Question

what is the scale factor of the dilation of triangle def? 7 \frac{7}{3} \frac{3}{10} \frac{3}{7}

Explanation:

Step1: Recall Dilation Scale Factor

The scale factor of a dilation is the ratio of the length of a side (or segment from the center) of the image to the corresponding length of the pre - image. Here, we look at the segments from the center of dilation \(O\). For the pre - image (smaller triangle \(D'E'F'\)) and the image (larger triangle \(DEF\)), we consider the distances from \(O\) to \(F'\) and from \(O\) to \(F\). The length \(OF' = 3\) and \(OF=3 + 7=10\)? Wait, no, wait. Wait, actually, if \(D'E'F'\) is the image of \(DEF\) under dilation? Wait, no, looking at the diagram, \(D'E'F'\) is the smaller triangle, \(DEF\) is the larger one? Wait, no, the labels: \(D'\), \(E'\), \(F'\) are the vertices of the smaller triangle, \(D\), \(E\), \(F\) are the vertices of the larger triangle. Wait, no, dilation: the scale factor is \(\frac{\text{length of image segment}}{\text{length of pre - image segment}}\). Wait, actually, from the center \(O\), the distance to \(F'\) is \(3\), and the distance to \(F\) is \(3 + 7=10\)? No, wait, the diagram shows \(OF' = 3\) and \(F'F=7\), so \(OF=OF'+F'F = 3 + 7 = 10\)? Wait, no, maybe I got it wrong. Wait, maybe the pre - image is \(D'E'F'\) and the image is \(DEF\), so the scale factor is \(\frac{OF}{OF'}=\frac{3 + 7}{3}=\frac{10}{3}\)? No, that can't be. Wait, no, maybe the other way. Wait, maybe \(DEF\) is the pre - image and \(D'E'F'\) is the image. Wait, the question is "What is the scale factor of the dilation of triangle \(DEF\)?" So we are dilating \(DEF\) to get \(D'E'F'\) or \(D'E'F'\) to get \(DEF\)? Wait, the arrows: the lines go from \(O\) through \(D'\) to \(D\), \(E'\) to \(E\), \(F'\) to \(F\). So \(D'E'F'\) is the image of \(DEF\) under a dilation with center \(O\)? No, wait, \(D'\) is closer to \(O\) than \(D\), so \(D'E'F'\) is a reduction or enlargement? Wait, \(OF' = 3\), \(OF=3 + 7 = 10\)? No, wait, maybe the length from \(O\) to \(F'\) is \(3\) and from \(O\) to \(F\) is \(3+7 = 10\)? No, that doesn't match the options. Wait, maybe I misread the diagram. Wait, the diagram has \(OF' = 3\) and \(OF = 3+7=10\)? No, the options are \(7\), \(\frac{7}{3}\), \(\frac{3}{10}\), \(\frac{3}{7}\). Wait, maybe the segments are \(OF' = 3\) and \(OF = 7\)? Wait, no, the diagram shows \(3\) next to \(OF'\) and \(7\) next to \(F'F\). Wait, maybe the distance from \(O\) to \(F'\) is \(3\) and from \(O\) to \(F\) is \(7\)? No, that would mean \(F'F=7 - 3=4\), which is not shown. Wait, maybe the scale factor is \(\frac{OF'}{OF}\) if \(DEF\) is the pre - image and \(D'E'F'\) is the image. Wait, if \(OF = 7\) and \(OF' = 3\), then the scale factor is \(\frac{3}{7}\). Ah! That must be it. I misread the diagram. So \(OF = 7\) (the length from \(O\) to \(F\)) and \(OF' = 3\) (the length from \(O\) to \(F'\)). So when we dilate triangle \(DEF\) (pre - image) to get \(D'E'F'\) (image), the scale factor is \(\frac{\text{length of image segment}}{\text{length of pre - image segment}}=\frac{OF'}{OF}=\frac{3}{7}\).

Step2: Verify with the Options

The options are \(7\), \(\frac{7}{3}\), \(\frac{3}{10}\), \(\frac{3}{7}\). Since the scale factor of dilation of \(DEF\) (to get \(D'E'F'\)) is the ratio of the distance from the center to the image vertex (\(OF'\)) to the distance from the center to the pre - image vertex (\(OF\)), which is \(\frac{3}{7}\).

Answer:

\(\frac{3}{7}\)