Step-by-step Solution:

To find the foot of the perpendicular from the point \( P(2,4) \) to the line \( x + y = 1 \), we use the formula: \[ \left( \frac{x - a}{m} = \frac{y - b}{-1} \right) \] The given line equation is \( x + y = 1 \), so its slope is \( m = -1 \). \[\] The perpendicular line will have a slope that is the negative reciprocal, i.e., \( 1 \). \[\] The equation of the perpendicular line through \( (2,4) \) with slope 1 is: \[\] \[ y - 4 = 1(x - 2) \] \[ y = x + 2 \] Now, solve for the intersection of the lines: \[ x + y = 1 \] \[ y = x + 2 \] Substituting \( y = x + 2 \) in \( x + y = 1 \): \[ x + (x + 2) = 1 \] \[ 2x + 2 = 1 \] \[ 2x = -1 \] \[ x = -\frac{1}{2} \] Find \( y \): \[ y = x + 2 = -\frac{1}{2} + 2 = \frac{3}{2} \] Thus, the foot of the perpendicular is: \[ {\left( -\frac{1}{2}, \frac{3}{2} \right)} \]