Finding Coordinates When A Point Divides A Line Segment In A 1:3 Ratio

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Introduction

In coordinate geometry, a fundamental concept is finding the coordinates of a point that divides a line segment in a given ratio. This is particularly useful in various mathematical and real-world applications, from computer graphics to engineering. This article addresses the question of determining the coordinates of a point C that divides a line segment in the ratio 1:3. We will explore the section formula, its derivation, and step-by-step methods to solve this type of problem. Understanding the section formula is essential for students and professionals dealing with geometric problems in a coordinate plane. Let’s delve into the details and understand how to find the coordinates of point C.

Understanding the Section Formula

The section formula is a crucial tool in coordinate geometry for finding the coordinates of a point that divides a line segment in a given ratio. To effectively use this formula, it's important to understand its components and underlying principles. The section formula is applied when we have a line segment with endpoints at known coordinates, and we want to find the coordinates of a point that divides this segment in a specific ratio. The ratio represents how the line segment is divided, indicating the proportional distances from the dividing point to each endpoint.

What is the Section Formula?

The section formula provides a method for determining the coordinates of a point that divides a line segment joining two points in a given ratio internally. If we have two points, say A(x₁, y₁) and B(x₂, y₂), and a point C divides the line segment AB in the ratio m:n, then the coordinates of point C (x, y) can be found using the following formulas:

  • x = (m * x₂ + n * x₁) / (m + n)
  • y = (m * y₂ + n * y₁) / (m + n)

Here, m and n represent the parts into which the line segment is divided. The ratio m:n signifies that the segment AC is m parts while the segment CB is n parts. The section formula is derived from the principles of similar triangles and proportional division in the coordinate plane. Understanding this derivation can provide deeper insight into how the formula works and why it is effective.

Derivation of the Section Formula

The derivation of the section formula involves geometric principles and coordinate geometry concepts. Consider two points A(x₁, y₁) and B(x₂, y₂) in the coordinate plane. Let C(x, y) be the point that divides the line segment AB in the ratio m:n. This means that the ratio of the length AC to CB is m/n.

To derive the formula, we draw perpendiculars from points A, B, and C to the x-axis, labeling the feet of these perpendiculars as P, Q, and R, respectively. Similarly, we draw perpendiculars from A and C to lines CR and BQ, naming the intersection points S and T, respectively. From this construction, we create two similar triangles: ΔACS and ΔCTB. The similarity of these triangles allows us to set up proportions between their corresponding sides.

Using the properties of similar triangles, we can establish the following relationships:

  • AC/CB = AS/CT = CS/BT
  • AS = x - x₁
  • CT = x₂ - x
  • CS = y - y₁
  • BT = y₂ - y

Since AC/CB = m/n, we can write the proportional relationships as:

  • (x - x₁) / (x₂ - x) = m/n
  • (y - y₁) / (y₂ - y) = m/n

Solving these equations for x and y, we derive the section formula:

  • x = (m * x₂ + n * x₁) / (m + n)
  • y = (m * y₂ + n * y₁) / (m + n)

This derivation highlights the mathematical foundation of the section formula, illustrating how it arises from geometric principles and the properties of similar triangles.

Internal vs. External Division

The section formula primarily deals with internal division, where the point C lies between the points A and B on the line segment. However, there is also the concept of external division, where point C lies outside the segment AB, on the extended line. In external division, the ratio is considered in terms of directed distances, and one of the ratio parts is negative.

For external division, if point C divides the line segment AB externally in the ratio m:n, the coordinates of C (x, y) are given by:

  • x = (m * x₂ - n * x₁) / (m - n)
  • y = (m * y₂ - n * y₁) / (m - n)

The key difference in the formula for external division is the subtraction in the numerator and denominator, which accounts for the point C being outside the line segment. It’s essential to understand whether a problem involves internal or external division, as this will determine which version of the section formula to apply.

Step-by-Step Solution to Finding Coordinates

To efficiently find the coordinates of a point dividing a line segment in a given ratio, it’s essential to follow a systematic approach. This step-by-step solution ensures accuracy and helps in understanding the underlying concepts. In this section, we will break down the process into manageable steps, providing clarity and ease of application.

Step 1: Identify the Given Information

The first step in solving any coordinate geometry problem is to identify the given information accurately. This involves noting down the coordinates of the endpoints of the line segment and the ratio in which the segment is divided. Let's assume we have two points, A(x₁, y₁) and B(x₂, y₂), and a point C divides the line segment AB in the ratio m:n. In our specific question, the ratio is 1:3.

