Point P partitions the directed line segment from A to B into the ratio $3:4$. Will P be closer to A or B? Why?
A. P will be closer to A because it will be $\frac{3}{7}$ the distance from A to B.
B. P will be closer to A because it will be $\frac{4}{7}$ the distance from A to B.
C. P will be closer to B because it will be $\frac{3}{7}$ the distance from B to A.
D. P will be closer to B because it will be $\frac{4}{7}$ the distance from B to A
Answer (2)
Point P divides the line segment AB in the ratio 3:4.
The distance from A to P is 7 3 of the total distance.
The distance from P to B is 7 4 of the total distance.
Since 7 3 < 7 4 , P is closer to A. P is closer to A
Explanation
Problem Analysis Let's analyze the problem. Point P divides the directed line segment from A to B in the ratio 3:4. This means that the distance from A to P is 3 parts, and the distance from P to B is 4 parts. We need to determine whether P is closer to A or B.
Distance Calculation Let the distance from A to P be 3 x and the distance from P to B be 4 x . The total distance from A to B is the sum of these distances, which is 3 x + 4 x = 7 x .
Fraction Comparison To determine which point P is closer to, we can compare the fractions of the total distance. The distance from A to P is 7 x 3 x = 7 3 of the total distance from A to B. The distance from P to B is 7 x 4 x = 7 4 of the total distance from A to B.
Conclusion Since 7 3 < 7 4 , the distance from A to P is less than the distance from P to B. Therefore, point P is closer to A.
Examples
In city planning, if you're designing a bus route between two main points, A and B, and you want a bus stop (P) that divides the route in a 3:4 ratio for accessibility, this calculation helps determine whether the bus stop is closer to the starting point (A) or the destination (B). This ensures more efficient service based on population distribution.
Point P divides the directed line segment from A to B in a ratio of 3:4, meaning it's closer to A. This is because P is 7 3 of the distance from A to B, while it is 7 4 from P to B. Therefore, the correct answer is A.
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