To find the slope of line \( q \), we first need to calculate the slope of line \( p \).
The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the given points \((92, 21)\) and \((4, 63)\):
\[ x_1 = 92, \quad y_1 = 21, \quad x_2 = 4, \quad y_2 = 63 \]
Substituting these values into the slope formula, we have:
\[ m_p = \frac{63 - 21}{4 - 92} \]
Calculating the differences in the numerator and denominator:
\[ 63 - 21 = 42 \] \[ 4 - 92 = -88 \]
Thus, the slope of line \( p \) is:
\[ m_p = \frac{42}{-88} \]
Simplifying this fraction:
\[ m_p = \frac{42 \div 2}{-88 \div 2} = \frac{21}{-44} = -\frac{21}{44} \]
Now, the slope of line \( q \), which is perpendicular to line \( p \), is the negative reciprocal of the slope of line \( p \):
\[ m_q = -\frac{1}{m_p} = -\frac{1}{-\frac{21}{44}} = \frac{44}{21} \]
So the slope of line \( q \) is:
\[ \boxed{\frac{44}{21}} \]