Finding Coordinates Of A Point On A Directed Line Segment

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In mathematics, particularly in coordinate geometry, we often encounter the problem of finding the coordinates of a point that lies on a line segment. This problem becomes especially interesting when we consider directed line segments, where the direction from one endpoint to the other matters. This article delves into the specifics of determining the coordinates of a point $P$ on a directed line segment from point $A$ to point $B$, where $P$ is located at a specific fraction of the total length of the segment. Specifically, we will address the scenario where $P$ is $\frac{2}{3}$ the length of the line segment from $A$ to $B$. This involves understanding how to apply the section formula, a crucial concept in coordinate geometry, and how it helps us find the $x$ and $y$ coordinates of the desired point. Let's explore the underlying principles and the step-by-step approach to solving this type of problem.

The Section Formula: A Foundation

At the heart of finding the coordinates of point $P$ lies the section formula. This formula provides a method to calculate the coordinates of a point that divides a line segment in a given ratio. Imagine a line segment connecting two points, $A(x_1, y_1)$ and $B(x_2, y_2)$. If a point $P(x, y)$ divides this segment in the ratio $m:n$, then the coordinates of $P$ can be found using the following formulas:

• $x = \frac{m x_2 + n x_1}{m + n}$
• $y = \frac{m y_2 + n y_1}{m + n}$

These formulas are derived from the principles of similar triangles and proportional division. They essentially state that the coordinates of the dividing point are a weighted average of the coordinates of the endpoints, where the weights are determined by the ratio $m:n$. Understanding and applying the section formula correctly is crucial for solving problems related to line segments and coordinate geometry. The ratio $m:n$ represents the division of the line segment. For instance, if $m = 1$ and $n = 1$, the point $P$ is the midpoint of the segment. If $m = 2$ and $n = 1$, the point $P$ is located two-thirds of the way from $A$ to $B$. Let's see how we can adapt the section formula to the specific problem of finding a point at a given fraction of the length of a line segment.

Applying the Section Formula to Our Problem

In our case, we are given that point $P$ is $\frac{2}{3}$ the length of the line segment from $A$ to $B$. This means that the ratio in which $P$ divides the segment is $2:1$. To see this, consider that if $P$ is $\frac{2}{3}$ of the way from $A$ to $B$, then the remaining portion of the segment from $P$ to $B$ is $\frac{1}{3}$ of the total length. Therefore, the lengths of the segments $AP$ and $PB$ are in the ratio of $2:1$. We can now use the section formula with $m = 2$ and $n = 1$ to find the coordinates of point $P$.

Let $A$ have coordinates $(x_1, y_1)$ and $B$ have coordinates $(x_2, y_2)$. Plugging $m = 2$ and $n = 1$ into the section formula, we get:

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

These formulas give us the $x$ and $y$ coordinates of point $P$ in terms of the coordinates of points $A$ and $B$. To make these formulas even more insightful, we can rewrite them in a slightly different form, which highlights the concept of weighted averages and the fraction of the distance.

Rewriting the Formula: A Weighted Average Perspective

The formulas we derived above can be rearranged to provide a more intuitive understanding of how the coordinates of point $P$ are determined. We can rewrite the formula for the $x$-coordinate as follows:

$x = \frac{2x_2 + x_1}{3} = \frac{2}{3}x_2 + \frac{1}{3}x_1$

Similarly, for the $y$-coordinate, we have:

$y = \frac{2y_2 + y_1}{3} = \frac{2}{3}y_2 + \frac{1}{3}y_1$

These rewritten formulas show that the $x$-coordinate of $P$ is a weighted average of the $x$-coordinates of $A$ and $B$, where the weights are $\frac{1}{3}$ and $\frac{2}{3}$, respectively. The same applies to the $y$-coordinates. This perspective is crucial because it allows us to interpret the coordinates of $P$ as a blend of the coordinates of the endpoints, with the weights reflecting the fraction of the distance $P$ is from each endpoint. Now, let's rewrite the x-coordinate equation to align more closely with the initial equation presented in the problem statement. This will help to illustrate the connection between the general section formula and the specific case we are addressing.

Connecting to the Given Formula: A Transformation

The original problem statement provides a formula for the $x$-coordinate of point $P$ in the form:

$x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1$

Let's demonstrate how this formula is equivalent to the one we derived using the section formula. We start with our result:

$x = \frac{2x_2 + x_1}{3}$

Since we established that $P$ is $\frac{2}{3}$ of the way from $A$ to $B$, we have $m = 2$ and $n = 1$. Plugging these values into the given formula, we get:

$x = \left(\frac{2}{2+1}\right)(x_2 - x_1) + x_1$

Simplifying this, we have:

$x = \frac{2}{3}(x_2 - x_1) + x_1$

Expanding the term, we get:

$x = \frac{2}{3}x_2 - \frac{2}{3}x_1 + x_1$

Combining the terms with $x_1$, we have:

$x = \frac{2}{3}x_2 + \left(1 - \frac{2}{3}\right)x_1$

Simplifying further:

$x = \frac{2}{3}x_2 + \frac{1}{3}x_1$

This is precisely the formula we derived earlier using the section formula. This equivalence demonstrates the consistency of the section formula and its adaptability to different forms of expression. The key takeaway here is that both formulas describe the same mathematical relationship, just expressed in slightly different ways. The first form highlights the proportional division of the segment, while the second form emphasizes the weighted average perspective. The formula we've just manipulated provides a direct method for calculating the $x$-coordinate, and a similar approach can be used for the $y$-coordinate. Let's now formulate the complete solution for both coordinates.

