Finding Coordinates On A Line Segment Using Section Formula A Step-by-Step Guide

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In coordinate geometry, determining the coordinates of a point that divides a line segment in a specific ratio is a common problem. This article will walk you through the process of finding the coordinates of point G, which lies on the directed line segment from point F (-1, -1) to point H (-8, 20) and partitions the segment in the ratio of 5:2. We will explore the section formula, which is the key to solving this type of problem, and apply it step-by-step to arrive at the solution.

Understanding the Section Formula

The section formula is a fundamental tool in coordinate geometry that allows us to find the coordinates of a point that divides a line segment in a given ratio. Let's say we have two points, A(x₁, y₁) and B(x₂, y₂), and a point P(x, y) that divides the line segment AB in the ratio m:n. The coordinates of point P can be calculated using the following formulas:

  • x = (mx₂ + nx₁) / (m + n)
  • y = (my₂ + ny₁) / (m + n)

In essence, the section formula is a weighted average of the coordinates of the endpoints, where the weights are determined by the ratio in which the segment is divided. The ratio m:n represents the proportion in which the point P divides the line segment AB. A ratio of m:n means that the distance from A to P is m times some constant, and the distance from P to B is n times the same constant.

The formula works by considering the weighted contribution of each endpoint to the coordinates of the dividing point. The x-coordinate of the dividing point is found by taking a weighted average of the x-coordinates of the endpoints, and similarly for the y-coordinate. The weights are determined by the ratio m:n, which specifies how close the dividing point is to each endpoint. For example, if m is much larger than n, the dividing point will be closer to endpoint B, and its coordinates will be more heavily influenced by the coordinates of B. Conversely, if n is much larger than m, the dividing point will be closer to endpoint A, and its coordinates will be more heavily influenced by the coordinates of A.

Applying the Section Formula to Find Point G

Now, let's apply the section formula to our specific problem. We are given:

  • Point F: (-1, -1) which is (x₁, y₁)
  • Point H: (-8, 20) which is (x₂, y₂)
  • Ratio: 5:2 which is m:n

We want to find the coordinates of point G(x, y) that divides the line segment FH in the ratio 5:2. Using the section formula, we can plug in the given values:

  • x = (5 * -8 + 2 * -1) / (5 + 2)
  • y = (5 * 20 + 2 * -1) / (5 + 2)

Let's calculate the x-coordinate first:

x = (5 * -8 + 2 * -1) / (5 + 2) = (-40 - 2) / 7 = -42 / 7 = -6

Now, let's calculate the y-coordinate:

y = (5 * 20 + 2 * -1) / (5 + 2) = (100 - 2) / 7 = 98 / 7 = 14

Therefore, the coordinates of point G are (-6, 14).

Detailed Calculation Breakdown

To further clarify the application of the section formula, let's break down the calculations step by step:

  1. Identify the coordinates of the endpoints: We have F(-1, -1) and H(-8, 20). These are our (x₁, y₁) and (x₂, y₂) respectively.
  2. Determine the ratio: The ratio is given as 5:2, which means m = 5 and n = 2. This ratio indicates that the segment FG is 5 parts, while the segment GH is 2 parts of the total segment FH.
  3. Apply the section formula for the x-coordinate:
    • x = (mx₂ + nx₁) / (m + n)
    • x = (5 * (-8) + 2 * (-1)) / (5 + 2)
    • x = (-40 - 2) / 7
    • x = -42 / 7
    • x = -6
  4. Apply the section formula for the y-coordinate:
    • y = (my₂ + ny₁) / (m + n)
    • y = (5 * 20 + 2 * (-1)) / (5 + 2)
    • y = (100 - 2) / 7
    • y = 98 / 7
    • y = 14
  5. Combine the x and y coordinates: The coordinates of point G are (-6, 14).

This step-by-step breakdown illustrates how the section formula systematically combines the coordinates of the endpoints and the given ratio to determine the coordinates of the dividing point. By understanding each step, you can confidently apply this formula to solve similar problems.

Analyzing the Solution

Our calculations show that the coordinates of point G are (-6, 14). Now, let's analyze this solution to ensure it makes sense within the context of the problem. We can do this by considering the relative positions of the points and the given ratio.

Point G's Location:

Point G lies on the directed line segment from F(-1, -1) to H(-8, 20). The x-coordinate of G (-6) is between the x-coordinates of F (-1) and H (-8), and the y-coordinate of G (14) is between the y-coordinates of F (-1) and H (20). This confirms that G indeed lies on the line segment FH.

Ratio 5:2:

The ratio 5:2 indicates that the segment FG is longer than the segment GH. Specifically, the length of FG is 5/7 of the total length of FH, while the length of GH is 2/7 of the total length of FH. This means that G should be closer to H than it is to F. We can visually imagine this by sketching a line segment from F to H and placing G such that it is further along the segment towards H.

