To find the slope of line \( u \), which is perpendicular to line \( t \), we first need to determine the slope of line \( t \).
The slope \( m \) of a line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Given the points \((8, 8)\) and \((10, 3)\), we assign:
- \((x_1, y_1) = (8, 8)\)
- \((x_2, y_2) = (10, 3)\)
Now, we substitute these values into the slope formula:
\[ m_t = \frac{3 - 8}{10 - 8} = \frac{-5}{2} \]
The slope of line \( t \) is \( -\frac{5}{2} \).
Next, to find the slope of line \( u \) which is perpendicular to line \( t \), we use the property that the slopes of two perpendicular lines are negative reciprocals of each other. If the slope of line \( t \) is \( m_t \), then the slope of line \( u \), denoted as \( m_u \), is given by:
\[ m_u = -\frac{1}{m_t} \]
Substituting \( m_t = -\frac{5}{2} \):
\[ m_u = -\frac{1}{-\frac{5}{2}} = \frac{2}{5} \]
Thus, the slope of line \( u \) is:
\[ \boxed{\frac{2}{5}} \]