To find the coordinates of point [tex]\(Q\)[/tex], given that point [tex]\(R\)[/tex] divides the segment [tex]\(PQ\)[/tex] in the ratio [tex]\(PR: RQ = 2: 3\)[/tex], and given the coordinates of points [tex]\(P\)[/tex] and [tex]\(R\)[/tex], we will use the section formula.
The coordinates of point [tex]\(P\)[/tex] are [tex]\((-10, 3)\)[/tex], and the coordinates of point [tex]\(R\)[/tex] are [tex]\((4, 7)\)[/tex]. Let the coordinates of point [tex]\(Q\)[/tex] be [tex]\((Q_x, Q_y)\)[/tex].
The section formula for a point dividing a line segment in a given ratio is:
[tex]\[R_x = \frac{m_1 Q_x + m_2 P_x}{m_1 + m_2}\][/tex]
[tex]\[R_y = \frac{m_1 Q_y + m_2 P_y}{m_1 + m_2}\][/tex]
We know:
[tex]\[R_x = 4, R_y = 7\][/tex]
[tex]\[P_x = -10, P_y = 3\][/tex]
[tex]\[m_1 = 2, m_2 = 3\][/tex]
First, we solve for [tex]\(Q_x\)[/tex]:
[tex]\[
4 = \frac{2Q_x + 3(-10)}{2 + 3}
\][/tex]
[tex]\[
4 = \frac{2Q_x - 30}{5}
\][/tex]
Multiplying both sides by 5:
[tex]\[
20 = 2Q_x - 30
\][/tex]
Adding 30 to both sides:
[tex]\[
50 = 2Q_x
\][/tex]
Dividing by 2:
[tex]\[
Q_x = 25
\][/tex]
Next, we solve for [tex]\(Q_y\)[/tex]:
[tex]\[
7 = \frac{2Q_y + 3(3)}{2 + 3}
\][/tex]
[tex]\[
7 = \frac{2Q_y + 9}{5}
\][/tex]
Multiplying both sides by 5:
[tex]\[
35 = 2Q_y + 9
\][/tex]
Subtracting 9 from both sides:
[tex]\[
26 = 2Q_y
\][/tex]
Dividing by 2:
[tex]\[
Q_y = 13
\][/tex]
Thus, the coordinates of point [tex]\(Q\)[/tex] are [tex]\((25, 13)\)[/tex].
The correct answer is:
C. [tex]\((25, 13)\)[/tex]
