Point R: Finding It On A Number Line

Hey mathletes! Let's dive into a cool geometry problem today, guys. We're going to tackle how to find a specific point on a number line when we know the endpoints and the ratio it divides the segment into. Our mission, should we choose to accept it, is to locate point R on the directed line segment from Q to S. We're given that Q is at -2 and S is at 6. This line segment is then partitioned by point R in a 3:2 ratio. This means that the distance from Q to R is 3 parts, and the distance from R to S is 2 parts, for a total of 5 parts. We'll be using the section formula, a super handy tool in coordinate geometry, to figure out exactly where R lands on this number line. So grab your calculators, sharpen those pencils, and let's get this math party started!

Understanding the Number Line and Directed Segments

Alright, let's really get a grip on what we're dealing with here. When we talk about a number line, think of it as a straight, infinite line where every point represents a real number. It's like a ruler, but it goes on forever in both directions. We've got negative numbers to the left and positive numbers to the right, with zero right in the middle. Now, a directed line segment is a piece of that line that has a specific starting point and a specific ending point, and the direction matters. In our case, the directed line segment goes from Q to S. This means we start at Q and move towards S. The fact that it's directed tells us the order is important. We are starting at Q (-2) and moving towards S (6). This is crucial because if it were the segment from S to Q, our calculations might look a little different depending on how we apply the formulas, although the geometric location of the segment would be the same. The endpoints are given: Q is located at the coordinate -2, and S is at the coordinate 6. So, we have a segment on the number line starting at -2 and ending at 6. The total length of this segment is the difference between the endpoint coordinates: $6 - (-2) = 6 + 2 = 8$ units. This means our segment stretches across 8 units on the number line.

The Power of Ratios in Geometry

Now, let's talk about this ratio 3:2. This is where things get really interesting. Point R doesn't just randomly sit on the segment QS; it divides it in a very specific way. The ratio 3:2 means that if you were to chop up the entire segment QS into smaller, equal pieces, R would be positioned such that the segment QR is made up of 3 of these pieces, and the segment RS is made up of 2 of these pieces. So, the whole segment QS is divided into $3 + 2 = 5$ equal parts. The ratio tells us about the relative lengths of the two sub-segments. QR is longer than RS because it has 3 parts compared to RS's 2 parts. This information is key to using the section formula. Without this ratio, we wouldn't know where to place R. The section formula is specifically designed to work with these kinds of ratio divisions. It's a general formula that works in any dimension – whether it's a 1D number line, a 2D Cartesian plane, or even 3D space. For our 1D case, it simplifies quite nicely, but the principle is the same. The ratio allows us to precisely pinpoint R's location by weighting the coordinates of the endpoints. Think of it like this: R is closer to S than it is to Q because the ratio 3:2 implies that R is further away from Q (3 parts) and closer to S (2 parts). Wait, let me rephrase that! A ratio of 3:2 partitioning the segment from Q to S means that the segment QR is 3 parts and the segment RS is 2 parts. Therefore, R is closer to S (2 parts away) than it is to Q (3 parts away). This is a common point of confusion, so it's good we caught that! The ratio dictates the proportional distances. It's all about how R splits the total length of the segment QS.

Introducing the Section Formula

The section formula is a mathematical tool that allows us to find the coordinates of a point that divides a line segment in a given ratio. For a line segment in one dimension (like our number line), with endpoints $x_1$ and $x_2$, if a point $x$ divides the segment in the ratio $m:n$ (meaning the segment from $x_1$ to $x$ is $m$ parts and the segment from $x$ to $x_2$ is $n$ parts), the formula is:

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

Let's break this down for our specific problem. We have the directed line segment from Q to S. So, our first endpoint, $x_1$, is the coordinate of Q, which is -2. Our second endpoint, $x_2$, is the coordinate of S, which is 6. The point R partitions this segment in a 3:2 ratio. This means that the segment QR corresponds to $m=3$ parts, and the segment RS corresponds to $n=2$ parts. Notice how $m$ is associated with the second point ($x_2$) in the denominator ($m+n$), and $n$ is associated with the first point ($x_1$). This is because $m$ represents the ratio away from the first point, and $n$ represents the ratio away from the second point. So, when we plug these values into the section formula, we get:

