To determine the scale factor used in the dilation of line \( AB \) with center point at the origin, we can use the coordinates of the original points \( A \) and \( B \) and their dilated counterparts \( A' \) and \( B' \).
The coordinates are given as follows:
- Original point \( A(2, 2) \) dilates to \( A'(10, 10) \)
- Original point \( B(1, 1) \) dilates to \( B'(5, 5) \)
The scale factor \( k \) for a dilation can be calculated by the formula:
\[ k = \frac{ \text{coordinate of dilated point} }{ \text{coordinate of original point} } \]
Let's calculate the scale factor for point \( A \):
\[ k = \frac{10}{2} = 5 \]
Now, let's calculate the scale factor for point \( B \):
\[ k = \frac{5}{1} = 5 \]
Both points give us the same scale factor. Therefore, the scale factor used in the dilation of line segment \( AB \) is:
\[ \boxed{5} \]
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