To find the coordinates of point [tex]\( R \)[/tex] that divides the line segment [tex]\( PQ \)[/tex] in the ratio [tex]\( 3:2 \)[/tex], we use the section formula. The section formula states that if a point [tex]\( R \)[/tex] divides the line segment joining two points [tex]\( P(x_1, y_1) \)[/tex] and [tex]\( Q(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( R \)[/tex] are given by:
[tex]\[
R\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\][/tex]
Given:
- [tex]\( P = (6, -5) \)[/tex]
- [tex]\( Q = (-2, 4) \)[/tex]
- Ratio [tex]\( m:n = 3:2 \)[/tex]
We can now apply the section formula to find [tex]\( R \)[/tex].
### Calculating the x-coordinate of [tex]\( R \)[/tex]:
[tex]\[
R_x = \frac{m x_2 + n x_1}{m + n}
\][/tex]
Substituting the values:
[tex]\[
R_x = \frac{3(-2) + 2(6)}{3 + 2}
\][/tex]
Simplifying:
[tex]\[
R_x = \frac{3(-2) + 2(6)}{5} \\
R_x = \frac{-6 + 12}{5} \\
R_x = \frac{6}{5}
\][/tex]
### Calculating the y-coordinate of [tex]\( R \)[/tex]:
[tex]\[
R_y = \frac{m y_2 + n y_1}{m + n}
\][/tex]
Substituting the values:
[tex]\[
R_y = \frac{3(4) + 2(-5)}{3 + 2}
\][/tex]
Simplifying:
[tex]\[
R_y = \frac{3(4) + 2(-5)}{5} \\
R_y = \frac{12 - 10}{5} \\
R_y = \frac{2}{5}
\][/tex]
Hence, the coordinates of point [tex]\( R \)[/tex] are:
[tex]\[
R \left(\frac{6}{5}, \frac{2}{5}\right)
\][/tex]
From the given choices, the correct option is:
A. [tex]\(\left(\frac{6}{5}, \frac{2}{5}\right)\)[/tex]
