Point [tex]P[/tex] partitions the directed line segment from [tex]A[/tex] to [tex]B[/tex] into the ratio [tex]3:4[/tex]. Will [tex]P[/tex] be closer to [tex]A[/tex] or [tex]B[/tex]? Why?

A. [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].
B. [tex]P[/tex] will be closer to [tex]A[/tex] because it will be [tex]\frac{4}{7}[/tex] the distance from [tex]A[/tex] to [tex]B[/tex].
C. [tex]P[/tex] will be closer to [tex]B[/tex] because it will be [tex]\frac{3}{7}[/tex] the distance from [tex]B[/tex] to [tex]A[/tex].
D. [tex]P[/tex] will be closer to [tex]B[/tex] because it will be [tex]\frac{4}{7}[/tex] the distance from [tex]B[/tex] to [tex]A[/tex].



Answer :

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].