Find Point Q On A Line Segment With A Given Ratio

Aug 7, 2025 by Kenji Nakamura 50 views

Decoding the Coordinates of Point Q on a Directed Line Segment

Hey everyone! Today, we're diving into a fun geometry problem involving directed line segments and ratios. We've got a line segment PQ, and we know the coordinates of point P and a point R that lies on the segment. The point R divides the segment PQ in a specific ratio. Our mission, should we choose to accept it, is to find the coordinates of point Q. Sounds like a plan? Let's break it down step by step.

Understanding Directed Line Segments and Ratios

Before we jump into the calculations, let's make sure we're all on the same page about what directed line segments and ratios mean in this context. A directed line segment, like PQ, is a line segment that has a specific direction. This means that the order of the points matters. The segment PQ starts at point P and ends at point Q. This directionality is crucial when we're dealing with ratios.

Now, let's talk about the ratio. We're given that point R divides segment PQ in the ratio PR : RQ = 2 : 3. This means that the distance from P to R is two parts, while the distance from R to Q is three parts. Imagine the segment PQ divided into five equal parts (2 + 3 = 5). Point R sits two-fifths of the way from P to Q. This understanding of the ratio is the key to unlocking the coordinates of point Q. We need to understand that the ratio is not just about lengths; it also applies to the change in the x-coordinates and the change in the y-coordinates as we move along the directed line segment. In essence, we will be using the section formula, but let's think about it conceptually first to make sure we grasp the underlying principles.

The Section Formula: A Quick Recap

For those who are familiar with it, the section formula provides a direct way to calculate the coordinates of a point that divides a line segment in a given ratio. If we have a line segment with endpoints extitP}(x₁, y₁) and extit{Q}(x₂, y₂), and a point extit{R} divides the segment in the ratio m n, then the coordinates of point R ( extit{x, extit{y}) can be found using the following formulas:

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

In our case, we know the coordinates of P and R, the ratio PR : RQ, and we want to find the coordinates of Q. So, we'll need to manipulate these formulas a bit to solve for the unknowns. But before we get bogged down in algebra, let's visualize the problem and think about how the coordinates change as we move along the line segment. This will help us build a solid understanding and avoid making simple mistakes.

Visualizing the Problem

It often helps to visualize these types of problems on a coordinate plane. We can plot the points P(-10, 3) and R(4, 7). We know that Q lies somewhere on the same line, further away from R than P is. Since the ratio PR : RQ is 2 : 3, the segment RQ is longer than the segment PR. This gives us a sense of where Q should be located – further along the line in the same direction from R as P is but more so.

Thinking about the changes in the x and y coordinates can be insightful. To get from P to R, the x-coordinate changes from -10 to 4, which is an increase of 14. The y-coordinate changes from 3 to 7, which is an increase of 4. These changes correspond to the '2' part of the ratio. Now, to get from R to Q, we need to make changes that correspond to the '3' part of the ratio. This means the changes in x and y will be 1.5 times the changes we observed from P to R (since 3 is 1.5 times 2). This intuitive understanding will guide us as we set up the equations and solve for the coordinates of Q.

Setting Up the Equations

Now that we have a solid understanding of the problem, let's translate that understanding into mathematical equations. Let's denote the coordinates of point Q as (x, y). We know that point R(4, 7) divides the segment PQ in the ratio 2 : 3. This means we can use the section formula, but we'll need to rearrange it slightly to solve for the coordinates of Q.

Using the section formula, we can write two equations:

For the x-coordinate: 4 = (2x + 3(-10)) / (2 + 3)
For the y-coordinate: 7 = (2y + 33) / (2 + 3)

These equations represent the relationship between the coordinates of P, R, and Q based on the given ratio. The left-hand side of each equation represents the coordinate of point R, while the right-hand side represents the weighted average of the coordinates of P and Q, weighted according to the ratio. Our next step is to simplify these equations and solve for x and y. This will give us the coordinates of point Q. Don't be intimidated by the algebra; we'll take it one step at a time.

Solving for the x-coordinate

Let's tackle the equation for the x-coordinate first: 4 = (2x + 3(-10)) / (2 + 3). The goal here is to isolate x on one side of the equation. To do that, we'll perform a series of algebraic manipulations. First, we can simplify the denominator: 4 = (2x - 30) / 5. Next, we can multiply both sides of the equation by 5 to get rid of the fraction: 20 = 2x - 30. Now, we can add 30 to both sides to isolate the term with x: 50 = 2*x. Finally, we divide both sides by 2 to solve for x: x = 25. So, the x-coordinate of point Q is 25. We're halfway there! Now, let's move on to the y-coordinate.

Solving for the y-coordinate

Now, let's work on the equation for the y-coordinate: 7 = (2y + 33) / (2 + 3). This process will be very similar to what we did for the x-coordinate. First, simplify the denominator: 7 = (2y + 9) / 5. Multiply both sides by 5: 35 = 2y + 9. Subtract 9 from both sides: 26 = 2*y. Divide both sides by 2: y = 13. So, the y-coordinate of point Q is 13. We've found both coordinates!

The Coordinates of Point Q

Alright, we've done the math, and we've arrived at our answer! The coordinates of point Q are (25, 13). That's it! We successfully navigated through the problem using our understanding of directed line segments, ratios, and a bit of algebra. It's always satisfying to solve a geometry puzzle, isn't it? Remember, the key to these problems is to break them down into smaller, manageable steps, visualize the situation, and don't be afraid to use the tools you have, like the section formula. But more importantly, understand the concepts behind the formulas. Guys, fantastic job working through this with me! Keep practicing, and you'll become geometry masters in no time.

Verification and Conclusion

Before we wrap things up, it's always a good idea to verify our answer. We can do this by plugging the coordinates of P ( -10, 3), Q (25, 13), and the ratio 2:3 back into the section formula to see if we get the coordinates of R (4, 7). This step ensures that we haven't made any calculation errors along the way and gives us confidence in our solution. Using the section formula, the x-coordinate of R should be (2 * 25 + 3 * -10) / (2 + 3) = (50 - 30) / 5 = 20 / 5 = 4, which matches the given x-coordinate of R. Similarly, the y-coordinate of R should be (2 * 13 + 3 * 3) / (2 + 3) = (26 + 9) / 5 = 35 / 5 = 7, which matches the given y-coordinate of R. Since our calculated coordinates of R match the given coordinates, we can confidently say that our solution for the coordinates of Q (25, 13) is correct.

In conclusion, by understanding the concept of directed line segments and ratios, applying the section formula, and carefully solving the resulting equations, we successfully determined the coordinates of point Q. Remember, visualization and verification are crucial steps in problem-solving, especially in geometry. Keep practicing these concepts, and you'll find yourself tackling even more challenging problems with ease!