Segment Division: Find Ratios And Coordinates

by Omar Yusuf 46 views

Hey guys! Let's dive into a fun and super practical math concept: dividing a line segment in a given ratio. This might sound a bit intimidating at first, but trust me, it's like pie – once you get the hang of it, it's super easy to slice and serve (or, in this case, solve!). We're going to break down a specific problem step-by-step, and by the end, you'll be a pro at tackling these types of questions.

Problem Overview: Understanding Segment Division

Before we jump into the nitty-gritty, let's understand the basic idea. Imagine you have a line segment – a straight line between two points, let’s call them P and Q. Now, suppose you have another point, R, somewhere on that line. This point R divides the segment PQ into two smaller segments: PR and RQ. The ratio in which R divides PQ tells us the relative lengths of these two segments. Think of it like this: if the ratio is 1:1, then R is right in the middle, dividing PQ into two equal parts. If the ratio is 2:1, then PR is twice as long as RQ. Make sense?

Why is this important, you ask? Well, this concept pops up everywhere in geometry, coordinate geometry, and even in more advanced math like vector algebra. So, grasping this is like unlocking a secret level in your math journey! And hey, it's not just about math class. Understanding ratios and proportions is crucial in real life too, whether you're scaling a recipe, designing a room layout, or even understanding financial charts. Trust me, this is a skill worth mastering.

Now, in our specific problem, we're given that point R divides the segment PQ in the ratio λ = 2/5. This means that the length of segment PR is 2/5 times the length of segment RQ. We're also given the coordinates of points P and Q, and our mission is to find a few things:

  • The ratio in which point Q divides the segment RP.
  • The ratio in which point P divides the segment QR.
  • The coordinates of point R.

Sounds like a challenge? Absolutely! But don't worry, we'll break it down piece by piece.

(a) Finding the Ratio in which Q Divides RP

Okay, let's start with the first part: determining the ratio in which point Q divides the segment RP. This might seem a bit tricky at first because we're flipping the perspective. We initially knew how R divided PQ, but now we need to figure out how Q divides RP. Think of it like looking at the same picture from a different angle – the elements are the same, but the way we relate them changes.

To tackle this, let’s visualize the situation. Imagine the line segment RP. Point Q lies somewhere on this segment (or its extension). The ratio in which Q divides RP tells us how the lengths of segments RQ and QP compare. Remember, a ratio is just a way of expressing the relative sizes of two quantities. So, if the ratio is, say, 3:2, it means that the first segment is 3/2 times the length of the second segment.

Now, we know that R divides PQ in the ratio λ = 2/5. This means PR/RQ = 2/5. This is our starting point, our known fact. But we need to find RQ/QP (or QR/PQ). Notice how we're kind of flipping things around? That's the key here – we need to manipulate the given information to get to what we want.

The trick here is to think about the relationships between the segments. We know PR/RQ = 2/5. To find the ratio in which Q divides RP, we need to find RQ/RP. We can start by using the given ratio to express PR in terms of RQ or vice versa. From PR/RQ = 2/5, we can say that PR = (2/5)RQ.

But, we need the ratio in terms of RP. To find that, we need to express RP in terms of RQ. The length of RP can be expressed as the sum of RQ and QP (or QR, same thing). To relate RQ and RP, we'll need to introduce the entire segment PQ into the equation.

Remember, we know how R divides PQ, and that gives us a relationship between PR and RQ. We need to leverage this relationship to find the relationship between RQ and RP. We can consider the entire length of the segment PQ as the sum of its parts: PQ = PR + RQ. This seemingly simple equation is a powerful tool in solving this problem.

By substituting PR = (2/5)RQ into the equation PQ = PR + RQ, we get PQ = (2/5)RQ + RQ. Now we can factor out RQ to have PQ = RQ(2/5 + 1), which simplifies to PQ = (7/5)RQ. From this, we derive RQ = (5/7)PQ.

Now, we need the length of RP in terms of PQ, for the target ratio RQ/RP. We can start by expressing RP as the difference of PQ and RQ. So RP = PQ - RQ. Substitute RQ = (5/7)PQ, we get RP = PQ - (5/7)PQ. Simplifying gives us RP = (2/7)PQ.

Finally, we want to express the ratio in which Q divides RP, which is RQ/RP. We have found expressions for both RQ and RP in terms of PQ. So we substitute those in: RQ/RP = ((5/7)PQ) / ((2/7)PQ) = (5/7) / (2/7). This simplifies to RQ/RP = 5/2. So the ratio in which Q divides RP is 5:2.

Therefore, point Q divides the segment RP in the ratio of 5:2. We did it! It might have seemed like a long journey, but we broke it down step-by-step, and now we have our answer. Feel like a math ninja yet?

(b) Determining the Ratio in which P Divides QR

Alright, let's keep the momentum going! Now we need to figure out the ratio in which point P divides the segment QR. Notice how we're shifting our perspective again? This is a common theme in these types of problems, so getting comfortable with this