To find the coordinates of point [tex]\( Q \)[/tex], we can use the section formula, which helps determine the coordinates of a point that divides a line segment internally in a given ratio. Given:
- Point [tex]\( P \)[/tex] at [tex]\( (-10, 3) \)[/tex]
- Point [tex]\( R \)[/tex] at [tex]\( (4, 7) \)[/tex]
- Ratio [tex]\( PR : RQ = 2 : 3 \)[/tex]
Let's denote the coordinates of point [tex]\( Q \)[/tex] as [tex]\( (x, y) \)[/tex].
The section formula for a point [tex]\( R \)[/tex] that divides a line segment [tex]\( PQ \)[/tex] internally in the ratio [tex]\( m : n \)[/tex] is given by:
[tex]\[
R\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\][/tex]
Since point [tex]\( R \)[/tex] divides the segment [tex]\( PQ \)[/tex] in the ratio [tex]\( 2 : 3 \)[/tex]:
[tex]\[
(4, 7) = \left( \frac{2x + 3(-10)}{2+3}, \frac{2y + 3(3)}{2+3} \right)
\][/tex]
Simplifying the x-coordinate:
[tex]\[
4 = \frac{2x + 3(-10)}{5}
\][/tex]
Multiplying both sides by 5:
[tex]\[
20 = 2x + 3(-10)
\][/tex]
[tex]\[
20 = 2x - 30
\][/tex]
Adding 30 to both sides:
[tex]\[
50 = 2x
\][/tex]
Dividing by 2:
[tex]\[
x = 25
\][/tex]
Now for the y-coordinate:
[tex]\[
7 = \frac{2y + 3(3)}{5}
\][/tex]
[tex]\[
7 = \frac{2y + 9}{5}
\][/tex]
Multiplying both sides by 5:
[tex]\[
35 = 2y + 9
\][/tex]
Subtracting 9 from both sides:
[tex]\[
26 = 2y
\][/tex]
Dividing by 2:
[tex]\[
y = 13
\][/tex]
Therefore, the coordinates of point [tex]\( Q \)[/tex] are [tex]\( (25, 13) \)[/tex].
So, the correct answer is:
B. [tex]\( (25, 13) \)[/tex]