For instance, suppose we have a line segment joining points A(1, 2) and B(9, 10), and point C divides this segment in the ratio 1:3. Here, we identify:

  • A(x₁, y₁) = (1, 2)
  • B(x₂, y₂) = (9, 10)
  • Ratio m:n = 1:3

Accurately identifying the given information is crucial because it forms the basis for applying the section formula correctly. Any error in this step can lead to an incorrect solution. Therefore, double-check the coordinates and the ratio before proceeding further.

Step 2: Apply the Section Formula

Once we have correctly identified the given information, the next step is to apply the section formula. As we learned earlier, the section formula is used to find the coordinates of a point that divides a line segment internally in a given ratio. For a point C(x, y) dividing the line segment joining A(x₁, y₁) and B(x₂, y₂) in the ratio m:n, the coordinates of C are given by:

  • x = (m * x₂ + n * x₁) / (m + n)
  • y = (m * y₂ + n * y₁) / (m + n)

Using our example where A(1, 2), B(9, 10), and the ratio is 1:3, we substitute these values into the formula:

  • x = (1 * 9 + 3 * 1) / (1 + 3)
  • y = (1 * 10 + 3 * 2) / (1 + 3)

This substitution is a critical step. Make sure you are placing the values correctly in the formula to avoid any calculation errors. Taking the time to double-check the substitution will ensure the accuracy of the next step.

Step 3: Calculate the Coordinates

After applying the section formula, the next step involves performing the necessary arithmetic to calculate the coordinates of point C. This is where we simplify the expressions obtained in the previous step to find the numerical values of x and y.

Continuing with our example, we have:

  • x = (1 * 9 + 3 * 1) / (1 + 3) = (9 + 3) / 4 = 12 / 4 = 3
  • y = (1 * 10 + 3 * 2) / (1 + 3) = (10 + 6) / 4 = 16 / 4 = 4

Thus, the coordinates of point C are (3, 4). This step requires careful arithmetic to avoid any mistakes. It's advisable to double-check your calculations, especially if dealing with complex numbers or fractions. Accurate calculations are essential to arrive at the correct coordinates of the dividing point.

Step 4: Verify the Solution

Once the coordinates of point C are calculated, the final step is to verify the solution. This ensures that the calculated coordinates are correct and consistent with the given information. One way to verify is to use the distance formula to check if the ratio of the lengths AC to CB matches the given ratio m:n.

In our example, we found C(3, 4). Let’s verify this solution. First, calculate the lengths AC and CB using the distance formula:

  • Distance AC = √[(3 - 1)² + (4 - 2)²] = √[2² + 2²] = √8
  • Distance CB = √[(9 - 3)² + (10 - 4)²] = √[6² + 6²] = √72

Now, find the ratio AC/CB:

  • AC/CB = √8 / √72 = √(8/72) = √(1/9) = 1/3

The calculated ratio matches the given ratio of 1:3, which confirms that our solution is correct. Another method to verify is to visually plot the points on a coordinate plane and check if the position of point C appears to divide the line segment AB in the given ratio. Verification is a crucial step to ensure the accuracy of your solution and to reinforce your understanding of the section formula.

Applying the Solution to the Given Problem

Now that we have a clear understanding of the section formula and the step-by-step solution, let's apply it to the given problem. The question asks: If point C divides the line segment in the ratio 1:3, what are the coordinates of C? To solve this, we need additional information, specifically the coordinates of the endpoints of the line segment. Without these, we cannot directly apply the section formula. However, we can discuss a general approach and then consider hypothetical endpoints to demonstrate the solution.

General Approach

To find the coordinates of point C, we need the coordinates of the endpoints, say A(x₁, y₁) and B(x₂, y₂). Once we have these, we can use the section formula as follows:

Given:

  • Ratio m:n = 1:3
  • Coordinates of A(x₁, y₁) and B(x₂, y₂)

Apply the section formula:

  • x = (m * x₂ + n * x₁) / (m + n)
  • y = (m * y₂ + n * y₁) / (m + n)

Substitute the given values into the formula and calculate the coordinates of point C(x, y). This general approach is applicable to any problem where a point divides a line segment in a given ratio.

