To find the coordinates of point [tex]\( B \)[/tex] that partitions the segment with endpoints [tex]\( A \)[/tex] and [tex]\( C \)[/tex] in a [tex]\(1:3\)[/tex] ratio, we can use the section formula for dividing a line segment internally.
Given:
- Coordinates of point [tex]\( A \)[/tex] are [tex]\( (2, -2) \)[/tex]
- Coordinates of point [tex]\( C \)[/tex] are [tex]\( (4, 1) \)[/tex]
- The ratio in which point [tex]\( B \)[/tex] divides the segment [tex]\( AC \)[/tex] is [tex]\(1:3\)[/tex]
The section formula for a point that internally divides the line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex] is given by:
[tex]\[
\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Substituting the given values into the formula:
- [tex]\( x_1 = 2 \)[/tex]
- [tex]\( y_1 = -2 \)[/tex]
- [tex]\( x_2 = 4 \)[/tex]
- [tex]\( y_2 = 1 \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 3 \)[/tex]
We find the coordinates of point [tex]\( B \)[/tex]:
For the x-coordinate:
[tex]\[
B_x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} = \frac{1 \cdot 4 + 3 \cdot 2}{1 + 3} = \frac{4 + 6}{4} = \frac{10}{4} = 2.5
\][/tex]
For the y-coordinate:
[tex]\[
B_y = \frac{m \cdot y_2 + n \cdot y_1}{m + n} = \frac{1 \cdot 1 + 3 \cdot (-2)}{1 + 3} = \frac{1 - 6}{4} = \frac{-5}{4} = -1.25
\][/tex]
Therefore, the coordinates of point [tex]\( B \)[/tex] are [tex]\( (2.5, -1.25) \)[/tex].
Thus, the correct option is:
[tex]\[ (2.5, -1.25) \][/tex]
