To find the coordinates of point [tex]\( Q \)[/tex], we can use the section formula. The section formula helps in finding the coordinates of a point that divides a line segment internally in a given ratio.
Given:
- [tex]\( P = (-10, 3) \)[/tex]
- [tex]\( R = (4, 7) \)[/tex]
- The ratio [tex]\( PR : RQ = 2 : 3 \)[/tex]
First, let's summarize the information:
- Coordinates of [tex]\( P \)[/tex] are [tex]\( (P_x, P_y) = (-10, 3) \)[/tex].
- Coordinates of [tex]\( R \)[/tex] are [tex]\( (R_x, R_y) = (4, 7) \)[/tex].
- Ratio [tex]\( m : n = 2 : 3 \)[/tex] where [tex]\( m = 2 \)[/tex] and [tex]\( n = 3 \)[/tex].
Here, [tex]\( Q \)[/tex] is the point that we need to find, and we use the section formula where [tex]\( (x, y) \)[/tex] are:
[tex]\[ x = \frac{m(x_2) + n(x_1)}{m + n} \][/tex]
[tex]\[ y = \frac{m(y_2) + n(y_1)}{m + n} \][/tex]
In this problem, the coordinates of [tex]\( Q \)[/tex] can be calculated as follows:
1. Calculate the x-coordinate of [tex]\( Q \)[/tex]:
[tex]\[ Q_x = \frac{2 \cdot 4 + 3 \cdot (-10)}{2 + 3} \][/tex]
[tex]\[ Q_x = \frac{8 - 30}{5} \][/tex]
[tex]\[ Q_x = \frac{-22}{5} \][/tex]
2. Calculate the y-coordinate of [tex]\( Q \)[/tex]:
[tex]\[ Q_y = \frac{2 \cdot 7 + 3 \cdot 3}{2 + 3} \][/tex]
[tex]\[ Q_y = \frac{14 + 9}{5} \][/tex]
[tex]\[ Q_y = \frac{23}{5} \][/tex]
So, the coordinates of point [tex]\( Q \)[/tex] are:
[tex]\[ Q\left(-\frac{22}{5}, \frac{23}{5}\right) \][/tex]
Therefore, the correct answer is:
D. [tex]\(\left(-\frac{22}{5}, \frac{23}{5}\right)\)[/tex]
