To find the coordinates of point [tex]\( B \)[/tex] that divides the segment connecting points [tex]\( A \)[/tex] and [tex]\( C \)[/tex] in a ratio of [tex]\( 1:3 \)[/tex], we can use the section formula for internal division.
The section formula states:
[tex]\[ B = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right) \][/tex]
Here, the coordinates of [tex]\( A \)[/tex] are [tex]\( (x_1, y_1) = (2, -1) \)[/tex], and the coordinates of [tex]\( C \)[/tex] are [tex]\( (x_2, y_2) = (4, 2) \)[/tex]. The ratio [tex]\( m : n \)[/tex] is [tex]\( 1 : 3 \)[/tex], so [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex].
Using the section formula, we calculate the coordinates of [tex]\( B \)[/tex] as follows:
1. Calculate the x-coordinate of [tex]\( B \)[/tex]:
[tex]\[ x_B = \frac{m x_2 + n x_1}{m + n} = \frac{1 \cdot 4 + 3 \cdot 2}{1 + 3} = \frac{4 + 6}{4} = \frac{10}{4} = 2.5 \][/tex]
2. Calculate the y-coordinate of [tex]\( B \)[/tex]:
[tex]\[ y_B = \frac{m y_2 + n y_1}{m + n} = \frac{1 \cdot 2 + 3 \cdot (-1)}{1 + 3} = \frac{2 + (-3)}{4} = \frac{2 - 3}{4} = \frac{-1}{4} = -0.25 \][/tex]
Therefore, the coordinates of point [tex]\( B \)[/tex] are [tex]\( \left(2.5, -0.25\right) \)[/tex].
Thus, the correct answer is:
[tex]\[ (2.5, -0.25) \][/tex]
So, the correct option is:
[tex]\[ \boxed{(2.5, -0.25)} \][/tex]
