Partitioning A Line Segment: Finding The X-Coordinate

Welcome back to our math corner! Today, we're diving into a fascinating concept in coordinate geometry: partitioning a directed line segment. Imagine you have a line segment, and you want to find a specific point on that segment that divides it into two smaller segments with a particular ratio. This is exactly what Gabrielle was working on, and we're here to break down her process and understand it thoroughly. Our focus today will be on finding the x-coordinate of such a point, using the endpoints Q (-10, 7) and R (1, -4) and a ratio of 2:3. This skill is incredibly useful in various mathematical applications, from understanding vectors to designing graphics.

Understanding the Partition Formula

Before we jump into Gabrielle's work, let's establish the foundation. The partition formula is your best friend when dealing with these types of problems. It allows us to calculate the coordinates of a point that divides a line segment internally in a given ratio. If we have a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$, and a point $(x, y)$ partitions this segment in the ratio $m:n$, then the coordinates of this point are given by:

$$x = \frac{nx_1 + mx_2}{m+n}$$

$$y = \frac{ny_1 + my_2}{m+n}$$

Here, $m$ and $n$ represent the parts of the ratio. It's crucial to remember which part of the ratio corresponds to which endpoint. The ratio $m:n$ means that the distance from the first endpoint $(x_1, y_1)$ to the point $(x, y)$ is $m$ parts, and the distance from the point $(x, y)$ to the second endpoint $(x_2, y_2)$ is $n$ parts. In Gabrielle's problem, the directed line segment is $\overline{QR}$, with $Q (-10, 7)$ as the first endpoint and $R (1, -4)$ as the second endpoint. The ratio is $2:3$. This means $m=2$ and $n=3$. Since we are only interested in the x-coordinate, we will use the formula for $x$: $x = \frac{nx_1 + mx_2}{m+n}$. It's important to correctly identify $(x_1, y_1)$ and $(x_2, y_2)$. In this case, $Q(-10, 7)$ corresponds to $(x_1, y_1)$, so $x_1 = -10$ and $y_1 = 7$. The point $R(1, -4)$ corresponds to $(x_2, y_2)$, so $x_2 = 1$ and $y_2 = -4$. With these values, we can now plug them into the formula. The ratio $2:3$ means $m=2$ and $n=3$. So, the $x$-coordinate will be calculated as $x = \frac{3(-10) + 2(1)}{2+3}$. Understanding this formula and correctly assigning the values is the first, and perhaps most important, step in solving problems like this. It's the blueprint for our calculations, ensuring we're on the right track to finding that precise partitioning point.

Gabrielle's Steps and the Calculation

Gabrielle's approach, as presented, aims to apply the partition formula to find the x-coordinate. Let's walk through her potential steps, assuming she correctly identified the components needed for the calculation. First, she would need to identify the coordinates of the endpoints. These are given as $Q(-10, 7)$ and $R(1, -4)$. So, $x_1 = -10$ and $x_2 = 1$. Next, she needs to identify the ratio in which the segment is partitioned. The problem states the ratio is $2:3$. This means $m=2$ and $n=3$. The '2' corresponds to the segment from $Q$ to the partition point, and the '3' corresponds to the segment from the partition point to $R$. With these values in hand, Gabrielle would then substitute them into the x-coordinate partition formula: $x = \frac{nx_1 + mx_2}{m+n}$. Plugging in our values, we get $x = \frac{(3)(-10) + (2)(1)}{2+3}$. This is where the calculation begins. The numerator involves multiplying $n$ by $x_1$ and $m$ by $x_2$, and then adding the results. So, $(3)(-10) = -30$ and $(2)(1) = 2$. Adding these gives $-30 + 2 = -28$. The denominator is simply the sum of the ratio parts, $m+n$, which is $2+3 = 5$. Therefore, the x-coordinate is $x = \frac{-28}{5}$. This fraction can also be expressed as a decimal, $-5.6$. It's essential to perform each step accurately, paying close attention to signs and arithmetic. Gabrielle's success hinges on correctly applying this formula, and her work should reflect these steps. If her final answer matches $-28/5$ or $-5.6$, she has successfully found the x-coordinate of the partitioning point. This systematic approach ensures that no detail is missed and that the final answer is derived logically from the given information.

Verifying the Result and Its Significance

Now that we've determined the x-coordinate of the partitioning point to be $-28/5$ (or $-5.6$), it's always a good practice to verify the result and understand its significance. Does this value make sense in the context of our line segment? Our segment starts at $x_1 = -10$ and ends at $x_2 = 1$. The ratio $2:3$ means the partition point is closer to the starting point $Q$ than to the ending point $R$. Visually, imagine the number line for the x-coordinates. We have $-10$ and $1$. The total distance between these x-coordinates is $1 - (-10) = 11$ units. The partition point should divide this total distance into two parts, where the first part (from $x_1$ to $x$) is $2/5$ of the total distance and the second part (from $x$ to $x_2$) is $3/5$ of the total distance. Let's check if our calculated $x = -5.6$ satisfies this. The distance from $x_1$ to $x$ is $-5.6 - (-10) = -5.6 + 10 = 4.4$. The distance from $x$ to $x_2$ is $1 - (-5.6) = 1 + 5.6 = 6.6$. Now, let's check the ratio of these distances: $4.4 : 6.6$. To simplify this ratio, we can divide both numbers by their greatest common divisor, or simply divide them by a common factor. Let's convert them to fractions: $4.4 = 44/10 = 22/5$ and $6.6 = 66/10 = 33/5$. The ratio is $(22/5) : (33/5)$. Multiplying both sides by 5 gives $22 : 33$. Dividing both by 11, we get $2:3$. This perfectly matches the given ratio! This verification confirms that our calculated $x$-coordinate is indeed correct. The significance of this point is that it divides the directed line segment $\overline{QR}$ such that the distance from $Q$ to this point is $2/5$ of the total length of $\overline{QR}$, and the distance from this point to $R$ is $3/5$ of the total length. This concept is fundamental in understanding weighted averages, barycentric coordinates, and vector operations, making it a cornerstone in many areas of mathematics and its applications, from physics to computer graphics. It allows us to precisely locate points along a line based on relative distances, a powerful tool for any aspiring mathematician or scientist.

Conclusion

In conclusion, finding the x-coordinate of a point that partitions a directed line segment is a methodical process that relies heavily on the partition formula. Gabrielle's work, once completed correctly, would demonstrate a clear application of this formula. By identifying the endpoints $Q(-10, 7)$ and $R(1, -4)$, and the ratio $2:3$, we substituted these values into $x = \frac{nx_1 + mx_2}{m+n}$. This yielded the result $x = \frac{3(-10) + 2(1)}{2+3} = \frac{-30+2}{5} = \frac{-28}{5}$ or $-5.6$. Our verification confirmed that this $x$-coordinate indeed divides the segment in the specified ratio $2:3$. This process highlights the elegance and power of coordinate geometry in solving precise problems. Remember, accuracy in identifying coordinates and the ratio, and careful arithmetic, are key to success in these calculations. This skill is a building block for more advanced mathematical concepts.

For further exploration into coordinate geometry and line segment partitioning, you can refer to resources like Khan Academy, which offers comprehensive explanations and practice problems.