Finding Coordinates Point P On A Line Segment 1/3 The Distance From A To B

In the realm of coordinate geometry, determining the coordinates of a point that divides a line segment in a given ratio is a fundamental concept. This article delves into the intricacies of finding the coordinates of a point $P$ on a directed line segment from point $A$ to point $B$, such that the distance from $A$ to $P$ is a specific fraction of the total length of the segment from $A$ to $B$. Specifically, we will explore the scenario where point $P$ is located one-third of the way along the directed line segment from $A$ to $B$.

To solve this problem, we will utilize the section formula, a powerful tool that enables us to calculate the coordinates of a point dividing a line segment in a given ratio. The section formula provides a systematic approach to determine the $x$- and $y$-coordinates of point $P$ based on the coordinates of the endpoints $A$ and $B$, as well as the given ratio. By applying this formula, we can accurately pinpoint the location of point $P$ on the line segment.

Understanding Directed Line Segments

Before we delve into the calculations, it's crucial to understand the concept of a directed line segment. A directed line segment is a line segment with a specific direction associated with it. In this case, the directed line segment goes from point $A$ to point $B$, implying that the direction matters. This direction is essential because it dictates the order in which we consider the coordinates of the endpoints when applying the section formula.

Imagine a line segment as a path connecting two points. The direction tells us which point is the starting point and which is the ending point. In our case, $A$ is the starting point, and $B$ is the ending point. This direction influences how we divide the line segment and determine the position of point $P$.

The Section Formula

The section formula is the cornerstone of solving this problem. It provides a mathematical framework for determining the coordinates of a point that divides a line segment in a specific ratio. Let's say we have two points, $A(x_1, y_1)$ and $B(x_2, y_2)$, and we want to find the coordinates of a point $P(x, y)$ that divides the line segment $AB$ in the ratio $m:n$. This means that the distance from $A$ to $P$ is $m$ times some constant, and the distance from $P$ to $B$ is $n$ times the same constant. The section formula then states:

• x-coordinate of P: x = (m / (m + n)) * (x₂ - x₁) + x₁
• y-coordinate of P: y = (m / (m + n)) * (y₂ - y₁) + y₁

These formulas provide a direct way to calculate the coordinates of point $P$ based on the coordinates of $A$ and $B$ and the ratio $m:n$. The formula essentially calculates a weighted average of the coordinates of the endpoints, where the weights are determined by the ratio.

In our specific problem, we are given that point $P$ is one-third of the way along the directed line segment from $A$ to $B$. This implies that the ratio $m:n$ is $1:2$, because the distance from $A$ to $P$ is one part, and the distance from $P$ to $B$ is two parts, making the total distance three parts. We can now substitute these values into the section formula to find the coordinates of $P$.

Applying the Section Formula to Our Problem

Now, let's apply the section formula to our problem where point $P$ is $\frac{1}{3}$ the length of the line segment from $A$ to $B$. As we established earlier, this means the ratio $m:n$ is $1:2$. So, $m = 1$ and $n = 2$. Let the coordinates of point $A$ be $(x_1, y_1)$ and the coordinates of point $B$ be $(x_2, y_2)$. Substituting these values into the section formula, we get:

• x-coordinate of P: x = (1 / (1 + 2)) * (x₂ - x₁) + x₁ = (1/3) * (x₂ - x₁) + x₁
• y-coordinate of P: y = (1 / (1 + 2)) * (y₂ - y₁) + y₁ = (1/3) * (y₂ - y₁) + y₁

These formulas provide us with the x- and y-coordinates of point $P$ in terms of the coordinates of points $A$ and $B$. Notice that the formulas essentially take one-third of the difference in the x- and y-coordinates between $B$ and $A$, and add it to the corresponding coordinate of $A$. This makes intuitive sense, as we are moving one-third of the way from $A$ to $B$.

Simplifying the Formulas

We can further simplify the formulas for the x- and y-coordinates of point $P$. Let's start with the x-coordinate:

x = (1/3) * (x₂ - x₁) + x₁

To simplify this, we can distribute the (1/3) and then combine like terms:

x = (1/3) * x₂ - (1/3) * x₁ + x₁

x = (1/3) * x₂ + (2/3) * x₁

Similarly, we can simplify the formula for the y-coordinate:

y = (1/3) * (y₂ - y₁) + y₁

y = (1/3) * y₂ - (1/3) * y₁ + y₁

y = (1/3) * y₂ + (2/3) * y₁

These simplified formulas provide a more concise way to calculate the coordinates of point $P$. They clearly show that the x-coordinate of $P$ is a weighted average of the x-coordinates of $A$ and $B$, with weights 2/3 and 1/3, respectively. The same holds true for the y-coordinate.

Numerical Example

To illustrate the application of these formulas, let's consider a numerical example. Suppose point $A$ has coordinates (1, 2) and point $B$ has coordinates (7, 8). We want to find the coordinates of point $P$ that is one-third of the way along the directed line segment from $A$ to $B$.

Using the simplified formulas we derived earlier, we can substitute the coordinates of $A$ and $B$:

• x = (1/3) * 7 + (2/3) * 1 = 7/3 + 2/3 = 9/3 = 3
• y = (1/3) * 8 + (2/3) * 2 = 8/3 + 4/3 = 12/3 = 4

Therefore, the coordinates of point $P$ are (3, 4). This means that point $P$ is located at the point (3, 4) on the coordinate plane, which is one-third of the way along the line segment from $A(1, 2)$ to $B(7, 8)$.

Conclusion

In this article, we have explored the problem of finding the coordinates of a point $P$ on a directed line segment from $A$ to $B$ such that $P$ is $\frac{1}{3}$ the length of the line segment from $A$ to $B$. We have seen how the section formula provides a powerful tool for solving this type of problem. By understanding the concept of directed line segments and applying the section formula, we can accurately determine the coordinates of any point that divides a line segment in a given ratio.

The key takeaway is that the coordinates of point $P$ can be calculated as a weighted average of the coordinates of the endpoints $A$ and $B$. The weights are determined by the ratio in which point $P$ divides the line segment. In our specific case, where $P$ is one-third of the way from $A$ to $B$, the x-coordinate of $P$ is (1/3) * x₂ + (2/3) * x₁, and the y-coordinate of $P$ is (1/3) * y₂ + (2/3) * y₁. These formulas provide a clear and concise way to calculate the coordinates of point $P$ for any given points $A$ and $B$.

This concept has wide applications in various fields, including computer graphics, game development, and engineering, where determining the position of points along a line segment is crucial. Understanding the section formula and its applications is therefore a valuable skill for anyone working in these fields.

By mastering the techniques discussed in this article, you will be well-equipped to solve a wide range of problems involving directed line segments and ratios, enhancing your understanding of coordinate geometry and its applications.

By utilizing the section formula, a fundamental tool in coordinate geometry, we can efficiently compute the $x$- and $y$-coordinates of point $P$ located at a specific fraction of the directed line segment from $A$ to $B$. In this case, since $P$ is $\frac{1}{3}$ the length of the segment, we established that the ratio $m:n$ is $1:2$. Substituting this into the section formula yields the simplified expressions for the coordinates of $P$:

• $x = \frac{1}{3}x_2 + \frac{2}{3}x_1$
• $y = \frac{1}{3}y_2 + \frac{2}{3}y_1$

These formulas provide a direct and efficient method for determining the coordinates of point $P$, allowing us to accurately pinpoint its location on the directed line segment. Understanding and applying the section formula is a key skill in coordinate geometry, enabling us to solve a variety of problems involving line segments and ratios.