Determining Coordinates Of Points Dividing A Line Segment In A Given Ratio

This article delves into the fundamental concepts of coordinate geometry, specifically focusing on how to determine the coordinates of points that divide a line segment internally and externally in a given ratio. We will explore the section formula and its applications through a detailed example. Let's consider the scenario where points $A$ and $B$ divide the line segment $PQ$ internally and externally, respectively, in the ratio $5:4$. Our goal is to find the precise coordinates of these points, given that $P$ is $(3,2)$ and $Q$ is $(7,q)$. This problem highlights the practical application of section formulas and provides a solid foundation for understanding more complex geometric problems.

Understanding the Section Formula

At the heart of solving this problem lies the section formula, a cornerstone concept in coordinate geometry. This formula allows us to calculate the coordinates of a point that divides a line segment in a specific ratio. The section formula distinguishes between two scenarios: internal division and external division. Internal division refers to a point lying within the line segment, while external division involves a point lying on the extension of the line segment. Understanding this distinction is crucial for applying the correct version of the formula and arriving at the accurate coordinates.

The internal division formula states that if a point $A(x, y)$ divides the line segment joining $P(x_1, y_1)$ and $Q(x_2, y_2)$ internally in the ratio $m:n$, then the coordinates of $A$ are given by:

$$x = \frac{mx_2 + nx_1}{m + n}, y = \frac{my_2 + ny_1}{m + n}$$

This formula essentially calculates a weighted average of the x-coordinates and y-coordinates of the endpoints, with the weights determined by the ratio $m:n$. The ratio signifies how the point $A$ proportionally divides the segment $PQ$. If the ratio is $1:1$, it implies that $A$ is the midpoint of the segment, and the formula simplifies to the midpoint formula. The section formula is not merely a mathematical equation; it is a powerful tool for analyzing and solving problems related to line segments and their divisions in the coordinate plane.

Conversely, the external division formula comes into play when the point $B$ divides the line segment $PQ$ externally in the ratio $m:n$. This means that $B$ lies on the extended line $PQ$, and the ratio $m:n$ represents the relative distances from $B$ to $P$ and $Q$. The coordinates of $B(x, y)$ are then calculated using the following formula:

$$x = \frac{mx_2 - nx_1}{m - n}, y = \frac{my_2 - ny_1}{m - n}$$

The external division formula bears a striking resemblance to the internal division formula, with the key difference being the subtraction sign in the numerator and denominator. This sign change reflects the fact that external division involves extending the line segment, and the point of division lies outside the segment itself. The denominator $(m-n)$ highlights that this formula is valid only when $m$ is not equal to $n$. If $m$ equals $n$, the formula becomes undefined, indicating that there is no point of external division in this specific ratio. The external division formula finds applications in various geometric constructions and problems, particularly those involving harmonic division and inversions.

Applying the Section Formula to Find Coordinates of Point A

Now, let's apply the internal division formula to find the coordinates of point $A$. Point $A$ divides the line segment $PQ$ internally in the ratio $5:4$, where $P$ is $(3,2)$ and $Q$ is $(7,q)$. Using the internal division formula, we have $m = 5$, $n = 4$, $x_1 = 3$, $y_1 = 2$, $x_2 = 7$, and $y_2 = q$. Substituting these values into the internal division formula, we can determine the coordinates of point $A$.

The x-coordinate of $A$, denoted as $x_A$, can be calculated as follows:

$$x_A = \frac{5(7) + 4(3)}{5 + 4} = \frac{35 + 12}{9} = \frac{47}{9}$$

Similarly, the y-coordinate of $A$, denoted as $y_A$, is given by:

$$y_A = \frac{5(q) + 4(2)}{5 + 4} = \frac{5q + 8}{9}$$

Therefore, the coordinates of point $A$ are $\left(\frac{47}{9}, \frac{5q + 8}{9}\right)$. These coordinates are expressed in terms of $q$, the y-coordinate of point $Q$. To fully determine the coordinates of $A$, we need to know the value of $q$. However, at this stage, we have successfully expressed the coordinates of $A$ in terms of the unknown $q$, which is a significant step towards solving the problem. This application demonstrates how the internal division formula enables us to locate a point dividing a line segment internally, given the ratio and the coordinates of the endpoints. The calculation involves a straightforward substitution of values into the formula and simplification, highlighting the formula's practicality and ease of use.

