Finding Coordinates Of Point B Using Section Formula A Comprehensive Guide

In geometry, determining the coordinates of a point that divides a line segment in a given ratio is a fundamental concept. This article delves into a specific problem involving the points $A(-2, -5)$ and $C(13, 15)$ on a straight line ABC, where the ratio of AB to BC is 3:2. Our main objective is to work out the precise coordinates of point B. To effectively solve this, we'll employ the section formula, a vital tool in coordinate geometry. This formula enables us to find the coordinates of a point that divides a line segment internally in a known ratio. Before we tackle the problem head-on, it's crucial to thoroughly understand the section formula and its underlying principles. This article aims to provide a step-by-step solution, ensuring clarity and a comprehensive understanding of the methods involved in solving such geometric problems.

Understanding the Section Formula

The section formula is a cornerstone in coordinate geometry, providing a method to find the coordinates of a point that divides a line segment in a specific ratio. This formula is especially useful when dealing with problems involving the division of lines, whether in a plane or in three-dimensional space. The section formula comes in two primary forms: the internal division formula and the external division formula. For the purpose of this article and the problem at hand, we will focus on the internal division formula, as point B lies between points A and C.

Internal Division

The internal division formula is used when a point divides a line segment into two parts within the segment itself. Suppose we have two points, $A(x_1, y_1)$ and $C(x_2, y_2)$, and a point B that divides the line segment AC internally in the ratio m:n. The coordinates of point B, denoted as $(x, y)$, can be calculated using the following formulas:

$$x = \frac{mx_2 + nx_1}{m + n}$$

$$y = \frac{my_2 + ny_1}{m + n}$$

In these formulas, $x_1$ and $y_1$ are the coordinates of point A, $x_2$ and $y_2$ are the coordinates of point C, and m and n represent the ratio in which B divides AC. The ratio m:n signifies that the distance from A to B is m parts, while the distance from B to C is n parts. This understanding is crucial for correctly applying the section formula.

The significance of the internal division formula lies in its ability to precisely locate a point within a line segment given a specific ratio. It is widely used in various geometric problems, including finding the centroid of a triangle, determining the point of intersection of medians, and solving problems related to similar triangles. The formula is derived from the principles of similar triangles and proportional division of line segments, making it a robust and reliable tool in coordinate geometry.

Applying the Section Formula to Our Problem

Now, let’s apply this knowledge to our specific problem. We are given the coordinates of points A and C, and the ratio in which point B divides the line segment AC. By substituting the given values into the internal division formula, we can systematically calculate the coordinates of point B. This step-by-step application will not only solve the problem but also reinforce our understanding of how the section formula works in practice.

Problem Statement and Given Information

Before we dive into the solution, let’s clearly restate the problem and the given information. This will help ensure that we have a solid foundation for our calculations and reasoning.

We are given two points, $A(-2, -5)$ and $C(13, 15)$, which lie on the straight line ABC. Point B is located on the line segment AC such that the ratio of the length of AB to the length of BC is 3:2. Mathematically, this is expressed as $AB:BC = 3:2$. Our primary goal is to determine the coordinates of point B.

To summarize, we have:

• Coordinates of point A: $(-2, -5)$
• Coordinates of point C: $(13, 15)$
• Ratio $AB:BC = 3:2$

This information is crucial for applying the section formula, which, as discussed earlier, allows us to find the coordinates of a point dividing a line segment in a known ratio. In this case, point B divides the line segment AC internally in the ratio 3:2. Understanding the given information and its implications is the first step towards a successful solution. The coordinates of A and C provide the fixed points, while the ratio gives us the proportional relationship between the segments AB and BC. This relationship is key to using the section formula effectively.

By clearly defining the problem and the given information, we set the stage for a systematic application of the section formula. The next step involves substituting these values into the formula and performing the necessary calculations to find the coordinates of point B. This methodical approach ensures accuracy and a clear understanding of the problem-solving process.

