Find point \( Q \) on \( \overline{R S} \) that is \( \frac{5}{8} \) of the distance from \( R \) to \( S \).

To find point \( Q \) on line segment \( \overline{RS} \) that is \( \frac{5}{8} \) of the distance from point \( R \) to point \( S \), we can use the concept of dividing a segment in a given ratio. Let \( R \) and \( S \) be represented by their coordinates in a coordinate system. Assume: - \( R = (x_1, y_1) \) - \( S = (x_2, y_2) \) The distance from \( R \) to \( S \) can be expressed as: \[ \text{Distance} = S - R = (x_2 - x_1, y_2 - y_1) \] To find point \( Q \) that is \( \frac{5}{8} \) of the way from \( R \) to \( S \), we can use the section formula. The coordinates of point \( Q \) can be calculated as follows: \[ Q = R + \frac{5}{8}(S - R) \] Substituting the coordinates of \( R \) and \( S \): \[ Q = (x_1, y_1) + \frac{5}{8}((x_2 - x_1), (y_2 - y_1)) \] This simplifies to: \[ Q = \left( x_1 + \frac{5}{8}(x_2 - x_1), y_1 + \frac{5}{8}(y_2 - y_1) \right) \] Now, we can further simplify the coordinates of \( Q \): \[ Q = \left( x_1 + \frac{5}{8}x_2 - \frac{5}{8}x_1, y_1 + \frac{5}{8}y_2 - \frac{5}{8}y_1 \right) \] \[ Q = \left( \frac{3}{8}x_1 + \frac{5}{8}x_2, \frac{3}{8}y_1 + \frac{5}{8}y_2 \right) \] Thus, the coordinates of point \( Q \) that is \( \frac{5}{8} \) of the distance from \( R \) to \( S \) are: \[ Q = \left( \frac{3}{8}x_1 + \frac{5}{8}x_2, \frac{3}{8}y_1 + \frac{5}{8}y_2 \right) \] This gives us the exact location of point \( Q \) on the segment \( \overline{RS} \).