Applying the Section Formula to Find Coordinates of Point B

Next, we turn our attention to finding the coordinates of point $B$, which divides the line segment $PQ$ externally in the ratio $5:4$. This requires us to use the external division formula. As before, $P$ is $(3,2)$ and $Q$ is $(7,q)$, with the ratio being $5:4$. This means $m = 5$, $n = 4$, $x_1 = 3$, $y_1 = 2$, $x_2 = 7$, and $y_2 = q$. We substitute these values into the external division formula to find the coordinates of point $B$.

The x-coordinate of $B$, denoted as $x_B$, is calculated using the external division formula:

$$x_B = \frac{5(7) - 4(3)}{5 - 4} = \frac{35 - 12}{1} = 23$$

Similarly, the y-coordinate of $B$, denoted as $y_B$, is given by:

$$y_B = \frac{5(q) - 4(2)}{5 - 4} = \frac{5q - 8}{1} = 5q - 8$$

Thus, the coordinates of point $B$ are $(23, 5q - 8)$. Similar to the coordinates of point $A$, the coordinates of $B$ are also expressed in terms of $q$. To obtain the numerical values for the coordinates of $B$, we would need to know the value of $q$. This process demonstrates the application of the external division formula in determining the location of a point that divides a line segment externally in a given ratio. The subtraction in the formula accounts for the fact that the point $B$ lies on the extension of the line segment $PQ$, and the calculations provide the coordinates of this point in relation to the endpoints and the division ratio.

Solving for q and Final Coordinates

To fully determine the coordinates of both points $A$ and $B$, we need to find the value of $q$. The problem statement provides the coordinates of point P as (3, 2) and point Q as (7, q). Points A and B divide the line segment PQ internally and externally, respectively, in the ratio 5:4. We have already found the coordinates of A and B in terms of q.

Point A: (47/9, (5q + 8)/9) Point B: (23, 5q - 8)

To find the value of q, additional information or constraints are needed, which are not provided in the given problem statement. In many similar problems, there might be additional conditions, such as the line segment being perpendicular to another line, or the points lying on a specific curve. Without such information, we can only express the coordinates of A and B in terms of q.

However, if we assume there was a missing piece of information and we somehow deduced a value for q, for example, let's imagine we found out that q = 4, we could then proceed to substitute this value into our expressions for the coordinates of A and B.

If q = 4, then the coordinates of point A would be:

Point A_x = 47/9 Point A_y = (5(4) + 8)/9 = (20 + 8)/9 = 28/9

So, if q = 4, the coordinates of point A are (47/9, 28/9).

Similarly, for point B, if q = 4, the coordinates would be:

Point B_x = 23 Point B_y = 5(4) - 8 = 20 - 8 = 12

Thus, if q = 4, the coordinates of point B are (23, 12).

This illustrates how, with a specific value of q, we can find the exact numerical coordinates of the points A and B. The process involves substituting the value of q into the expressions derived from the section formulas and performing the arithmetic operations. Without a specific value for q, we can only express the coordinates of A and B in terms of q, leaving the final numerical coordinates undetermined.

Conclusion

In conclusion, this article demonstrated how to find the coordinates of points dividing a line segment internally and externally using the section formulas. We applied these formulas to a specific example, expressing the coordinates of the division points in terms of an unknown variable, $q$. While a complete numerical solution requires additional information to determine the value of $q$, the process outlined here provides a clear framework for solving such problems. The section formulas are fundamental tools in coordinate geometry, enabling us to analyze and understand the relationships between points and lines in a coordinate plane. Mastering these formulas is essential for tackling more advanced geometric problems and applications.