Partitioning A Line Segment: Find Point P Coordinates

by Alex Johnson 54 views

Let's dive into a fun problem involving line segments and ratios! We're given two points, A and B, and we want to find a point P that lies on the line segment AB, dividing it into two smaller segments, AP and PB, with a specific ratio. This is a classic problem in coordinate geometry, and it's super useful in various fields like computer graphics and spatial analysis. So, let’s break it down step by step.

Understanding the Problem

Before we jump into the calculations, let's make sure we understand what the problem is asking. We have a line segment AB, with the coordinates of point A being (1, 3) and the coordinates of point B being (11, -7). Our goal is to find a point P that sits on this line segment such that the ratio of the length of segment AP to the length of segment PB is 4:5. In other words, AP is 4/9 of the total length of AB, and PB is 5/9 of the total length of AB. Understanding this ratio is key to solving the problem correctly. Visualizing the line segment and the position of point P can also be helpful.

To solve this, we will use the section formula, which helps us find the coordinates of a point that divides a line segment in a given ratio. This formula is derived from the concept of similar triangles and proportional division of the line segment. The section formula is a powerful tool that allows us to find the exact location of point P without having to measure distances or draw precise diagrams. It's all about applying the formula correctly and plugging in the given values.

The beauty of this problem lies in its simplicity and direct application of a well-established formula. Once you grasp the concept of dividing a line segment in a given ratio, you can apply this technique to various geometric problems and real-world scenarios. So, let's get ready to roll up our sleeves and calculate the coordinates of point P using the section formula.

The Section Formula

The section formula is our primary tool for solving this problem. It states that if a point P(x, y) divides the line segment joining points A(x1, y1) and B(x2, y2) in the ratio m:n, then the coordinates of P are given by:

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

In our case, A(1, 3) is (x1, y1), B(11, -7) is (x2, y2), and the ratio m:n is 4:5. It's crucial to correctly identify and assign these values before plugging them into the formula. A mistake here can lead to incorrect coordinates for point P.

The section formula is based on the idea that the x-coordinate and y-coordinate of point P are weighted averages of the x-coordinates and y-coordinates of points A and B, respectively. The weights are determined by the ratio m:n. This formula is a direct result of similar triangles formed by the line segment AB and the lines parallel to the x and y axes passing through points A, B, and P.

The formula is very versatile and can be used to find the coordinates of any point that divides a line segment in a given ratio, whether the point is internal or external to the line segment. In our case, point P is internal, meaning it lies between points A and B. However, the formula can also be used to find the coordinates of a point that lies outside the line segment, in which case the ratio m:n would be negative.

Understanding the derivation and application of the section formula is fundamental to solving problems involving line segments and ratios. It's a cornerstone concept in coordinate geometry and has numerous applications in various fields, including computer graphics, engineering, and physics.

Applying the Formula

Now, let's plug in the values into the section formula to find the coordinates of point P. We have:

  • x1 = 1, y1 = 3
  • x2 = 11, y2 = -7
  • m = 4, n = 5

Using the formula for the x-coordinate:

x = (4 * 11 + 5 * 1) / (4 + 5) = (44 + 5) / 9 = 49 / 9 = 5 4/9

And for the y-coordinate:

y = (4 * (-7) + 5 * 3) / (4 + 5) = (-28 + 15) / 9 = -13 / 9 = -1 4/9

So, the coordinates of point P are (5 4/9, -1 4/9). Double-checking your calculations is always a good idea to avoid errors. Make sure you've substituted the values correctly and performed the arithmetic accurately.

Applying the formula correctly is essential for obtaining the right answer. Pay close attention to the order of operations and ensure that you are using the correct values for x1, y1, x2, y2, m, and n. A common mistake is to mix up the values or to perform the calculations incorrectly. To avoid these errors, it's helpful to write down the values clearly and to double-check your work as you go along.

Once you have calculated the coordinates of point P, it's a good idea to verify that the point lies on the line segment AB and that it divides the segment in the correct ratio. You can do this by calculating the distances AP and PB and checking that their ratio is indeed 4:5. This will give you confidence that you have solved the problem correctly.

The Solution

Therefore, the point P that divides the line segment AB in the ratio 4:5 is (5 4/9, -1 4/9).

It is important to present the solution clearly and concisely, stating the coordinates of point P in a way that is easy to understand. You can also include a brief explanation of how you arrived at the solution, summarizing the steps you took and the formulas you used. This will help the reader to understand your reasoning and to follow your solution more easily.

In addition to stating the coordinates of point P, it's also helpful to provide a visual representation of the problem and the solution. You can do this by drawing a diagram of the line segment AB and marking the position of point P. This will make it easier for the reader to visualize the problem and to understand the relationship between the points A, B, and P.

Finally, it's a good idea to check your answer by plugging the coordinates of point P back into the section formula and verifying that they satisfy the given conditions. This will give you extra confidence that you have solved the problem correctly and that your solution is accurate.

Conclusion

We've successfully found the coordinates of point P using the section formula. This problem demonstrates a fundamental concept in coordinate geometry, and it's a great example of how mathematical formulas can be applied to solve real-world problems. Remember, practice makes perfect! The more you work with these types of problems, the more comfortable and confident you'll become.

For further exploration on coordinate geometry and related topics, you can visit Khan Academy's Geometry Section. This external resource provides comprehensive lessons and practice exercises to enhance your understanding of geometric concepts.