To determine the position of point [tex]\( P \)[/tex] that partitions the line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] in a 3:4 ratio, let’s first understand what this ratio signifies.
### Step-by-Step Solution:
1. Understanding the Ratio:
- The ratio 3:4 means the entire length of the segment [tex]\( AB \)[/tex] is divided into two parts such that the part closer to [tex]\( A \)[/tex] is 3 units long and the part closer to [tex]\( B \)[/tex] is 4 units long.
2. Finding the Total Length:
- Total parts = 3 parts + 4 parts = 7 parts.
- Hence, the total length of the segment [tex]\( AB \)[/tex] can be considered as 7 equal parts.
3. Distance from [tex]\( A \)[/tex] to [tex]\( P \)[/tex]:
- If [tex]\( P \)[/tex] partitions the segment in the 3:4 ratio, the distance from [tex]\( A \)[/tex] to [tex]\( P \)[/tex] is given by:
[tex]\[
\text{Distance from } A \text{ to } P = \frac{3}{7} \times \text{Total length of } AB
\][/tex]
4. Distance from [tex]\( B \)[/tex] to [tex]\( P \)[/tex]:
- Conversely, the distance from [tex]\( B \)[/tex] to [tex]\( P \)[/tex] is given by:
[tex]\[
\text{Distance from } B \text{ to } P = \frac{4}{7} \times \text{Total length of } AB
\][/tex]
5. Comparing the Distances:
- [tex]\(\frac{3}{7}\)[/tex] of the distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] is approximately 0.4286 (rounded to four decimal places).
- [tex]\(\frac{4}{7}\)[/tex] of the distance from [tex]\( B \)[/tex] to [tex]\( A \)[/tex] is approximately 0.5714 (rounded to four decimal places).
6. Conclusion:
- Since [tex]\( \frac{3}{7} \)[/tex] (0.4286) is less than [tex]\( \frac{4}{7} \)[/tex] (0.5714), point [tex]\( P \)[/tex] is closer to [tex]\( A \)[/tex].
Therefore, [tex]\( P \)[/tex] is closer to [tex]\( A \)[/tex] because it will be [tex]\(\frac{3}{7}\)[/tex] the distance from [tex]\( A \)[/tex] to [tex]\( B \)[/tex]. This confirms that the correct answer is:
P 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].
