Partitioning Line Segments Understanding Ratios And Directed Segments

In geometry, partitioning a line segment involves dividing it into specific ratios. This concept is fundamental in various geometric problems, especially when dealing with coordinate geometry and vector algebra. When we talk about a directed segment, the order of the endpoints matters, and the ratio indicates the proportion in which the segment is divided. In this article, we delve into a specific scenario where two points, P and Q, partition a line segment AB in a 1:3 ratio, but in opposite directions. We aim to determine whether these points, P and Q, coincide or are distinct. This exploration will provide valuable insights into the intricacies of directed segments and ratios in geometry.

Problem Statement: Partitioning Directed Segments AB and BA

Consider two points, A and B, which define a line segment. Point P partitions the directed segment from A to B in a 1:3 ratio, while point Q partitions the directed segment from B to A also in a 1:3 ratio. The crucial question we address is: Are P and Q the same point? To answer this, we must carefully analyze what it means to partition a directed segment in a given ratio and how the direction of the segment affects the position of the partitioning point. The phrase "directed segment" is key here, as it implies that the order of the points matters. Partitioning AB in a 1:3 ratio means that the distance from A to P is one-third the distance from P to B. Conversely, partitioning BA in a 1:3 ratio means the distance from B to Q is one-third the distance from Q to A. This distinction in direction is critical to the problem. By meticulously examining these relationships, we will uncover whether P and Q occupy the same location on the line segment or if they are indeed distinct points.

Analytical Approach: Unraveling the Partitioning Points

To rigorously determine whether P and Q are the same point, we will employ an analytical approach using ratios and distances. Let's denote the position vectors of points A and B as a and b, respectively. When point P partitions the directed segment AB in a 1:3 ratio, this implies that AP:PB = 1:3. Using the section formula, which is a cornerstone in coordinate geometry, we can express the position vector p of point P in terms of a and b. The section formula states that if a point divides a line segment in the ratio m:n, then its position vector can be calculated as a weighted average of the position vectors of the endpoints. In this case, the formula is applied as follows:

p = (3a + 1b) / (1 + 3) = (3a + b) / 4

Similarly, when point Q partitions the directed segment BA in a 1:3 ratio, it means BQ:QA = 1:3. Now, we express the position vector q of point Q using the section formula, but this time, considering the direction from B to A:

q = (3b + 1a) / (1 + 3) = (a + 3b) / 4

By deriving these expressions for p and q, we now have a concrete mathematical representation of the positions of points P and Q. The next step is to compare these expressions to see if they are equivalent, which would imply that P and Q are the same point. If the expressions differ, then P and Q are distinct points on the line segment. This analytical method provides a clear and concise way to resolve the problem by leveraging the fundamental principles of ratios and vector algebra.

Deriving the Positions of P and Q Using the Section Formula

To definitively answer whether points P and Q are the same, we apply the section formula to calculate their positions. The section formula is a vital tool in coordinate geometry, enabling us to find the coordinates (or position vector) of a point that divides a line segment in a given ratio. This formula is derived from the principles of similar triangles and proportional division of line segments. It provides a direct relationship between the position vectors of the endpoints of the segment and the ratio in which the segment is divided. Let's consider the directed segment AB. Point P divides this segment in the ratio 1:3, meaning that the distance from A to P is one-fourth of the total length of AB, and the distance from P to B is three-fourths. Using the section formula, we can express the position vector of P, denoted as p, in terms of the position vectors of A and B, denoted as a and b, respectively. The calculation is as follows:

p = (3a + 1b) / (1 + 3) = (3a + b) / 4

This equation tells us that the position of P is a weighted average of the positions of A and B, where A is weighted by 3 and B is weighted by 1. Now, let's consider the directed segment BA. Point Q divides this segment in the ratio 1:3, but this time, the direction is reversed. This means that the distance from B to Q is one-fourth of the total length of BA, and the distance from Q to A is three-fourths. Applying the section formula again, we find the position vector of Q, denoted as q:

q = (3b + 1a) / (1 + 3) = (a + 3b) / 4

By methodically using the section formula for both points P and Q, we have now obtained explicit expressions for their positions. The next crucial step is to compare these expressions to determine if they are identical. If the expressions for p and q are the same, then P and Q occupy the same position on the line segment. If they are different, then P and Q are distinct points.

Comparing Position Vectors: Are P and Q the Same Point?

Having derived the position vectors p and q for points P and Q, respectively, we now proceed to compare these vectors. Recall that the position vector p of point P, which divides the directed segment AB in a 1:3 ratio, is given by:

p = (3a + b) / 4

And the position vector q of point Q, which divides the directed segment BA in a 1:3 ratio, is given by:

q = (a + 3b) / 4

A straightforward comparison of these two expressions reveals that they are not identical. The coefficients of the position vectors a and b are interchanged between the expressions for p and q. In p, the position vector a is multiplied by 3, and b is multiplied by 1, whereas in q, a is multiplied by 1, and b is multiplied by 3. This difference in coefficients signifies that the positions of P and Q are distinct on the line segment. To further illustrate this, let's consider a numerical example. Suppose point A has coordinates (0, 0) and point B has coordinates (4, 0). Then, the position vector a is (0, 0), and b is (4, 0). Substituting these values into the expressions for p and q, we get:

p = (3(0, 0) + (4, 0)) / 4 = (1, 0) q = ((0, 0) + 3(4, 0)) / 4 = (3, 0)

As we can see, the coordinates of P are (1, 0), and the coordinates of Q are (3, 0). These points are clearly different, confirming our analytical conclusion. This comparison definitively shows that P and Q are not the same point. They occupy different positions on the line segment AB due to the reversed direction of partitioning and the specific 1:3 ratio.

Conclusion: P and Q are Distinct Points

In conclusion, our analysis demonstrates unequivocally that points P and Q are not the same. Point P, which partitions the directed segment AB in a 1:3 ratio, is located at a different position compared to point Q, which partitions the directed segment BA in the same 1:3 ratio. The key factor contributing to this distinction is the direction of the segments being partitioned. The order of endpoints A and B matters significantly when dealing with directed segments. When P divides AB in the ratio 1:3, it is closer to point A than to point B. Conversely, when Q divides BA in the ratio 1:3, it is closer to point B than to point A. This difference in proximity to the endpoints results in P and Q occupying distinct locations on the line segment. Our application of the section formula, a fundamental tool in coordinate geometry, provided a rigorous mathematical framework to confirm this conclusion. By expressing the position vectors of P and Q in terms of the position vectors of A and B, we were able to directly compare their positions. The expressions (3a + b) / 4 for p and (a + 3b) / 4 for q clearly illustrate that the positions are different unless a and b are the same, which would mean A and B are the same point, an uninteresting case. Therefore, the answer to the question