Example with Hypothetical Endpoints

Let’s consider a hypothetical line segment with endpoints A(1, 1) and B(9, 9). Point C divides this segment in the ratio 1:3. We can now apply the section formula:

  • A(x₁, y₁) = (1, 1)
  • B(x₂, y₂) = (9, 9)
  • Ratio m:n = 1:3

Using the section formula:

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

So, the coordinates of point C are (3, 3). This example illustrates how to apply the section formula when the endpoints are known. By following this approach, you can solve any similar problem.

Addressing the Multiple-Choice Options

Without the specific coordinates of the endpoints, we cannot definitively choose one of the multiple-choice options (A. (7,3), B. (5,-1), C. (-1,5), D. (3,7)). To select the correct answer, we would need the endpoint coordinates and then apply the section formula to see which option matches our calculated coordinates for point C.

For demonstration purposes, let’s assume the endpoints are such that option A, (7, 3), is the correct answer. This implies that if we were given certain endpoints, applying the section formula with a ratio of 1:3 would result in C(7, 3). To illustrate, we can work backward from the section formula to find possible endpoints that would yield this result. However, without the initial coordinates, we cannot confirm this assumption.

Real-World Applications of the Section Formula

The section formula is not just a theoretical concept; it has numerous practical applications in various fields. Understanding its real-world relevance can enhance appreciation for its importance and utility. From computer graphics to urban planning, the section formula provides a valuable tool for solving geometric problems in a coordinate system.

Computer Graphics

In computer graphics, the section formula is used extensively for tasks such as line interpolation and object transformations. When creating animations or 3D models, it’s often necessary to calculate intermediate points along a line segment. For example, if you want to draw a smooth line or curve between two points, you can use the section formula to find equally spaced points along that line. This is crucial for rendering graphics smoothly and realistically.

Object transformations, such as scaling, rotation, and translation, also rely on the section formula. When an object is scaled, the coordinates of its vertices need to be recalculated. The section formula can be used to determine the new positions of these vertices relative to a fixed point or another part of the object. Similarly, in rotations, the formula helps in calculating the new coordinates of points after the rotation transformation.

Navigation and Mapping

The section formula finds applications in navigation and mapping systems. For instance, determining intermediate locations along a route can be done using this formula. If a map shows two locations and you need to find a specific point along the direct path between them, given a certain proportion of the distance, the section formula can provide the coordinates of that point.

Surveying and geographic information systems (GIS) also utilize the section formula. Surveyors often need to divide land or map out specific areas based on given ratios. The formula helps in accurately determining the coordinates of boundary points or landmarks. GIS applications use similar principles to analyze spatial data and determine positions of interest based on proportional distances.

Engineering and Construction

In engineering and construction, the section formula is used for planning and design purposes. When constructing bridges, roads, or buildings, engineers need to accurately calculate the positions of structural elements. For example, if a support beam needs to be placed at a specific point along a main beam, dividing it in a certain ratio, the section formula can be used to find the exact coordinates for placement.

Architectural design also benefits from this formula. When creating floor plans or designing layouts, architects often need to position elements proportionally within a space. The section formula aids in determining the coordinates of walls, fixtures, or other design features, ensuring that they are placed according to the intended proportions and layout.

Urban Planning

Urban planners use the section formula to make informed decisions about city layouts, transportation routes, and zoning. Determining the optimal location for a new facility, such as a school or hospital, often involves considering its accessibility to different parts of the city. By treating the city as a coordinate plane, planners can use the section formula to find locations that divide the distances between key areas in a desired ratio.

Public transportation routes can also be optimized using this formula. When designing bus routes or subway lines, planners need to identify intermediate stops that are proportionally spaced along the route. The section formula helps in determining the coordinates of these stops, ensuring that they are conveniently located for the population they serve.

Conclusion

In conclusion, understanding and applying the section formula is essential for solving problems related to coordinate geometry. This formula enables us to find the coordinates of a point that divides a line segment in a given ratio, a concept that has broad applications across various fields. From the step-by-step process of identifying given information, applying the formula, calculating coordinates, and verifying the solution, we've seen how to approach these problems systematically.

While our initial question lacked specific endpoint coordinates, we explored a general approach and a hypothetical example to illustrate how the section formula is used. We also highlighted real-world applications in computer graphics, navigation, engineering, and urban planning, demonstrating the practical significance of this mathematical tool. Mastering the section formula not only enhances problem-solving skills in mathematics but also provides a foundation for tackling geometric challenges in diverse professional domains.