Let's analyze the positions of points [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] based on the given information.
1. Point [tex]\( P \)[/tex]: Point [tex]\( P \)[/tex] partitions the directed segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] into a [tex]\( 1:3 \)[/tex] ratio.
- This means [tex]\( P \)[/tex] is [tex]\( \frac{1}{4} \)[/tex] of the way from [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
- Mathematically, if [tex]\( A \)[/tex] has the coordinate 0 and [tex]\( B \)[/tex] has the coordinate 1, then the position of [tex]\( P \)[/tex] is:
[tex]\[
P = \frac{1 \cdot B + 3 \cdot A}{1+3} = \frac{1 \cdot 1 + 3 \cdot 0}{1+3} = \frac{1}{4}
\][/tex]
2. Point [tex]\( Q \)[/tex]: Point [tex]\( Q \)[/tex] partitions the directed segment from [tex]\( B \)[/tex] to [tex]\( A \)[/tex] in a [tex]\( 1:3 \)[/tex] ratio.
- This means [tex]\( Q \)[/tex] is [tex]\( \frac{1}{4} \)[/tex] of the way from [tex]\( B \)[/tex] to [tex]\( A \)[/tex].
- Again, mathematically, if [tex]\( A \)[/tex] has the coordinate 0 and [tex]\( B \)[/tex] has the coordinate 1, then from [tex]\( B \)[/tex] to [tex]\( A \)[/tex], we reverse the direction. Hence, the position of [tex]\( Q \)[/tex] is:
[tex]\[
Q = \frac{1 \cdot A + 3 \cdot B}{1+3} = \frac{1 \cdot 0 + 3 \cdot 1}{1+3} = \frac{3}{4}
\][/tex]
From these calculations:
- [tex]\( P \)[/tex] is located at [tex]\( \frac{1}{4} \)[/tex] the distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
- [tex]\( Q \)[/tex] is located at [tex]\( \frac{3}{4} \)[/tex] the distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
Therefore, even though both [tex]\( P \)[/tex] and [tex]\( Q \)[/tex] are defined by a [tex]\( 1:3 \)[/tex] ratio, they are not the same points because they are located at different positions along the segment. Specifically, [tex]\( P \)[/tex] is [tex]\( 1/4 \)[/tex] the distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex], whereas [tex]\( Q \)[/tex] is [tex]\( 1/4 \)[/tex] the distance from [tex]\( B \)[/tex] to [tex]\( A \)[/tex], meaning [tex]\( Q \)[/tex] is [tex]\( 3/4 \)[/tex] from [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
Thus, the correct answer is:
[tex]\[
\text{No, } P \text{ is } \frac{1}{4} \text{ the distance from } A \text{ to } B, \text{ and } Q \text{ is } \frac{1}{4} \text{ the distance from } B \text{ to } A.
\][/tex]
