To determine the coordinates of point P, which divides the line segment AB in the ratio 3:2, we can use the section formula. The section formula helps find the coordinates of a point that divides a line segment internally in a given ratio.
Given:
- Coordinates of point A: [tex]\( A(-6, -5) \)[/tex]
- Coordinates of point B: [tex]\( B(4, 0) \)[/tex]
- The ratio [tex]\( AP:PB = 3:2 \)[/tex]
According to the section formula, if a point P divides the line segment joining points [tex]\( (x_1, y_1) \)[/tex] and [tex]\( (x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of P, i.e., [tex]\( P(x, y) \)[/tex] are given by:
[tex]\[
x = \frac{m x_2 + n x_1}{m + n}
\][/tex]
[tex]\[
y = \frac{m y_2 + n y_1}{m + n}
\][/tex]
Where:
- [tex]\( (x_1, y_1) \)[/tex] are the coordinates of point A
- [tex]\( (x_2, y_2) \)[/tex] are the coordinates of point B
- [tex]\( m \)[/tex] and [tex]\( n \)[/tex] are the given ratio parts
Substituting the given values:
- [tex]\( x_1 = -6 \)[/tex]
- [tex]\( y_1 = -5 \)[/tex]
- [tex]\( x_2 = 4 \)[/tex]
- [tex]\( y_2 = 0 \)[/tex]
- [tex]\( m = 3 \)[/tex]
- [tex]\( n = 2 \)[/tex]
For [tex]\( x \)[/tex]:
[tex]\[
x = \frac{3 \cdot 4 + 2 \cdot (-6)}{3 + 2} = \frac{12 - 12}{5} = \frac{0}{5} = 0
\][/tex]
For [tex]\( y \)[/tex]:
[tex]\[
y = \frac{3 \cdot 0 + 2 \cdot (-5)}{3 + 2} = \frac{0 - 10}{5} = \frac{-10}{5} = -2
\][/tex]
Therefore, the coordinates of point P are:
[tex]\[
P(0, -2)
\][/tex]
