30}=\frac{5}{6}\). So the scale factor is \(\frac{5}{6}\)