Let's carefully analyze the situation to determine the correct statement.
First, let's understand the partition ratio. We are given that point [tex]\( P \)[/tex] partitions the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in the ratio [tex]\( 3:4 \)[/tex]. This means the segment from [tex]\( A \)[/tex] to [tex]\( P \)[/tex] is 3 parts long, and the segment from [tex]\( P \)[/tex] to [tex]\( B \)[/tex] is 4 parts long.
Next, let's calculate the fractions of the total distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] that [tex]\( P \)[/tex] is away from [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
The total number of parts from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is:
[tex]\[ 3 + 4 = 7 \text{ parts} \][/tex]
The fraction of the distance from [tex]\( A \)[/tex] to [tex]\( P \)[/tex] relative to the total distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is:
[tex]\[ \frac{3}{7} \][/tex]
The fraction of the distance from [tex]\( P \)[/tex] to [tex]\( B \)[/tex] relative to the total distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is:
[tex]\[ \frac{4}{7} \][/tex]
Now, let's compare these distances:
- The distance from [tex]\( A \)[/tex] to [tex]\( P \)[/tex] is [tex]\( \frac{3}{7} \)[/tex] of the total distance.
- The distance from [tex]\( P \)[/tex] to [tex]\( B \)[/tex] is [tex]\( \frac{4}{7} \)[/tex] of the total distance.
Since [tex]\( \frac{3}{7} \approx 0.4286 \)[/tex] and [tex]\( \frac{4}{7} \approx 0.5714 \)[/tex], we see that [tex]\( \frac{3}{7} \)[/tex] is less than [tex]\( \frac{4}{7} \)[/tex]. Therefore, [tex]\( P \)[/tex] is closer to [tex]\( A \)[/tex] than to [tex]\( B \)[/tex].
So, the correct statement is:
[tex]\( P \)[/tex] will be closer to [tex]\( A \)[/tex] because it will be [tex]\( \frac{3}{7} \)[/tex] the distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex].
