To find the coordinates of point [tex]\( B \)[/tex] that divides the segment [tex]\( \overline{AC} \)[/tex] into a ratio of [tex]\( 1:3 \)[/tex], we can use the section formula. The section formula helps us find the coordinates of a point that divides a line segment into a specific ratio.
Given:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (2, -1) \)[/tex]
- Point [tex]\( C \)[/tex] has coordinates [tex]\( (4, 2) \)[/tex]
- The ratio in which [tex]\( B \)[/tex] divides [tex]\( \overline{AC} \)[/tex] is [tex]\( 1:3 \)[/tex]
Let [tex]\( A = (x_1, y_1) = (2, -1) \)[/tex] and [tex]\( C = (x_2, y_2) = (4, 2) \)[/tex].
The ratio [tex]\( \frac{m}{n} = \frac{1}{3} \)[/tex].
The section formula in the case where a point [tex]\( (x, y) \)[/tex] divides a line segment joining [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] in the ratio [tex]\( \frac{m}{n} \)[/tex] is given by:
[tex]\[
(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Substitute the known values into the formula:
1. For the x-coordinate of [tex]\( B \)[/tex]:
[tex]\[
B_x = \frac{(1 \cdot 4) + (3 \cdot 2)}{1 + 3}
= \frac{4 + 6}{4}
= \frac{10}{4}
= 2.5
\][/tex]
2. For the y-coordinate of [tex]\( B \)[/tex]:
[tex]\[
B_y = \frac{(1 \cdot 2) + (3 \cdot -1)}{1 + 3}
= \frac{2 + (-3)}{4}
= \frac{2 - 3}{4}
= \frac{-1}{4}
= -0.25
\][/tex]
Thus, the coordinates of point [tex]\( B \)[/tex] are [tex]\( (2.5, -0.25) \)[/tex].
So, the point [tex]\( B \)[/tex] that divides the segment [tex]\( \overline{AC} \)[/tex] in the ratio [tex]\( 1:3 \)[/tex] is [tex]\( (2.5, -0.25) \)[/tex].
