To find point [tex]\( Q \)[/tex] that divides the line segment [tex]\( PR \)[/tex] in the ratio [tex]\( 4:5 \)[/tex], we will use the section formula. The coordinates of point [tex]\( Q \)[/tex] can be determined by the formula:
[tex]\[ Qx = \frac{m \cdot Rx + n \cdot Px}{m + n} \][/tex]
[tex]\[ Qy = \frac{m \cdot Ry + n \cdot Py}{m + n} \][/tex]
where [tex]\( P(x_1, y_1) = (-10,7) \)[/tex], [tex]\( R(x_2, y_2) = (8,-5) \)[/tex], and the ratio [tex]\( PQ : QR = 4:5 \)[/tex], which means [tex]\( m = 4 \)[/tex] and [tex]\( n = 5 \)[/tex].
First, let's find the x-coordinate of [tex]\( Q \)[/tex]:
[tex]\[ Qx = \frac{4 \cdot 8 + 5 \cdot (-10)}{4 + 5} \][/tex]
[tex]\[ Qx = \frac{32 - 50}{9} \][/tex]
[tex]\[ Qx = \frac{-18}{9} \][/tex]
[tex]\[ Qx = -2 \][/tex]
Next, we find the y-coordinate of [tex]\( Q \)[/tex]:
[tex]\[ Qy = \frac{4 \cdot (-5) + 5 \cdot 7}{4 + 5} \][/tex]
[tex]\[ Qy = \frac{-20 + 35}{9} \][/tex]
[tex]\[ Qy = \frac{15}{9} \][/tex]
[tex]\[ Qy = \frac{5}{3} \][/tex]
So, the coordinates of point [tex]\( Q \)[/tex] are [tex]\( \left(-2, \frac{5}{3}\right) \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{\left(-2, \frac{5}{3}\right)} \][/tex]
