To find the coordinates of point [tex]\( B \)[/tex] that divides the segment [tex]\( AC \)[/tex] in a ratio of 1:3, we will use the section formula. The coordinates of [tex]\( A \)[/tex] are [tex]\( (2, -2) \)[/tex] and the coordinates of [tex]\( C \)[/tex] are [tex]\( (4, 1) \)[/tex]. Given that the ratio is 1:3, we can denote it as [tex]\( m:n = 1:3 \)[/tex].
The section formula states that if a point [tex]\( B \)[/tex] divides the segment joining two points [tex]\( A \)[/tex] and [tex]\( C \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( B \)[/tex] are:
[tex]\[
B\left( \frac{(nx_1 + mx_2)}{m+n}, \frac{(ny_1 + my_2)}{m+n} \right)
\][/tex]
Here, we have:
[tex]\[
x_1 = 2, \quad y_1 = -2, \quad x_2 = 4, \quad y_2 = 1, \quad m = 1, \quad n = 3
\][/tex]
Let's find the x-coordinate of [tex]\( B \)[/tex]:
[tex]\[
B_x = \frac{(n \cdot x_1 + m \cdot x_2)}{m+n} = \frac{(3 \cdot 2 + 1 \cdot 4)}{1+3} = \frac{(6 + 4)}{4} = \frac{10}{4} = 2.5
\][/tex]
Next, let's find the y-coordinate of [tex]\( B \)[/tex]:
[tex]\[
B_y = \frac{(n \cdot y_1 + m \cdot y_2)}{m+n} = \frac{(3 \cdot -2 + 1 \cdot 1)}{1+3} = \frac{(-6 + 1)}{4} = \frac{-5}{4} = -1.25
\][/tex]
Hence, the coordinates of point [tex]\( B \)[/tex] are:
[tex]\[
B(2.5, -1.25)
\][/tex]
So, the answer is [tex]\((2.5, -1.25)\)[/tex].
