To find the point [tex]\( Q \)[/tex] on the line segment [tex]\( PR \)[/tex] that partitions it into segments [tex]\( PQ \)[/tex] and [tex]\( QR \)[/tex] in the ratio [tex]\( 4:5 \)[/tex], we use the section formula. Given points [tex]\( P(-10, 7) \)[/tex] and [tex]\( R(8, -5) \)[/tex], and the ratios [tex]\( m=4 \)[/tex] and [tex]\( n=5 \)[/tex], the section formula in two dimensions helps us determine the coordinates of [tex]\( Q \)[/tex]. The formula for the coordinates [tex]\((Q_x, Q_y)\)[/tex] is given by:
[tex]\[
Q_x = \frac{m \cdot R_x + n \cdot P_x}{m + n}
\][/tex]
[tex]\[
Q_y = \frac{m \cdot R_y + n \cdot P_y}{m + n}
\][/tex]
Substituting the given values:
1. Calculate [tex]\( Q_x \)[/tex]:
[tex]\[
Q_x = \frac{4 \cdot 8 + 5 \cdot (-10)}{4 + 5}
\][/tex]
[tex]\[
= \frac{32 - 50}{9}
\][/tex]
[tex]\[
= \frac{-18}{9}
\][/tex]
[tex]\[
= -2
\][/tex]
2. Calculate [tex]\( Q_y \)[/tex]:
[tex]\[
Q_y = \frac{4 \cdot (-5) + 5 \cdot 7}{4 + 5}
\][/tex]
[tex]\[
= \frac{-20 + 35}{9}
\][/tex]
[tex]\[
= \frac{15}{9}
\][/tex]
[tex]\[
= \frac{5}{3}
\][/tex]
Thus, the coordinates of the point [tex]\( Q \)[/tex] are:
[tex]\[
Q = \left(-2, \frac{5}{3}\right)
\][/tex]
Therefore, the correct option is:
C. [tex]\(\left(-2, \frac{5}{3}\right)\)[/tex]
