Find point \( W \) on \( \overline{U V} \) that is \( \frac{1}{7} \) of the distance from \( U \) to \( V \).

To find point \( W \) on line segment \( \overline{UV} \) that is \( \frac{1}{7} \) of the distance from point \( U \) to point \( V \), we can use the concept of dividing a line segment in a given ratio. Let the coordinates of points \( U \) and \( V \) be represented as follows: - Let \( U = (x_U, y_U) \) - Let \( V = (x_V, y_V) \) The distance from \( U \) to \( V \) can be expressed as: \[ \text{Distance} = \sqrt{(x_V - x_U)^2 + (y_V - y_U)^2} \] To find point \( W \) that is \( \frac{1}{7} \) of the distance from \( U \) to \( V \), we can use the section formula. The coordinates of point \( W \) can be calculated using the formula for dividing a line segment in the ratio \( m:n \), where \( m \) is the part from \( U \) and \( n \) is the part from \( V \). In this case, we want \( W \) to be \( \frac{1}{7} \) of the way from \( U \) to \( V \). This means that the ratio \( m:n \) is \( 1:6 \) (since \( 1 + 6 = 7 \)). Using the section formula, the coordinates of point \( W \) are given by: \[ W = \left( \frac{mx_V + nx_U}{m+n}, \frac{my_V + ny_U}{m+n} \right) \] Substituting \( m = 1 \) and \( n = 6 \): \[ W = \left( \frac{1 \cdot x_V + 6 \cdot x_U}{1 + 6}, \frac{1 \cdot y_V + 6 \cdot y_U}{1 + 6} \right) \] This simplifies to: \[ W = \left( \frac{x_V + 6x_U}{7}, \frac{y_V + 6y_U}{7} \right) \] Thus, the coordinates of point \( W \) that is \( \frac{1}{7} \) of the distance from \( U \) to \( V \) are: \[ W = \left( \frac{x_V + 6x_U}{7}, \frac{y_V + 6y_U}{7} \right) \]