To find point [tex]\( Q \)[/tex] on the line segment [tex]\( PR \)[/tex] that divides it in the ratio [tex]\( 4:5 \)[/tex], we can use the section formula. The section formula states that if a point [tex]\( Q \)[/tex] divides a line segment [tex]\( PR \)[/tex], where [tex]\( P(x_1, y_1) \)[/tex] and [tex]\( R(x_2, y_2) \)[/tex], in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( Q(x,y) \)[/tex] are given by:
[tex]\[
Q_x = \frac{m x_2 + n x_1}{m + n}, \quad Q_y = \frac{m y_2 + n y_1}{m + n}
\][/tex]
Here, [tex]\( P = (-10, 7) \)[/tex], [tex]\( R = (8, -5) \)[/tex], and the ratio [tex]\( m:n = 4:5 \)[/tex].
We substitute these values into the section formula.
To find [tex]\( Q_x \)[/tex]:
[tex]\[
Q_x = \frac{4 \cdot 8 + 5 \cdot (-10)}{4 + 5} = \frac{32 - 50}{9} = \frac{-18}{9} = -2
\][/tex]
To find [tex]\( Q_y \)[/tex]:
[tex]\[
Q_y = \frac{4 \cdot (-5) + 5 \cdot 7}{4 + 5} = \frac{-20 + 35}{9} = \frac{15}{9} = \frac{5}{3}
\][/tex]
So, the coordinates of point [tex]\( Q \)[/tex] are [tex]\( Q \left( -2, \frac{5}{3} \right) \)[/tex].
Therefore, the correct choice is:
C. [tex]\(\left(-2, \frac{5}{3}\right)\)[/tex]
