To find the coordinates of point [tex]\( Q \)[/tex] given the points [tex]\( P \)[/tex] and [tex]\( R \)[/tex] and the ratio [tex]\( PR:RQ = 2:3 \)[/tex], use the section formula. Given the coordinates of [tex]\( P \)[/tex] as [tex]\( (-10, 3) \)[/tex] and [tex]\( R \)[/tex] as [tex]\( (4, 7) \)[/tex] and the ratio [tex]\( m:n = 2:3 \)[/tex], you need to find the coordinates of [tex]\( Q \)[/tex].
The section formula states that if a point [tex]\( R \)[/tex] divides the line segment joining 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]\( Q \)[/tex] are given by:
[tex]\[
Q = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right).
\][/tex]
In our case, we need to find [tex]\( Q(x_2, y_2) \)[/tex]:
[tex]\[
R = \left( \frac{2x_2 + 3(-10)}{2+3}, \frac{2y_2 + 3(3)}{2+3} \right).
\][/tex]
Given that [tex]\( R = (4, 7) \)[/tex], equate the coordinates:
[tex]\[
4 = \frac{2x_2 - 30}{5}
\][/tex]
[tex]\[
7 = \frac{2y_2 + 9}{5}.
\][/tex]
Let's solve the equations one by one:
For the x-coordinate:
[tex]\[
4 = \frac{2x_2 - 30}{5}.
\][/tex]
Multiply both sides by 5:
[tex]\[
20 = 2x_2 - 30.
\][/tex]
Add 30 to both sides:
[tex]\[
50 = 2x_2.
\][/tex]
Divide by 2:
[tex]\[
x_2 = 25.
\][/tex]
For the y-coordinate:
[tex]\[
7 = \frac{2y_2 + 9}{5}.
\][/tex]
Multiply both sides by 5:
[tex]\[
35 = 2y_2 + 9.
\][/tex]
Subtract 9 from both sides:
[tex]\[
26 = 2y_2.
\][/tex]
Divide by 2:
[tex]\[
y_2 = 13.
\][/tex]
So, the coordinates of point [tex]\( Q \)[/tex] are:
[tex]\[
Q = (25, 13).
\][/tex]
Thus, the correct answer is:
C. [tex]\((25, 13)\)[/tex].
