Finding Coordinates On A Directed Line Segment A Comprehensive Guide

by Admin 69 views

Introduction

In mathematics, particularly in coordinate geometry, determining the coordinates of a point on a directed line segment is a fundamental concept. This article delves into the method of finding the coordinates of a point P on a directed line segment from point A to point B, such that P is located at a specific fraction of the distance from A to B. Specifically, we will explore how to find the x- and y-coordinates of point P when it is located at 23{\frac{2}{3}} the length of the line segment from A to B. This concept is crucial in various fields, including computer graphics, engineering, and physics, where precise positioning and division of line segments are essential.

Understanding Directed Line Segments

A directed line segment, unlike a regular line segment, has a specific direction and magnitude. It starts at an initial point and ends at a terminal point. The direction is crucial because it indicates the order in which the points are considered. In our case, the directed line segment goes from point A to point B, making A the initial point and B the terminal point. The magnitude is simply the length of the segment. To find a point P that divides this segment in a given ratio, we need to consider both the coordinates of A and B and the fraction representing the desired division. The formula provided, x=(mm+n)(x2−x1)+x1{x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1}, is a key component in this process, allowing us to calculate the x-coordinate of point P based on the coordinates of A and B and the ratio mm+n{\frac{m}{m+n}}. This article will provide step-by-step explanations and examples to clarify how this formula, and its counterpart for the y-coordinate, are applied in practice.

The Section Formula

The section formula is a cornerstone in coordinate geometry for determining the coordinates of a point that divides a line segment in a given ratio. When dealing with a directed line segment, this formula becomes particularly useful. Let's consider two points, A with coordinates (x1,y1){(x_1, y_1)} and B with coordinates (x2,y2){(x_2, y_2)}. We want to find the coordinates of a point P that lies on the line segment AB such that the ratio of AP to PB is m:n. In other words, P divides the line segment AB internally in the ratio m:n. The coordinates of point P, denoted as (x,y){(x, y)}, can be calculated using the section formula:

  • x=(mm+n)(x2−x1)+x1{x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1}
  • y=(mm+n)(y2−y1)+y1{y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1}

These equations are derived from the principles of similar triangles and proportional division. The term mm+n{\frac{m}{m+n}} represents the fraction of the distance from A to B at which point P is located. The differences (x2−x1){(x_2 - x_1)} and (y2−y1){(y_2 - y_1)} represent the change in x and y coordinates from A to B, respectively. By multiplying these changes by the fraction mm+n{\frac{m}{m+n}}, we find the change in coordinates from A to P. Adding these changes to the coordinates of A gives us the coordinates of P. This method ensures that P lies on the line segment AB and divides it in the desired ratio. Understanding and applying the section formula is essential for solving problems involving directed line segments and coordinate geometry.

Calculating Coordinates of Point P

To find the coordinates of point P, which is 23{\frac{2}{3}} the length of the line segment from A to B, we apply the section formula. This involves identifying the values for x1{x_1}, y1{y_1}, x2{x_2}, y2{y_2}, and the ratio mm+n{\frac{m}{m+n}}. The coordinates of point A are represented as (x1,y1){(x_1, y_1)}, and the coordinates of point B are represented as (x2,y2){(x_2, y_2)}. The ratio mm+n{\frac{m}{m+n}} is given as 23{\frac{2}{3}}, indicating that point P is located two-thirds of the way from A to B. This means that for every 2 units of distance from A to P, there is 1 unit of distance from P to B. Substituting these values into the section formula will allow us to calculate the x- and y-coordinates of point P. The process involves a straightforward application of the formulas, ensuring that the correct values are substituted for each variable. The result will be the precise location of point P on the directed line segment AB.

Applying the Formula for x-coordinate

Let's delve into the process of calculating the x-coordinate of point P. The formula we use is:

x=(mm+n)(x2−x1)+x1{x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1}

Here, x1{x_1} is the x-coordinate of point A, and x2{x_2} is the x-coordinate of point B. The ratio mm+n{\frac{m}{m+n}} is given as 23{\frac{2}{3}}. To apply this formula, we substitute the known values. For instance, if we assume point A has coordinates (1, 2) and point B has coordinates (4, 5), then x1=1{x_1 = 1} and x2=4{x_2 = 4}. Plugging these values into the formula gives us:

x=(23)(4−1)+1{x = \left(\frac{2}{3}\right)(4 - 1) + 1}

Simplifying the expression, we first calculate the difference inside the parentheses:

x=(23)(3)+1{x = \left(\frac{2}{3}\right)(3) + 1}

Next, we multiply the fraction by the result:

x=2+1{x = 2 + 1}

Finally, we add the remaining value to find the x-coordinate of point P:

x=3{x = 3}

Thus, the x-coordinate of point P is 3. This step-by-step calculation demonstrates how the formula is applied to find the x-coordinate of a point on a directed line segment. Understanding this process is crucial for accurately determining the position of point P.

