Answer :
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]
- 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]
