ind point \( C \) on \( \overline{A B} \) that is \( \frac{1}{5} \) of the distance from \( A \) to \( B \).

To find point \( C \) on line segment \( \overline{AB} \) that is \( \frac{1}{5} \) of the distance from \( A \) to \( B \), we can use the concept of dividing a line segment in a given ratio. Let's denote the coordinates of points \( A \) and \( B \) as follows: - Let \( A = (x_A, y_A) \) - Let \( B = (x_B, y_B) \) The distance from \( A \) to \( B \) can be represented as a vector: \[ \overrightarrow{AB} = (x_B - x_A, y_B - y_A) \] To find point \( C \), which is \( \frac{1}{5} \) of the way from \( A \) to \( B \), we can calculate the coordinates of \( C \) using the formula for dividing a segment in a given ratio. Since \( C \) is \( \frac{1}{5} \) of the way from \( A \) to \( B \), we can express the coordinates of \( C \) as: \[ C = A + \frac{1}{5} \overrightarrow{AB} \] Substituting the vector \( \overrightarrow{AB} \): \[ C = A + \frac{1}{5} (B - A) \] \[ C = A + \frac{1}{5} (x_B - x_A, y_B - y_A) \] \[ C = (x_A, y_A) + \left(\frac{1}{5}(x_B - x_A), \frac{1}{5}(y_B - y_A)\right) \] \[ C = \left(x_A + \frac{1}{5}(x_B - x_A), y_A + \frac{1}{5}(y_B - y_A)\right) \] Thus, the coordinates of point \( C \) are: \[ C = \left(x_A + \frac{1}{5}(x_B - x_A), y_A + \frac{1}{5}(y_B - y_A)\right) \] This gives you the exact position of point \( C \) on the line segment \( \overline{AB} \) that is \( \frac{1}{5} \) of the distance from \( A \) to \( B \).