To find the coordinates of point [tex]\( B \)[/tex] on line segment [tex]\( \overline{AC} \)[/tex] such that the ratio [tex]\( AB : AC = 1 : 3 \)[/tex], we can use the section formula for internal division of a line segment.
Given:
- Coordinates of point [tex]\( A \)[/tex]: [tex]\( A(-2, 4) \)[/tex]
- Coordinates of point [tex]\( C \)[/tex]: [tex]\( C(4, 7) \)[/tex]
- Ratio [tex]\( AB : AC = 1 : 3 \)[/tex]
Let's denote the ratio as [tex]\( \frac{m}{n} \)[/tex], where [tex]\( m = 1 \)[/tex] and [tex]\( n = 3 \)[/tex].
The section formula for a point dividing a line segment internally in the ratio [tex]\( \frac{m}{n} \)[/tex] between points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] is:
[tex]\[
\left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)
\][/tex]
In this case:
- [tex]\( (x_1, y_1) = (-2, 4) \)[/tex]
- [tex]\( (x_2, y_2) = (4, 7) \)[/tex]
- [tex]\( m = 1 \)[/tex]
- [tex]\( n = 3 \)[/tex]
Plugging these values into the formula:
[tex]\[
B_x = \frac{(1)(4) + (3)(-2)}{1 + 3} = \frac{4 + (-6)}{4} = \frac{-2}{4} = -0.5
\][/tex]
[tex]\[
B_y = \frac{(1)(7) + (3)(4)}{1 + 3} = \frac{7 + 12}{4} = \frac{19}{4} = 4.75
\][/tex]
Therefore, the coordinates of point [tex]\( B \)[/tex] are [tex]\( (-0.5, 4.75) \)[/tex].
