Question

what is the scale factor of the dilation of triangle def?
○ \\(\frac{3}{10}\\)
○ \\(\frac{3}{7}\\)
○ \\(\frac{7}{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 original figure. Here, we use the segments from the center \( O \) to the vertices. For \( F' \) (image) and \( F \) (original), \( OF' = 3 \) and \( OF = 3 + 7 = 10 \)? Wait, no, wait. Wait, the segments: \( OF' = 3 \), \( F'F = 7 \), so \( OF = OF' + F'F = 3 + 7 = 10 \)? Wait, no, maybe I misread. Wait, the small triangle is \( D'E'F' \), large is \( DEF \). The center is \( O \). So \( OF' \) is 3, \( OF \) is \( 3 + 7 = 10 \)? No, wait, maybe the lengths from \( O \) to \( F' \) is 3, and from \( O \) to \( F \) is \( 3 + 7 = 10 \)? Wait, no, the problem is dilation of \( DEF \) (so \( DEF \) is the original, and \( D'E'F' \) is the image? Wait, no, dilation: if \( D'E'F' \) is the image of \( DEF \) under dilation, then the scale factor is \( \frac{OF'}{OF} \). Wait, let's check the diagram. \( O \) is the center. \( OF' = 3 \), \( OF = 3 + 7 = 10 \)? No, wait, maybe \( OF' = 3 \), \( OF = 7 \)? Wait, no, the labels: \( O \) to \( F' \) is 3, \( F' \) to \( F \) is 7. So \( OF = OF' + F'F = 3 + 7 = 10 \)? Wait, but the options include \( \frac{3}{10} \), \( \frac{3}{7} \), \( \frac{7}{3} \), 7. Wait, maybe I got the original and image reversed. If \( DEF \) is the original, and \( D'E'F' \) is the image, then the scale factor is \( \frac{\text{length of image segment}}{\text{length of original segment}} \). So if \( OF' \) is the image segment (from \( O \) to \( F' \)) and \( OF \) is the original segment (from \( O \) to \( F \)), then scale factor is \( \frac{OF'}{OF} \). But \( OF' = 3 \), \( OF = 3 + 7 = 10 \)? No, that would be \( \frac{3}{10} \), but that's an option. Wait, maybe the diagram is such that \( OF' = 3 \), \( OF = 7 \)? Wait, no, the labels: \( O \) to \( F' \) is 3, \( F' \) to \( F \) is 7, so \( OF = 3 + 7 = 10 \). Wait, but the options have \( \frac{3}{10} \), \( \frac{3}{7} \), \( \frac{7}{3} \), 7. Wait, maybe I have it reversed. If the dilation is of \( DEF \) to get \( D'E'F' \), then the scale factor is \( \frac{OF'}{OF} \). But if \( DEF \) is the image, and \( D'E'F' \) is the original, then scale factor is \( \frac{OF}{OF'} = \frac{7 + 3}{3} = \frac{10}{3} \), but that's not an option. Wait, maybe the lengths from \( O \) to \( F' \) is 3, and from \( O \) to \( F \) is 7. So \( OF' = 3 \), \( OF = 7 \). Then scale factor is \( \frac{3}{7} \)? No, that's an option. Wait, no, maybe the original is \( D'E'F' \) and the image is \( DEF \), so scale factor is \( \frac{OF}{OF'} = \frac{7}{3} \)? Wait, no, the problem says "dilation of triangle \( DEF \)". So \( DEF \) is the figure being dilated, so the image is smaller or larger? The small triangle is \( D'E'F' \), so if \( DEF \) is dilated to \( D'E'F' \), then it's a reduction, scale factor less than 1. So \( OF' \) is the image, \( OF \) is original. So \( OF' = 3 \), \( OF = 3 + 7 = 10 \)? No, that would be \( \frac{3}{10} \). Wait, the options: first option is \( \frac{3}{10} \). Let's check the options. The options are \( \frac{3}{10} \), \( \frac{3}{7} \), \( \frac{7}{3} \), 7. So if the scale factor is \( \frac{3}{10} \), that would be the case if \( OF' = 3 \) and \( OF = 10 \). But maybe the diagram has \( OF' = 3 \) and \( OF = 10 \) (since \( O \) to \( F' \) is 3, \( F' \) to \( F \) is 7, so \( O \) to \( F \) is 10). So the scale factor of dilation of \( DEF \) (so \( DEF \) is the original, and we dilate it to get \( D'E'F' \)) is \( \frac{OF'}{OF} = \frac{3}{10} \)? Wait, but let's confirm. Dilation scale factor: if the center is \( O \), then for any point \( P \) on the original figure, the image \( P' \) is on the line \( OP \), and \( OP' = k \cdot OP \), where \( k \) is the scale factor. So if \( F \) is on \( DEF \), \( F' \) is on \( D'E'F' \), then \( OF' = k \cdot OF \). So \( k = \frac{OF'}{OF} \). If \( OF' = 3 \) and \( OF = 3 + 7 = 10 \), then \( k = \frac{3}{10} \). So that's the scale factor.

Step2: Identify the Correct Ratio

So the scale factor is the ratio of the length from the center to the image vertex (\( OF' \)) to the length from the center to the original vertex (\( OF \)). Here, \( OF' = 3 \) and \( OF = 3 + 7 = 10 \), so the scale factor is \( \frac{3}{10} \). Wait, but let's check the options. The first option is \( \frac{3}{10} \), which matches.

Answer:

\(\boxed{\frac{3}{10}}\)