To find the point [tex]\( Q \)[/tex] on the line segment [tex]\( PR \)[/tex] that partitions it into the segments [tex]\( PQ \)[/tex] and [tex]\( QR \)[/tex] in the ratio 4:5, we will use the Section Formula.
Given:
- [tex]\( P = (0, 0) \)[/tex]
- [tex]\( R = (5, 4) \)[/tex]
- Ratio [tex]\( PQ : QR = 4:5 \)[/tex]
The Section Formula for a line segment [tex]\( AB \)[/tex] that is divided by a point [tex]\( C \)[/tex] in the ratio [tex]\( m:n \)[/tex] is:
[tex]\[
C = \left( \frac{m B_x + n A_x}{m + n}, \frac{m B_y + n A_y}{m + n} \right)
\][/tex]
Applying this formula to our points and ratio:
- [tex]\( P = A = (0, 0) \)[/tex]
- [tex]\( R = B = (5, 4) \)[/tex]
- [tex]\( m = 4 \)[/tex]
- [tex]\( n = 5 \)[/tex]
Calculate the x-coordinate of [tex]\( Q \)[/tex]:
[tex]\[
Q_x = \frac{(4 \cdot 5) + (5 \cdot 0)}{4 + 5} = \frac{20}{9}
\][/tex]
Calculate the y-coordinate of [tex]\( Q \)[/tex]:
[tex]\[
Q_y = \frac{(4 \cdot 4) + (5 \cdot 0)}{4 + 5} = \frac{16}{9}
\][/tex]
Thus, the coordinates of [tex]\( Q \)[/tex] are:
[tex]\[
Q = \left( \frac{20}{9}, \frac{16}{9} \right)
\][/tex]
We now need to match our calculated coordinates of [tex]\( Q \)[/tex] with one of the given options.
The options are:
A. [tex]\(\left(-2, \frac{5}{3}\right)\)[/tex]
B. [tex]\(\left(-2, -\frac{5}{3}\right)\)[/tex]
C. [tex]\(\left(-\frac{9}{2}, 3\right)\)[/tex]
D. [tex]\(\left(0, -\frac{1}{3}\right)\)[/tex]
The calculated coordinates do not match any of the given answer options exactly. Therefore, none of the provided options is the correct solution.
As a result, the correct answer is:
None of the given options match the coordinates of point [tex]\( Q \)[/tex].