Step-by-Step Solution Using the Section Formula

With a clear understanding of the problem and the section formula, we can now proceed with a step-by-step solution to find the coordinates of point B. This process involves substituting the given values into the formula and performing the arithmetic operations carefully.

Step 1: Identify the Values

First, we need to identify the values from the given information:

• $x_1 = -2$ (x-coordinate of point A)
• $y_1 = -5$ (y-coordinate of point A)
• $x_2 = 13$ (x-coordinate of point C)
• $y_2 = 15$ (y-coordinate of point C)
• $m = 3$ (part of the ratio corresponding to AB)
• $n = 2$ (part of the ratio corresponding to BC)

Step 2: Apply the Section Formula for the x-coordinate

Using the section formula for the x-coordinate, we have:

$$x = \frac{mx_2 + nx_1}{m + n}$$

Substitute the values:

$$x = \frac{3(13) + 2(-2)}{3 + 2}$$

Perform the calculations:

$$x = \frac{39 - 4}{5}$$

$$x = \frac{35}{5}$$

$$x = 7$$

Step 3: Apply the Section Formula for the y-coordinate

Similarly, using the section formula for the y-coordinate, we have:

$$y = \frac{my_2 + ny_1}{m + n}$$

Substitute the values:

$$y = \frac{3(15) + 2(-5)}{3 + 2}$$

Perform the calculations:

$$y = \frac{45 - 10}{5}$$

$$y = \frac{35}{5}$$

$$y = 7$$

Step 4: State the Coordinates of Point B

From our calculations, we have found that the x-coordinate of point B is 7 and the y-coordinate of point B is 7. Therefore, the coordinates of point B are $(7, 7)$.

This step-by-step solution demonstrates the clear and methodical application of the section formula. By breaking down the problem into smaller, manageable steps, we ensure accuracy and clarity in our calculations. The identification of values, substitution into the formula, and arithmetic operations are all performed systematically, leading us to the correct coordinates of point B.

Verification and Conclusion

To ensure the accuracy of our solution, it is always a good practice to verify the results. In this case, we can perform a quick check to see if the coordinates of point B make sense in the context of the given information. After verifying our calculations, we can confidently conclude the problem.

Verification

We found that the coordinates of point B are $(7, 7)$. To verify this, we can consider the relative positions of points A, B, and C. Point B should lie on the line segment AC, and the ratio of AB to BC should be 3:2. While a rigorous verification would involve calculating the distances AB and BC and confirming the ratio, a simple visual check can provide initial reassurance.

Point A is at $(-2, -5)$, and point C is at $(13, 15)$. Point B at $(7, 7)$ is indeed located between A and C. The x-coordinate of B (7) is greater than that of A (-2) and less than that of C (13). Similarly, the y-coordinate of B (7) is greater than that of A (-5) and less than that of C (15). This suggests that point B is correctly positioned on the line segment AC.

For a more precise verification, one could calculate the distances AB and BC using the distance formula and then check if their ratio is approximately 3:2. However, for the purpose of this article, we will rely on the logical consistency of the coordinates and the correct application of the section formula.

Conclusion

In conclusion, by applying the section formula to the given problem, we have successfully determined the coordinates of point B. Given the points $A(-2, -5)$ and $C(13, 15)$ on a straight line ABC, with the ratio $AB:BC = 3:2$, we found that the coordinates of point B are $(7, 7)$.

This problem highlights the importance and utility of the section formula in coordinate geometry. The formula provides a powerful tool for solving problems involving the division of line segments in a specific ratio. The step-by-step approach we used in this article demonstrates how to apply the formula methodically, ensuring accuracy and clarity in the solution. Understanding and mastering the section formula is essential for anyone studying coordinate geometry and related fields. The ability to solve such problems is not only valuable in academic settings but also in various practical applications where spatial relationships and proportional divisions are involved.

Coordinates of point B, Section Formula, Straight Line ABC, Ratio AB BC, Coordinate Geometry, Internal Division, Problem Solving, Step-by-Step Solution, Verification, Geometric Problems.