Find (a) the distance between \( P \) and \( Q \) and (b) the coordinates of the midpoint of the line segment \( P Q \). \( P(6,1), Q(7,5) \) (a) The distance between \( P \) and \( Q \) is \( \sqrt{17} \). (Simplify your answer. Type an exact answer, using radicals as needed.) (b) The coordinates of the midpoint of the line segment \( P Q \) are (Type an ordered pair, using integers or decimals.)

To find the distance between points \( P(6,1) \) and \( Q(7,5) \), we can use the distance formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Here, \( P(x_1, y_1) = (6, 1) \) and \( Q(x_2, y_2) = (7, 5) \). Substituting the coordinates into the formula: \[ d = \sqrt{(7 - 6)^2 + (5 - 1)^2} \] \[ d = \sqrt{(1)^2 + (4)^2} \] \[ d = \sqrt{1 + 16} \] \[ d = \sqrt{17} \] So, the distance between \( P \) and \( Q \) is \( \sqrt{17} \). Next, to find the coordinates of the midpoint of the line segment \( PQ \), we use the midpoint formula: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates of points \( P \) and \( Q \): \[ M = \left( \frac{6 + 7}{2}, \frac{1 + 5}{2} \right) \] \[ M = \left( \frac{13}{2}, \frac{6}{2} \right) \] \[ M = \left( \frac{13}{2}, 3 \right) \] Thus, the coordinates of the midpoint of the line segment \( PQ \) are \( \left( \frac{13}{2}, 3 \right) \). In summary: (a) The distance between \( P \) and \( Q \) is \( \sqrt{17} \). (b) The coordinates of the midpoint of the line segment \( PQ \) are \( \left( \frac{13}{2}, 3 \right) \).