Finding A Point On A Line Segment: Distance Ratio Problems

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Hey guys! Let's dive into a cool math problem that involves finding a specific point on a line segment. We're going to use the endpoints of a line segment and a distance ratio to pinpoint the exact location of a point. This is super useful in geometry and can even be applied in real-world scenarios. We'll break down the problem step-by-step, making it easy to understand. So, grab your pencils and let's get started!

Understanding the Problem

Alright, so the problem gives us a line segment, which we'll call L. This line segment has two endpoints: (-5, -2) and (-1, 4). Imagine this line on a graph. Now, we need to find a point p that lives somewhere on this line segment L. But here's the twist: the distance from the endpoint (-5, -2) to point p is six times the distance from point p to the other endpoint, (-1, 4). This means that point p is much closer to the endpoint (-1, 4). Our mission, should we choose to accept it, is to figure out the exact coordinates of this point p.

This kind of problem falls under the umbrella of coordinate geometry. It's about using the power of coordinates to describe and analyze geometric shapes. This particular problem is a classic example of section formula applications. The section formula provides a way to find the coordinates of a point that divides a line segment into a specific ratio. The given ratio here is 6:1 (the distance from one endpoint to p is 6 times the distance from p to the other). A clear visualization can often make the problem simpler. Sketching the line segment L on a coordinate plane and imagining the point p will help you conceptualize the problem better. This initial understanding of what we're trying to find is crucial. So, we're not just looking for any point; we're looking for a point with a very specific distance relationship to the endpoints. This introduction is key for building a solid foundation before we jump into the math.

Now, let's think about this problem a bit differently. We know the distance from (-5,-2) to p is six times the distance from p to (-1,4). That means, if we consider the entire length of the line segment from (-5,-2) to (-1,4) as a total of 7 parts (6 parts for the longer segment and 1 part for the shorter segment), the point p splits the line segment into these two parts. Understanding this ratio is super important because it helps us to find the point p using the section formula. We're essentially dividing the line segment in a ratio of 6:1, which allows us to find the position of p. The section formula is like a secret weapon in coordinate geometry! It is designed specifically for this type of problem. Once we learn how to correctly apply the section formula and the ratio that divides the line segment, the computation becomes much easier. It's all about systematically applying the right formulas. This understanding of ratios and how they relate to the section formula is vital for effectively solving this type of problem. Remember that practice is key, so the more you work through these problems, the more comfortable and adept you'll become in applying this strategy.

Applying the Section Formula

Okay, time for some math! To find the coordinates of point p, we'll use the section formula. This formula is like a magic recipe for finding a point that divides a line segment into a specific ratio. If a point P(x, y) divides the line segment joining the points A(x1, y1) and B(x2, y2) internally in the ratio m:n, the coordinates of P are given by:

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

In our case, the endpoints of the line segment are (-5, -2) and (-1, 4). Let's call (-5, -2) as (x1, y1) and (-1, 4) as (x2, y2). The ratio is 6:1 (m:n), where m = 6 and n = 1. Therefore, according to the question, the distance from (-5, -2) to p is 6 times the distance from p to (-1, 4). Now, it’s time to plug in all the numbers. We need to substitute the coordinates of the endpoints and the ratio into the formula to find the coordinates of point p.

Let’s start with the x-coordinate of p. Applying the section formula for the x-coordinate:

  • x = (6 * (-1) + 1 * (-5)) / (6 + 1)
  • x = (-6 - 5) / 7
  • x = -11/7

Next, let’s find the y-coordinate of p. Applying the section formula for the y-coordinate:

  • y = (6 * 4 + 1 * (-2)) / (6 + 1)
  • y = (24 - 2) / 7
  • y = 22/7

Therefore, the coordinates of point p are (-11/7, 22/7). That’s it! We’ve successfully found the point p on the line segment that satisfies the given distance ratio. Now the problem is almost done, just a couple more steps. We’ve found the x and y coordinates by applying the section formula and correctly plugging in all the numbers. It's a very systematic process, following the section formula. This is a very common type of question. The core skill here is correctly identifying and substituting the values into the section formula. Also, remember to double-check your calculations to avoid any silly mistakes. This result provides us with the precise location of p on the line segment.

Verifying the Solution

Great job, guys! Now, let's make sure our answer is correct. We can do this by checking if the distances from (-5, -2) to p and from p to (-1, 4) really do match the 6:1 ratio. We will use the distance formula to calculate the distance between two points. This gives us a good opportunity to review another fundamental concept. The distance formula is given as:

  • d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

Let's calculate the distance between (-5, -2) and p (-11/7, 22/7):

  • d1 = sqrt(((-11/7) - (-5))^2 + ((22/7) - (-2))^2)
  • d1 = sqrt(((-11/7) + (35/7))^2 + ((22/7) + (14/7))^2)
  • d1 = sqrt((24/7)^2 + (36/7)^2)
  • d1 = sqrt((576/49) + (1296/49))
  • d1 = sqrt(1872/49)

Now, let's calculate the distance between p (-11/7, 22/7) and (-1, 4):

  • d2 = sqrt(((-1) - (-11/7))^2 + ((4) - (22/7))^2)
  • d2 = sqrt(((-7/7) + (11/7))^2 + ((28/7) - (22/7))^2)
  • d2 = sqrt((4/7)^2 + (6/7)^2)
  • d2 = sqrt((16/49) + (36/49))
  • d2 = sqrt(52/49)

Now we need to check if d1 is 6 times d2:

  • d1 / d2 = (sqrt(1872/49)) / (sqrt(52/49))
  • d1 / d2 = sqrt(1872/52)
  • d1 / d2 = sqrt(36)
  • d1 / d2 = 6

So, d1 is indeed 6 times d2. This confirms that the point p we found is correct! Verifying the solution is super important. It gives us confidence that our answer is right and helps reinforce our understanding of the concepts. We applied the distance formula to compute distances, which ensured our ratio was correct. Therefore, the distance between the two points we have calculated confirms the solution and our correct calculations. Remember to always check your answers to catch any potential errors and ensure you fully grasp the problem.

Conclusion

Awesome work, everyone! We successfully found point p on the line segment by using the section formula and verified our answer with the distance formula. This problem illustrates how coordinate geometry can be used to solve real-world problems. We first understood the problem by visualizing it and understanding the ratio. Then, we used the section formula to find the coordinates. After this, we double-checked our solution using the distance formula. The section formula is a powerful tool for solving various geometry problems. Keep practicing and exploring different types of problems to become even better. By understanding and applying the section formula correctly, we've successfully addressed the problem! Keep up the excellent work. Hopefully, this explanation was helpful. If you have any questions, feel free to ask. Cheers!