Right Triangle LMN Translation Rule Finding The Shift

Hey guys! Let's dive into a fun geometry problem involving translations on the coordinate plane. We've got a right triangle LMN, and we need to figure out how it was moved from one spot to another. This is a classic transformation question, and by the end of this, you'll be a pro at solving these!

Problem Breakdown: Understanding the Translation

So, the core of this problem lies in understanding translations. In the coordinate plane, a translation is basically sliding a shape (in our case, a triangle) without rotating or flipping it. Think of it as picking up the triangle and moving it to a new location without changing its orientation. Mathematically, we describe a translation using a rule that tells us how each point (x, y) is shifted. This rule usually looks like (x, y) → (x + a, y + b), where 'a' tells us how much to move horizontally (left or right) and 'b' tells us how much to move vertically (up or down).

Now, let's get into the specifics of our triangle LMN. We're given the coordinates of its vertices: L (7, -3), M (7, -8), and N (10, -8). We're also told that the triangle is translated, and the new coordinates of vertex L are L' (-1, 8). The key here is to focus on what happened to point L. By comparing the original coordinates of L with the new coordinates of L', we can figure out the translation rule that applies to the entire triangle. Remember, in a translation, every point of the shape moves in the same way.

To determine the rule, we need to figure out how much the x-coordinate changed and how much the y-coordinate changed. The x-coordinate of L went from 7 to -1. What do we need to add to 7 to get -1? That's a decrease of 8, or adding -8. So, the 'a' in our rule is -8. Now let's look at the y-coordinate. It went from -3 to 8. To get from -3 to 8, we need to add 11. So, the 'b' in our rule is 11. Putting it all together, the translation rule looks like this: (x, y) → (x - 8, y + 11). This means every point in the triangle was moved 8 units to the left and 11 units up. To be absolutely sure, we could check this rule with the other vertices, M and N, but since we're pretty confident, let's move on to explaining how we arrived at this answer and why the other options might be incorrect.

Finding the Translation Rule: A Step-by-Step Guide

Okay, so let's break down exactly how we nail down that translation rule. Remember, the goal is to find the specific shift that moves every point of our triangle the exact same way. We're given that point L (7, -3) is translated to L' (-1, 8). This is our golden ticket because it tells us precisely how the x and y coordinates change.

1. Focus on the x-coordinate:

First, let's look at the x-coordinate. It goes from 7 to -1. The question we need to ask ourselves is: What do we add to 7 to get -1? To figure this out, we can set up a simple equation: 7 + a = -1. Solving for 'a', we subtract 7 from both sides, and we get a = -8. This means the x-coordinate is shifted -8 units, which is the same as moving 8 units to the left.

2. Now, the y-coordinate:

Next up, we look at the y-coordinate. It goes from -3 to 8. Again, we need to find the amount we add to -3 to get 8. We can set up another equation: -3 + b = 8. Solving for 'b', we add 3 to both sides, and we get b = 11. This means the y-coordinate is shifted 11 units, which is the same as moving 11 units up.

3. Putting it all together:

Now we have all the pieces of the puzzle! We know the x-coordinate is shifted by -8, and the y-coordinate is shifted by 11. This means our translation rule is (x, y) → (x - 8, y + 11). This rule tells us exactly how to move any point in the triangle to its new location after the translation.

4. Double-Checking (Optional but Recommended):

To be super sure, you could apply this rule to the other vertices, M and N, and see if you get the correct translated coordinates. However, once you've confirmed the rule using one point (like we did with L), you've usually got it!

This step-by-step process is the key to cracking these types of translation problems. By carefully analyzing the changes in the x and y coordinates, you can easily determine the translation rule. And remember, this rule applies to every single point on the shape, not just the vertices.

Why Other Options Are Wrong: Spotting the Distractors

It's just as important to understand why the wrong answers are wrong as it is to know why the correct answer is right. This helps you avoid common mistakes and truly grasp the concept. So, let's think about why some other potential translation rules might not work in our triangle LMN problem.

1. Incorrect x-shift:

Imagine an option that had the correct y-shift (adding 11 to the y-coordinate) but an incorrect x-shift. For example, let's say it proposed a rule like (x, y) → (x + 6, y + 11). This would mean the triangle is being shifted 6 units to the right in the x-direction. But we know that point L (7, -3) is translated to L' (-1, 8). To get from 7 to -1 in the x-coordinate, we need to move left, not right. So, any rule that adds a positive number to the x-coordinate is immediately suspect.

2. Incorrect y-shift:

Similarly, a rule with the correct x-shift (subtracting 8 from the x-coordinate) but a wrong y-shift wouldn't work. Let's say we had a rule like (x, y) → (x - 8, y + 5). This would shift the triangle 5 units up in the y-direction. But for L (-3) to become L' (8) in the y-coordinate, we need to move much further up than just 5 units. So, this rule is also incorrect.

3. Mixing it up:

Sometimes, incorrect options might try to mix up the x and y shifts. For example, a rule like (x, y) → (x + 11, y - 8) might look tempting at first glance because it uses the numbers 8 and 11. But it's applying the 11 shift to the x-coordinate and the -8 shift to the y-coordinate, which is the opposite of what we need. Remember, the x-shift is determined by the change in x-coordinates, and the y-shift is determined by the change in y-coordinates. Don't get them mixed up!

4. No shift at all:

Finally, be wary of options that suggest no shift in either the x or y direction (or both). This would mean the triangle isn't being translated at all, which contradicts the problem statement. If you see a rule like (x, y) → (x, y), that's a dead giveaway that it's wrong.

By carefully considering why these other options are incorrect, you strengthen your understanding of translations and improve your ability to spot the right answer quickly.

Conclusion: Mastering Coordinate Plane Translations

So, guys, we've really dug into the world of coordinate plane translations here! We tackled a problem with a right triangle LMN and figured out the precise rule that describes how it was moved. The key takeaway is that translations involve sliding a shape without changing its orientation, and we can represent this movement with a simple rule: (x, y) → (x + a, y + b). To find this rule, we focused on how the coordinates of a single point (like L) changed, and that gave us all the information we needed.

We also learned the importance of understanding why the wrong answers are wrong. By spotting common mistakes and distractors, we become more confident problem-solvers. Remember to carefully analyze the changes in both the x and y coordinates, and don't mix them up! With practice, you'll be able to breeze through these translation problems.

Keep practicing, and you'll be a master of coordinate plane transformations in no time! Geometry can be super fun once you get the hang of these core concepts. Good luck with your studies, and remember, math is awesome!