Find point \( D \) on \( \overline{A B} \) that is \( \frac{3}{4} \) of the distance from \( A \) to \( B \).

To find point \( D \) on line segment \( \overline{AB} \) that is \( \frac{3}{4} \) of the distance from point \( A \) to point \( B \), we can use the concept of dividing a segment in a given ratio. Let the coordinates of point \( A \) be \( (x_A, y_A) \) and the coordinates of point \( B \) be \( (x_B, y_B) \). The distance from \( A \) to \( B \) can be represented as: \[ \text{Distance} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} \] To find point \( D \), which is \( \frac{3}{4} \) of the way from \( A \) to \( B \), we can use the section formula. The coordinates of point \( D \) can be calculated as follows: \[ D = \left( \frac{3x_B + 1x_A}{3 + 1}, \frac{3y_B + 1y_A}{3 + 1} \right) \] This formula is derived from the fact that point \( D \) divides the segment \( \overline{AB} \) in the ratio \( 3:1 \). Now, substituting the values into the formula: \[ D = \left( \frac{3x_B + x_A}{4}, \frac{3y_B + y_A}{4} \right) \] Thus, the coordinates of point \( D \) are: \[ D = \left( \frac{3x_B + x_A}{4}, \frac{3y_B + y_A}{4} \right) \] This gives us the point \( D \) that is \( \frac{3}{4} \) of the distance from \( A \) to \( B \).