To determine the scale factor of the dilation, we'll compare the coordinates of the original point [tex]\( B \)[/tex] with the coordinates of its image [tex]\( B' \)[/tex].
1. Original coordinates of [tex]\( B \)[/tex]:
[tex]\[
B = (10, -15)
\][/tex]
2. Coordinates of the image [tex]\( B' \)[/tex]:
[tex]\[
B' = (2, -3)
\][/tex]
3. The scale factor of the dilation can be found by comparing the corresponding coordinates of [tex]\( B \)[/tex] and [tex]\( B' \)[/tex].
We need to look at the ratio of the x-coordinates:
[tex]\[
\text{Scale factor (x-coordinate)} = \frac{2}{10} = 0.2
\][/tex]
Similarly, we look at the ratio of the y-coordinates:
[tex]\[
\text{Scale factor (y-coordinate)} = \frac{-3}{-15} = 0.2
\][/tex]
4. Both the x and y coordinates give us the same scale factor, which is [tex]\( 0.2 \)[/tex].
5. We can see that the scale factor matches for both coordinates, so the uniform scale factor of dilation is [tex]\( 0.2 \)[/tex].
6. Converting [tex]\( 0.2 \)[/tex] to a fraction, we get:
[tex]\[
0.2 = \frac{1}{5}
\][/tex]
Thus, the scale factor of the dilation is [tex]\( \frac{1}{5} \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{1}{5}}
\][/tex]
