To determine the coordinates of point [tex]\( Q \)[/tex], we use the section formula which provides the coordinates of a point dividing a line segment in a given ratio.
Given:
- Coordinates of point [tex]\( P \)[/tex]: [tex]\( P(-10, 3) \)[/tex]
- Coordinates of point [tex]\( R \)[/tex]: [tex]\( R(4, 7) \)[/tex]
- The ratio [tex]\( P R : R Q = 2 : 3 \)[/tex]
We need to find the coordinates of point [tex]\( Q \)[/tex].
Using the section formula:
[tex]\[ R\left(\frac{m_1 \cdot x_2 + m_2 \cdot x_1}{m_1 + m_2}, \frac{m_1 \cdot y_2 + m_2 \cdot y_1}{m_1 + m_2}\right) \][/tex]
Here, [tex]\( R \)[/tex] divides [tex]\( P Q \)[/tex] in the ratio [tex]\( m_1 : m_2 \)[/tex]. For our problem, [tex]\( m_1 = 2 \)[/tex] and [tex]\( m_2 = 3 \)[/tex]:
[tex]\[ R_x = \frac{2 \cdot Q_x + 3 \cdot (-10)}{2 + 3} = 4 \][/tex]
[tex]\[ R_y = \frac{2 \cdot Q_y + 3 \cdot 3}{2 + 3} = 7 \][/tex]
Now, we need to solve for [tex]\( Q_x \)[/tex] and [tex]\( Q_y \)[/tex] from these equations.
Step 1: Solve for [tex]\( Q_x \)[/tex].
[tex]\[ \frac{2 \cdot Q_x + 3 \cdot (-10)}{5} = 4 \][/tex]
[tex]\[ 2 \cdot Q_x + 3 \cdot (-10) = 20 \][/tex]
[tex]\[ 2 \cdot Q_x - 30 = 20 \][/tex]
[tex]\[ 2 \cdot Q_x = 50 \][/tex]
[tex]\[ Q_x = \frac{50}{2} = 25 \][/tex]
Step 2: Solve for [tex]\( Q_y \)[/tex].
[tex]\[ \frac{2 \cdot Q_y + 3 \cdot 3}{5} = 7 \][/tex]
[tex]\[ 2 \cdot Q_y + 9 = 35 \][/tex]
[tex]\[ 2 \cdot Q_y = 26 \][/tex]
[tex]\[ Q_y = \frac{26}{2} = 13 \][/tex]
Therefore, the coordinates of point [tex]\( Q \)[/tex] are [tex]\( (25, 13) \)[/tex].
The correct answer is:
A. [tex]\( (25, 13) \)[/tex]
