To determine the position of point [tex]\(P\)[/tex] along the line segment from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] given that the segment is partitioned in the ratio [tex]\(3:4\)[/tex], let’s follow these steps:
1. Understand the Ratio: The ratio [tex]\(3:4\)[/tex] indicates that the entire segment is divided into two parts such that for every 3 units of distance on one side, there are 4 units of distance on the other.
2. Total Parts: Add the parts of the ratio together to find the total parts:
[tex]\[ 3 + 4 = 7 \][/tex]
So, the entire segment is divided into 7 equal parts.
3. Distance from [tex]\(A\)[/tex] to [tex]\(P\)[/tex]: [tex]\(P\)[/tex] is [tex]\(3/7\)[/tex] of the way from [tex]\(A\)[/tex] to [tex]\(B\)[/tex], meaning it covers 3 out of the 7 parts of the total length from [tex]\(A\)[/tex] to [tex]\(B\)[/tex].
[tex]\[ \text{Distance from } A \text{ to } P = 3/7 \][/tex]
4. Distance from [tex]\(P\)[/tex] to [tex]\(B\)[/tex]: Similarly, the remaining segment from [tex]\(P\)[/tex] to [tex]\(B\)[/tex] is [tex]\(4/7\)[/tex] of the total distance.
[tex]\[ \text{Distance from } P \text{ to } B = 4/7 \][/tex]
Since [tex]\(P\)[/tex] is positioned [tex]\(3/7\)[/tex] the distance from [tex]\(A\)[/tex] to [tex]\(B\)[/tex] (less than half the total distance) and [tex]\(4/7\)[/tex] the distance from [tex]\(P\)[/tex] to [tex]\(B\)[/tex] (more than half the total distance), [tex]\(P\)[/tex] is closer to [tex]\(A\)[/tex] than to [tex]\(B\)[/tex].
Thus, the correct answer is:
[tex]\[ P \text{ will be closer to } A \text{ because it will be } \frac{3}{7} \text{ the distance from } A \text{ to } B. \][/tex]
So, you should choose the first option:
[tex]\[ \text{P will be closer to } A \text{ because it will be } \frac{3}{7} \text{ the distance from } A \text{ to } B. \][/tex]