$$R = \frac{(2)(-2) + (3)(6)}{3+2}$$

Here, $x_1 = -2$, $x_2 = 6$, $m = 3$, and $n = 2$. The $n$ is multiplied by $x_1$, and the $m$ is multiplied by $x_2$. This is because the ratio $m:n$ means that the distance from the first point ($x_1$) to the dividing point ($x$) is $m$ parts, and the distance from the dividing point ($x$) to the second point ($x_2$) is $n$ parts. Therefore, the dividing point $x$ is closer to $x_2$ by $n$ parts and closer to $x_1$ by $m$ parts. Let's re-evaluate the formula application carefully. The ratio $m:n$ means that the segment from $x_1$ to the point is $m$ parts, and the segment from the point to $x_2$ is $n$ parts. So, the point is $m$ units away from $x_1$ and $n$ units away from $x_2$ in terms of ratio parts. When we use the formula x = rac{nx_1 + mx_2}{m+n}, the $n$ is multiplied by $x_1$ and the $m$ is multiplied by $x_2$. This structure ensures that the point's coordinate is a weighted average of the endpoints, where the weights are inversely proportional to the distance from the respective endpoints. In our case, $m=3$ and $n=2$. So, we have $n=2$ multiplying $x_1=-2$, and $m=3$ multiplying $x_2=6$. The sum of the ratio parts, $m+n$, is $3+2=5$, which represents the total number of equal parts the segment is divided into.

Applying the Formula Step-by-Step

Let's get down to business and calculate the position of point R. We have our values plugged into the section formula:

$$R = \frac{(2)(-2) + (3)(6)}{3+2}$$

First, let's handle the multiplications in the numerator. We have $(2)(-2)$, which equals -4. Next, we have $(3)(6)$, which equals 18. So, the numerator becomes $-4 + 18$.

$$R = \frac{-4 + 18}{3+2}$$

Now, let's sum the numbers in the numerator: $-4 + 18 = 14$. The denominator is $3 + 2 = 5$.

$$R = \frac{14}{5}$$

And there you have it! The coordinate of point R is rac{14}{5}. To make this a bit more intuitive, we can convert this fraction to a decimal. $14 ilde{A}· 5 = 2.8$. So, point R is located at 2.8 on the number line. This means that R is positioned at 2.8, which is between -2 and 6. It makes sense that R is not at an endpoint and lies somewhere along the segment QS. The value 2.8 is positive, which is expected since it's closer to 6 than to -2. If we were to check the distances: The distance from Q (-2) to R (2.8) is $2.8 - (-2) = 2.8 + 2 = 4.8$. The distance from R (2.8) to S (6) is $6 - 2.8 = 3.2$. Now, let's check if these distances are in the ratio 3:2. We can do this by dividing the distance QR by the distance RS: rac{4.8}{3.2}. If we multiply both the numerator and denominator by 10 to get rid of the decimals, we have rac{48}{32}. Both 48 and 32 are divisible by 16. $48 ilde{A}· 16 = 3$ and $32 ilde{A}· 16 = 2$. So, the ratio of the distances is indeed rac{3}{2}, or 3:2. This confirms our calculation is correct!

Verifying the Result

It's always a good idea to double-check our work, right? Let's verify that point R at 2.8 indeed partitions the segment QS (from -2 to 6) in a 3:2 ratio. We already calculated the distances: the distance from Q to R is 4.8 units, and the distance from R to S is 3.2 units.

Checking the Distances and Ratio

We found that the distance $QR = 4.8$ and the distance $RS = 3.2$. The ratio we are looking for is rac{QR}{RS}. So, we calculate rac{4.8}{3.2}. As we saw before, this simplifies to rac{48}{32}, which further simplifies to rac{3}{2}. This is exactly the ratio of 3:2 that was given in the problem. So, the point R at 2.8 perfectly divides the segment QS into two parts where the first part (QR) is 3 times some unit length and the second part (RS) is 2 times that same unit length.

Total Length Check

Another way to think about this is to consider the total length of the segment QS. The total length is $6 - (-2) = 8$ units. Our ratio is 3:2, meaning there are $3+2=5$ parts in total. So, each part should have a length of rac{8}{5} units. Let's check our distances using this value. The distance QR should be 3 parts, so 3 imes rac{8}{5} = rac{24}{5}. As a decimal, rac{24}{5} = 4.8. This matches our calculated distance QR. The distance RS should be 2 parts, so 2 imes rac{8}{5} = rac{16}{5}. As a decimal, rac{16}{5} = 3.2. This matches our calculated distance RS. And if we add these distances, $4.8 + 3.2 = 8$, which is the total length of the segment QS. Everything checks out perfectly!

Conclusion: Rachel's Success

So, guys, by applying the section formula, Rachel successfully found the location of point R on the number line. The directed line segment from Q at -2 to S at 6 is partitioned by point R in a 3:2 ratio. Using the formula $R = \frac{nx_1 + mx_2}{m+n}$ with $x_1 = -2$, $x_2 = 6$, $m=3$, and $n=2$, we calculated:

$$R = \frac{(2)(-2) + (3)(6)}{3+2} = \frac{-4 + 18}{5} = \frac{14}{5} = 2.8$$

Therefore, point R is located at 2.8 on the number line. This means that the segment QR has a length of 4.8 units, and the segment RS has a length of 3.2 units, and the ratio of these lengths is 4.8:3.2, which simplifies to 3:2. Way to go, Rachel! Math problems solved! Keep practicing, and you'll be a coordinate geometry whiz in no time!