Step-by-step Solution:

The midpoints of the diagonals in a parallelogram coincide, so we have: \[ \frac{P + R}{2} = \frac{Q + S}{2} \] Substituting the coordinates of points \( P(1, 2) \), \( R(5, 7) \), \( Q(4, 6) \), and \( S(a, b) \), we get: \[ \left( \frac{1 + 5}{2}, \frac{2 + 7}{2} \right) = \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) \] Simplifying both sides: \[ \left( \frac{6}{2}, \frac{9}{2} \right) = \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) \] \[ (3, 4.5) = \left( \frac{4 + a}{2}, \frac{6 + b}{2} \right) \] Now, equating the corresponding coordinates: \[ \frac{4 + a}{2} = 3 \quad \Rightarrow \quad 4 + a = 6 \quad \Rightarrow \quad a = 2 \] \[ \frac{6 + b}{2} = 4.5 \quad \Rightarrow \quad 6 + b = 9 \quad \Rightarrow \quad b = 3 \] Thus, the values are: \[ a = 2, \quad b = 3 \]