Find point \( Z \) on \( \overline{E F} \) such that the ratio of \( E Z \) to \( Z F \) is \( 2: 3 \).

To find point \( Z \) on \( \overline{E F} \) such that the ratio of \( E Z \) to \( Z F \) is \( 2: 3 \), we can use the section formula. The section formula states that for a line segment \( \overline{AB} \) with points \( A \) and \( B \), and a point \( P \) on the line segment such that the ratio of \( AP \) to \( PB \) is \( m:n \), the coordinates of point \( P \) can be calculated as: \[ P = \left( \frac{m \cdot x_B + n \cdot x_A}{m+n}, \frac{m \cdot y_B + n \cdot y_A}{m+n} \right) \] Given that the ratio of \( E Z \) to \( Z F \) is \( 2: 3 \), we can let \( m = 2 \) and \( n = 3 \). Let's denote the coordinates of points \( E \) and \( F \) as \( (x_E, y_E) \) and \( (x_F, y_F) \) respectively. Using the section formula, the coordinates of point \( Z \) can be calculated as: \[ Z = \left( \frac{2 \cdot x_F + 3 \cdot x_E}{2+3}, \frac{2 \cdot y_F + 3 \cdot y_E}{2+3} \right) \] Let's calculate the coordinates of point \( Z \) using the given information. Divide by following steps: - step0: Divide the numbers: \(\frac{\left(2x_{F}+3x_{E}\right)}{\left(2+3\right)}\) - step1: Calculate: \(\frac{2x_{F}+3x_{E}}{\left(2+3\right)}\) - step2: Calculate: \(\frac{2x_{F}+3x_{E}}{5}\) - step3: Calculate: \(\frac{2}{5}x_{F}+\frac{3}{5}x_{E}\) Calculate or simplify the expression \( (2*y_F + 3*y_E)/(2+3) \). Divide by following steps: - step0: Divide the numbers: \(\frac{\left(2y_{F}+3y_{E}\right)}{\left(2+3\right)}\) - step1: Calculate: \(\frac{2y_{F}+3y_{E}}{\left(2+3\right)}\) - step2: Calculate: \(\frac{2y_{F}+3y_{E}}{5}\) - step3: Calculate: \(\frac{2}{5}y_{F}+\frac{3}{5}y_{E}\) The coordinates of point \( Z \) on \( \overline{E F} \) such that the ratio of \( E Z \) to \( Z F \) is \( 2: 3 \) are: \[ Z = \left( \frac{2}{5}x_{F} + \frac{3}{5}x_{E}, \frac{2}{5}y_{F} + \frac{3}{5}y_{E} \right) \] These are the coordinates of point \( Z \) in terms of the coordinates of points \( E \) and \( F \).