Directed line segment \( K C \) has endpoints \( K(-4,-2) \) and \( C(1,8) \). Point \( E \) divides \( \overline{K C} \) such that \( K E: E C \) is \( 3: 2 \). What are the coordinates of point \( E \) ? \( \begin{array}{ll}\text { (1) }(-1,4) & \text { (3) }(-3,0) \\ \text { (2) }(-2,2) & \text { (4) }(0,6)\end{array} \)

To find the coordinates of point \( E \) that divides the directed line segment \( \overline{K C} \) in the ratio \( K E: E C = 3:2 \), we can use the section formula. The section formula states that if a point \( E \) divides the line segment joining points \( K(x_1, y_1) \) and \( C(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( E \) are given by: \[ E\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \] In this case, we have: - \( K(-4, -2) \) with coordinates \( (x_1, y_1) = (-4, -2) \) - \( C(1, 8) \) with coordinates \( (x_2, y_2) = (1, 8) \) - The ratio \( m:n = 3:2 \) (where \( m = 3 \) and \( n = 2 \)) Now we can substitute these values into the section formula: 1. Calculate the x-coordinate of \( E \): \[ E_x = \frac{3 \cdot 1 + 2 \cdot (-4)}{3 + 2} = \frac{3 - 8}{5} = \frac{-5}{5} = -1 \] 2. Calculate the y-coordinate of \( E \): \[ E_y = \frac{3 \cdot 8 + 2 \cdot (-2)}{3 + 2} = \frac{24 - 4}{5} = \frac{20}{5} = 4 \] Thus, the coordinates of point \( E \) are \( (-1, 4) \). The answer is \( \boxed{(-1, 4)} \).