The Complete Solution: $x$ and $y$ Coordinates

We've established the formula for the $x$-coordinate of point $P$. Now, let's derive the corresponding formula for the $y$-coordinate. Following the same logic, we can start with the general section formula and adapt it to our specific case where $P$ is $\frac{2}{3}$ of the way from $A$ to $B$.

Using the same approach, we can express the $y$-coordinate as:

$y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1$

Substituting $m = 2$ and $n = 1$, we get:

$y = \left(\frac{2}{2+1}\right)(y_2 - y_1) + y_1$

Simplifying, we have:

$y = \frac{2}{3}(y_2 - y_1) + y_1$

Expanding the term, we get:

$y = \frac{2}{3}y_2 - \frac{2}{3}y_1 + y_1$

Combining the terms with $y_1$, we have:

$y = \frac{2}{3}y_2 + \left(1 - \frac{2}{3}\right)y_1$

Simplifying further:

$y = \frac{2}{3}y_2 + \frac{1}{3}y_1$

Therefore, the complete solution for the coordinates of point $P$ is:

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

These formulas provide a straightforward method for finding the coordinates of point $P$ given the coordinates of points $A$ and $B$. By plugging in the coordinates of $A$ and $B$ into these formulas, we can easily calculate the coordinates of the point that divides the directed line segment in the specified ratio. Let's recap the key steps involved in solving this type of problem.

Recapping the Key Steps

To find the coordinates of a point $P$ on a directed line segment $AB$ such that $P$ is a fraction of the distance from $A$ to $B$, we follow these key steps:

1. Identify the ratio: Determine the ratio in which point $P$ divides the line segment $AB$. In our case, since $P$ is $\frac{2}{3}$ of the way from $A$ to $B$, the ratio is $2:1$.
2. Apply the section formula: Use the section formula to express the coordinates of $P$ in terms of the coordinates of $A$ and $B$ and the ratio. We have shown two forms of the section formula: the general form and the form provided in the problem statement.
3. Substitute the values: Plug in the given coordinates of points $A$ and $B$ and the values of $m$ and $n$ (from the ratio) into the formulas.
4. Simplify: Simplify the expressions to find the $x$ and $y$ coordinates of point $P$.

By following these steps, we can confidently solve problems involving directed line segments and fractional distances. This method is widely applicable in various areas of mathematics and has practical applications in fields such as computer graphics, engineering, and physics. Understanding the section formula and its applications is a fundamental skill in coordinate geometry. Let's consider a numerical example to solidify our understanding of the process.

Numerical Example: Putting Theory into Practice

To illustrate the application of the formulas we derived, let's consider a specific example. Suppose we have two points, $A(1, 2)$ and $B(4, 8)$. We want to find the coordinates of point $P$ that is $\frac{2}{3}$ of the way from $A$ to $B$. Using our formulas, we can proceed as follows:

1. Identify the coordinates: $x_1 = 1$, $y_1 = 2$, $x_2 = 4$, $y_2 = 8$.
2. Apply the formulas:
   • $x = \frac{2}{3}(x_2 - x_1) + x_1 = \frac{2}{3}(4 - 1) + 1$
   • $y = \frac{2}{3}(y_2 - y_1) + y_1 = \frac{2}{3}(8 - 2) + 2$
3. Simplify:
   • $x = \frac{2}{3}(3) + 1 = 2 + 1 = 3$
   • $y = \frac{2}{3}(6) + 2 = 4 + 2 = 6$

Therefore, the coordinates of point $P$ are $(3, 6)$. This example demonstrates how easily we can find the coordinates of a point dividing a line segment in a given ratio by applying the formulas we've discussed. By understanding the section formula and its variants, we can solve a wide range of problems in coordinate geometry and related fields. In conclusion, let's summarize the key takeaways from this exploration.

Conclusion: Key Takeaways

In this article, we have explored the method for finding the coordinates of a point $P$ on a directed line segment $AB$ such that $P$ is a fraction of the distance from $A$ to $B$. Specifically, we focused on the case where $P$ is $\frac{2}{3}$ of the way from $A$ to $B$. We've covered the following key points:

• The Section Formula: We introduced the section formula as the foundation for solving this type of problem.
• Ratio Identification: We emphasized the importance of correctly identifying the ratio in which the point divides the line segment.
• Formula Application: We demonstrated how to apply the section formula in different forms to find the coordinates of the point.
• Weighted Average Perspective: We highlighted the interpretation of the coordinates as a weighted average of the endpoints' coordinates.
• Numerical Example: We provided a numerical example to illustrate the practical application of the formulas.

By understanding these concepts and practicing the techniques, you can confidently tackle problems involving directed line segments and fractional distances in coordinate geometry. The ability to manipulate and apply these formulas is a valuable skill in mathematics and its applications. This exploration provides a comprehensive understanding of the topic and equips readers with the knowledge and tools to solve similar problems effectively.