To determine if three points are collinear, the slopes between any two points must be identical. The slope between \( (5, 5) \) and \( (-5, 1) \) is calculated as:
\[
\text{slope} = \frac{1 - 5}{-5 - 5} = \frac{-4}{-10} = \frac{2}{5}
\]
The slope between \( (5, 5) \) and \( (10, k) \) is:
\[
\text{slope} = \frac{k - 5}{10 - 5} = \frac{k - 5}{5}
\]
For collinearity, equate the slopes:
\[
\frac{k - 5}{5} = \frac{2}{5}
\]
Solve for *k*:
\[
k - 5 = 2 \quad \Rightarrow \quad k = 7
\]
Thus, the solution is \( k = 7 \).