95. \( (\sqrt{2}, 3 \sqrt{5}),(\sqrt{2},-2 \sqrt{5}) \)

To analyze the points \( (\sqrt{2}, 3 \sqrt{5}) \) and \( (\sqrt{2}, -2 \sqrt{5}) \), we can start by identifying their coordinates: 1. The first point is \( A = (\sqrt{2}, 3\sqrt{5}) \). 2. The second point is \( B = (\sqrt{2}, -2\sqrt{5}) \). ### Step 1: Determine the distance between the two points The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in a Cartesian plane is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Substituting the coordinates of points \( A \) and \( B \): - \( x_1 = \sqrt{2} \), \( y_1 = 3\sqrt{5} \) - \( x_2 = \sqrt{2} \), \( y_2 = -2\sqrt{5} \) Calculating the differences: \[ x_2 - x_1 = \sqrt{2} - \sqrt{2} = 0 \] \[ y_2 - y_1 = -2\sqrt{5} - 3\sqrt{5} = -5\sqrt{5} \] Now substituting these into the distance formula: \[ d = \sqrt{(0)^2 + (-5\sqrt{5})^2} = \sqrt{0 + 25 \cdot 5} = \sqrt{125} = 5\sqrt{5} \] ### Step 2: Determine the midpoint of the two points The midpoint \( M \) of two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \] Substituting the coordinates of points \( A \) and \( B \): \[ M = \left( \frac{\sqrt{2} + \sqrt{2}}{2}, \frac{3\sqrt{5} + (-2\sqrt{5})}{2} \right) = \left( \frac{2\sqrt{2}}{2}, \frac{3\sqrt{5} - 2\sqrt{5}}{2} \right) \] This simplifies to: \[ M = \left( \sqrt{2}, \frac{\sqrt{5}}{2} \right) \] ### Summary - The distance between the points \( (\sqrt{2}, 3\sqrt{5}) \) and \( (\sqrt{2}, -2\sqrt{5}) \) is \( 5\sqrt{5} \). - The midpoint of the two points is \( \left( \sqrt{2}, \frac{\sqrt{5}}{2} \right) \).