To find point B that partitions segment AC in a 1:3 ratio, we can use the section formula. The formula for finding the coordinates of a point that divides a segment in the ratio \( m:n \) is given by:
\[ B = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]
In this case, let \( A(1, -1) \) be point \( (x_1, y_1) \) and \( C(3, 2) \) be point \( (x_2, y_2) \). The ratio is \( m:n = 1:3 \) (meaning point B is closer to A).
So, \( m = 1 \) and \( n = 3 \).
Now substituting the coordinates and the ratio into the formula:
- Calculate the x-coordinate of B:
\[ x_B = \frac{1 \cdot 3 + 3 \cdot 1}{1 + 3} = \frac{3 + 3}{4} = \frac{6}{4} = 1.5 \]
- Calculate the y-coordinate of B:
\[ y_B = \frac{1 \cdot 2 + 3 \cdot (-1)}{1 + 3} = \frac{2 - 3}{4} = \frac{-1}{4} = -0.25 \]
Thus, the coordinates of point B are:
\[ B(1.5, -0.25) \]
Out of the provided options, the correct answer is:
(1.5, −0.25).