Visual Verification:

While we don't have a graph here, we can mentally visualize the points on a coordinate plane. F is in the third quadrant, H is in the first quadrant, and G is also in the second quadrant. The line segment FH has a steep positive slope, moving significantly upwards and to the left. Point G, with coordinates (-6, 14), appears to be positioned appropriately along this line segment, closer to H as the ratio suggests.

Alternative Method Verification:

To further verify our solution, we can use an alternative approach. We can find the vector FH, scale it by the appropriate fraction based on the ratio, and add it to the coordinates of point F. This method provides an independent check of our result.

The vector FH is found by subtracting the coordinates of F from the coordinates of H:

FH = H - F = (-8 - (-1), 20 - (-1)) = (-7, 21)

Since G divides FH in the ratio 5:2, the fraction of the vector FH that corresponds to FG is 5/(5+2) = 5/7. We scale the vector FH by this fraction:

(5/7) * FH = (5/7) * (-7, 21) = (-5, 15)

Now, we add this scaled vector to the coordinates of F to find the coordinates of G:

G = F + (5/7) * FH = (-1, -1) + (-5, 15) = (-6, 14)

This alternative method confirms our previous result, giving us further confidence in the correctness of our solution.

Conclusion

In conclusion, we have successfully found the coordinates of point G, which lies on the directed line segment from F(-1, -1) to H(-8, 20) and partitions the segment in the ratio of 5:2. By applying the section formula, we determined that the coordinates of point G are (-6, 14). We further analyzed this solution and verified it using an alternative method, solidifying our understanding of the process. This problem demonstrates the power and utility of the section formula in coordinate geometry for solving problems involving ratios and line segments.

Practice Problems

To solidify your understanding of the section formula, try solving these practice problems:

  1. Find the coordinates of the point that divides the line segment joining A(2, -3) and B(5, 6) in the ratio 2:1.
  2. Point P divides the line segment joining C(-4, 8) and D(6, -2) in the ratio 3:2. Find the coordinates of P.
  3. If point Q divides the line segment joining E(1, 5) and F(7, -1) in the ratio 1:3, find the coordinates of Q.

By working through these problems, you'll gain confidence in applying the section formula and develop a deeper understanding of its applications in coordinate geometry.

Real-World Applications of the Section Formula

While the section formula might seem like a purely mathematical concept, it has several real-world applications in fields such as computer graphics, navigation, and engineering. Let's explore some of these applications:

1. Computer Graphics:

In computer graphics, the section formula is used for various tasks, including:

  • Line Interpolation: When drawing lines or curves on a screen, computers often use interpolation techniques to fill in the pixels between two endpoints. The section formula can be used to calculate the coordinates of intermediate points along a line segment, ensuring a smooth and accurate representation.
  • 3D Modeling: In 3D modeling, objects are often represented as a collection of polygons. The vertices of these polygons are points in 3D space, and the edges are line segments. The section formula can be used to subdivide these edges, creating more detailed and smoother surfaces.
  • Animation: In animation, objects move along paths defined by keyframes. The section formula can be used to calculate the positions of the object at intermediate frames, ensuring a smooth and natural motion.

2. Navigation:

In navigation systems, the section formula can be used for:

  • Route Planning: When planning a route, navigation systems often break down the route into a series of line segments. The section formula can be used to calculate intermediate points along these segments, providing more accurate directions and guidance.
  • GPS Tracking: GPS devices track the location of a vehicle or person by calculating its coordinates. The section formula can be used to estimate the location of the device between GPS readings, providing a more continuous and accurate tracking history.

3. Engineering:

In engineering, the section formula is used in various applications, such as:

  • Structural Design: When designing structures, engineers need to calculate the forces and stresses acting on different parts of the structure. The section formula can be used to determine the location of points along structural members, allowing for accurate calculations.
  • Surveying: Surveyors use the section formula to calculate the coordinates of points in the field. This is essential for creating accurate maps and plans.
  • Robotics: In robotics, the section formula can be used to control the movement of robot arms and other robotic systems. By calculating the coordinates of intermediate points along a desired path, robots can perform precise and controlled movements.

These are just a few examples of the real-world applications of the section formula. By understanding the underlying mathematical principles and its practical uses, you can appreciate the versatility and importance of this concept in various fields.

Selecting the Correct Answer

Based on our calculations, the coordinates of point G are (-6, 14). Now, let's look at the given options:

A. (-6, 14) B. (-9, 19) C. (-4.5, 9.5) D. (-5, 15)

The correct answer is A. (-6, 14).