Applying the Formula for y-coordinate

Similarly, to find the y-coordinate of point P, we use the counterpart of the section formula:

y=(mm+n)(y2−y1)+y1{y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1}

In this equation, y1{y_1} represents the y-coordinate of point A, and y2{y_2} represents the y-coordinate of point B. Again, the ratio mm+n{\frac{m}{m+n}} is 23{\frac{2}{3}}. Using the same example points A(1, 2) and B(4, 5), we have y1=2{y_1 = 2} and y2=5{y_2 = 5}. Substituting these values into the formula, we get:

y=(23)(5−2)+2{y = \left(\frac{2}{3}\right)(5 - 2) + 2}

First, we simplify the expression inside the parentheses:

y=(23)(3)+2{y = \left(\frac{2}{3}\right)(3) + 2}

Then, we multiply the fraction by the result:

y=2+2{y = 2 + 2}

Finally, we add the remaining value to find the y-coordinate of point P:

y=4{y = 4}

Therefore, the y-coordinate of point P is 4. This calculation, mirroring the process for the x-coordinate, illustrates the application of the section formula to determine the vertical position of point P on the directed line segment. By combining the calculated x-coordinate and y-coordinate, we can precisely locate point P in the coordinate plane.

Example Calculation

Let's solidify our understanding with a concrete example. Suppose point A has coordinates (-2, 1) and point B has coordinates (4, 7). We want to find the coordinates of point P, which is 23{\frac{2}{3}} the distance from A to B. We will apply the section formula for both the x- and y-coordinates.

Calculating the x-coordinate

Using the formula:

x=(mm+n)(x2−x1)+x1{x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1}

We substitute the given values: x1=−2{x_1 = -2}, x2=4{x_2 = 4}, and mm+n=23{\frac{m}{m+n} = \frac{2}{3}}. Plugging these values into the equation, we get:

x=(23)(4−(−2))+(−2){x = \left(\frac{2}{3}\right)(4 - (-2)) + (-2)}

Simplifying the expression inside the parentheses:

x=(23)(6)+(−2){x = \left(\frac{2}{3}\right)(6) + (-2)}

Multiplying the fraction by the result:

x=4+(−2){x = 4 + (-2)}

Finally, adding the remaining values:

x=2{x = 2}

Thus, the x-coordinate of point P is 2.

Calculating the y-coordinate

Now, we calculate the y-coordinate using the formula:

y=(mm+n)(y2−y1)+y1{y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1}

Substituting the given values: y1=1{y_1 = 1}, y2=7{y_2 = 7}, and mm+n=23{\frac{m}{m+n} = \frac{2}{3}}. The equation becomes:

y=(23)(7−1)+1{y = \left(\frac{2}{3}\right)(7 - 1) + 1}

Simplifying the expression inside the parentheses:

y=(23)(6)+1{y = \left(\frac{2}{3}\right)(6) + 1}

Multiplying the fraction by the result:

y=4+1{y = 4 + 1}

Finally, adding the remaining values:

y=5{y = 5}

Therefore, the y-coordinate of point P is 5. Combining the x- and y-coordinates, we find that point P has coordinates (2, 5). This example illustrates the practical application of the section formula in determining the precise location of a point on a directed line segment.

Conclusion

In conclusion, finding the coordinates of a point P on a directed line segment from A to B, such that P is a fraction of the distance from A to B, is a crucial concept in coordinate geometry. By applying the section formula, we can accurately determine the x- and y-coordinates of point P. The formula x=(mm+n)(x2−x1)+x1{x = \left(\frac{m}{m+n}\right)(x_2 - x_1) + x_1} and its counterpart for the y-coordinate, y=(mm+n)(y2−y1)+y1{y = \left(\frac{m}{m+n}\right)(y_2 - y_1) + y_1}, provide a systematic approach to solving such problems. Understanding and utilizing these formulas is essential for various applications in mathematics, engineering, and computer graphics, where precise positioning and division of line segments are necessary. The examples and step-by-step calculations provided in this article serve to reinforce the practical application of the section formula, ensuring that readers can confidently tackle similar problems.