Triangle Translation: Finding The Rule

Hey there, math enthusiasts! Let's dive into a fascinating geometry problem involving a right triangle and its transformation on the coordinate plane. We've got triangle LMN, and it's been translated – that's math-speak for moved – to a new location. Our mission, should we choose to accept it (and we do!), is to figure out the exact rule that dictates this movement. It's like detective work, but with coordinates and triangles! So grab your thinking caps, and let's unravel this mathematical mystery together.

The Setup: Triangle LMN and Its Transformation

Our starting point is right triangle LMN, with its vertices neatly placed at the coordinates L(7, -3), M(7, -8), and N(10, -8). Imagine it sitting there on the coordinate plane, a perfect right-angled triangle just minding its own business. Then, something happens: a translation! The triangle gets shifted, and suddenly, vertex L finds itself at a brand new location, L'(-1, 8). The question that burns in our mathematical minds is this: what magic spell (or, more accurately, what mathematical rule) caused this transformation? To find the translation rule, we need to figure out how much the x-coordinate and the y-coordinate have changed. This involves a bit of coordinate geometry sleuthing, and we're here for it. We'll break down the shift in both the x and y directions to pinpoint the exact transformation rule. Remember, a translation is a rigid transformation, meaning the size and shape of the triangle remain unchanged; only its position alters. This tidbit will be crucial as we verify our rule later on. Stay tuned as we dissect the coordinate shifts and uncover the hidden rule behind this geometric maneuver. We're on the verge of cracking this puzzle, so let's keep those brain gears turning!

Deciphering the Rule: X and Y Shifts

The heart of this problem lies in understanding how the coordinates change during a translation. We know L(7, -3) has been translated to L'(-1, 8). So, let's dissect this coordinate shift step by step. First, we'll tackle the x-coordinate. It moved from 7 to -1. How do we get from 7 to -1? We subtract 8, right? Mathematically, this can be represented as 7 + (-8) = -1. So, there's our x-shift: -8 units. Now, let's focus on the y-coordinate. It journeyed from -3 to 8. To get from -3 to 8, we need to add 11. Expressed mathematically, -3 + 11 = 8. So, the y-shift is a positive 11 units. We've now pinpointed the individual shifts in both the x and y directions. The x-coordinate decreases by 8 units, and the y-coordinate increases by 11 units. This is a crucial step! It allows us to formulate the translation rule in a concise and mathematical way. Remember, these shifts are the key to unlocking the transformation rule, and we're about to put it all together. By carefully analyzing these coordinate changes, we're building a solid foundation for our solution. The pieces are falling into place, and the translation rule is becoming clearer with each step. Let's keep this momentum going!

Formulating the Translation Rule: The (x, y) Mapping

Now that we've meticulously calculated the shifts in both the x and y coordinates, it's time to put it all together and express the translation rule in a standard mathematical format. Remember, the x-coordinate decreased by 8 units, and the y-coordinate increased by 11 units. This means for any point (x, y) on the original triangle, the corresponding point on the translated triangle will be (x - 8, y + 11). This is our translation rule in action! We can express this rule more formally using mapping notation. In mapping notation, we use an arrow to show how a point (x, y) is transformed. So, our translation rule can be written as: (x, y) → (x - 8, y + 11). This concise notation elegantly captures the essence of the transformation. It tells us exactly how each point in the original triangle is moved to create the translated triangle. This rule is the key to understanding the entire transformation. We've essentially decoded the mathematical instructions that govern the triangle's movement. This mapping notation is not just a shorthand; it's a powerful tool for describing geometric transformations. It provides a clear and unambiguous way to represent how points are shifted in the coordinate plane. With this rule in hand, we're ready to verify its accuracy and confidently declare our solution. Let's move on to the final verification step and solidify our understanding.

Verifying the Rule: Testing with Point M

We've derived our translation rule: (x, y) → (x - 8, y + 11). But before we declare victory, it's crucial to verify our rule. A solid check will give us the confidence that we've cracked the code correctly. To verify, we'll apply our rule to another point on the triangle, say point M(7, -8), and see if it lands in the correct translated position, which we'll call M'. If our rule holds true for M, it adds strong evidence that it's the correct translation rule for the entire triangle. Let's put our rule to the test. Applying the rule (x, y) → (x - 8, y + 11) to point M(7, -8), we get: M' = (7 - 8, -8 + 11) = (-1, 3). So, according to our rule, the translated point M' should be at (-1, 3). This is a crucial step in our problem-solving process. We're not just blindly accepting the rule we've derived; we're actively testing it to ensure its validity. This verification process is a hallmark of good mathematical practice. By confirming our rule with an additional point, we're strengthening our solution and building a deeper understanding of the transformation. Now, let's check if this result aligns with the information given or implied in the problem statement. This will be the final piece of the puzzle!

The Final Answer: The Correct Translation Rule

After careful deduction and verification, we've arrived at the solution! We determined the translation rule by analyzing the shift of point L to L' and then confirmed it by applying the rule to point M. Our analysis revealed that the x-coordinate decreases by 8 units, and the y-coordinate increases by 11 units. This translates to the rule: (x, y) → (x - 8, y + 11). This rule accurately describes the transformation of triangle LMN on the coordinate plane. It dictates the precise movement of each point, ensuring that the triangle maintains its shape and size while shifting its position. We've successfully deciphered the mathematical code behind this geometric transformation. We started with a problem involving a translated triangle and a missing rule, and through careful analysis and verification, we've uncovered the hidden transformation. This problem highlights the power of coordinate geometry in describing and understanding geometric transformations. By breaking down the problem into smaller steps – analyzing coordinate shifts, formulating the rule, and verifying its accuracy – we've confidently arrived at the solution. This journey through the coordinate plane has not only given us the answer but also deepened our understanding of translations and geometric transformations. So, give yourselves a pat on the back, math detectives! We've cracked the case!

What is the translation rule used to move triangle LMN, where L(7,-3) becomes L'(-1,8)?

Triangle Translation Finding the Rule