\( (04.03 \mathrm{MC}) \) Point B on a segment with endpoints \( A(2,-2) \) and \( C(4,1) \) partitions the segment in a \( 1: 3 \) ratio. Find \( B \). (2.5,-1.25) (0.5, 0.75) \( (0.1 .25,2.5) \)

To find the coordinates of point \( B \) that partitions the segment \( AC \) in a \( 1:3 \) ratio, we can use the section formula. The section formula states that if a point \( B(x, y) \) divides the line segment joining points \( A(x_1, y_1) \) and \( C(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( B \) can be calculated as follows: \[ B_x = \frac{m \cdot x_2 + n \cdot x_1}{m+n} \] \[ B_y = \frac{m \cdot y_2 + n \cdot y_1}{m+n} \] In this case: - \( A(2, -2) \) corresponds to \( (x_1, y_1) \) - \( C(4, 1) \) corresponds to \( (x_2, y_2) \) - The ratio \( m:n = 1:3 \) means \( m = 1 \) and \( n = 3 \). Now, we can substitute the values into the formulas. 1. Calculate \( B_x \): \[ B_x = \frac{1 \cdot 4 + 3 \cdot 2}{1 + 3} = \frac{4 + 6}{4} = \frac{10}{4} = 2.5 \] 2. Calculate \( B_y \): \[ B_y = \frac{1 \cdot 1 + 3 \cdot (-2)}{1 + 3} = \frac{1 - 6}{4} = \frac{-5}{4} = -1.25 \] Thus, the coordinates of point \( B \) are \( (2.5, -1.25) \). The correct answer is \( (2.5, -1.25) \).