Translate Triangle LMN: Step-by-Step Solution

by Viktoria Ivanova 46 views

Hey guys! Ever wondered how geometric shapes move around on a coordinate plane? Today, we're diving deep into the fascinating world of translations, specifically focusing on a right triangle named LMN. We'll break down a problem step-by-step, making sure you grasp the core concepts and can tackle similar challenges with confidence. So, buckle up and let's get started!

The Problem: A Right Triangle's Journey

Let's set the stage. We have a right triangle, LMN, hanging out on a coordinate plane. Its vertices (those pointy corners) are located at the following coordinates:

  • L: (7, -3)
  • M: (7, -8)
  • N: (10, -9)

Now, imagine picking up this triangle and shifting it somewhere else on the plane without rotating or resizing it. That's precisely what a translation does! In our case, triangle LMN undergoes a translation, and the new location of vertex L, which we'll call L', is at (-1, 8). Our mission, should we choose to accept it, is to figure out the rule that dictates this translation. In other words, we need to find out how the x and y coordinates changed during the move.

To find the rule, our main keyword here is translation rule, we first have to understand what a translation is. A translation is a rigid transformation, meaning it moves a figure without changing its size or shape. This is key! We're only sliding the triangle around. The translation rule is expressed as (x, y) -> (x + a, y + b), where a represents the horizontal shift and b represents the vertical shift. Our goal is to determine the values of a and b.

Let's focus on point L, the main keyword here is point L, which moves from (7, -3) to L' (-1, 8). To find the horizontal shift (a), we compare the x-coordinates: 7 to -1. What do we need to add to 7 to get -1? The answer is -8. So, a = -8. This means the triangle shifted 8 units to the left.

Next, we look at the vertical shift (b). The y-coordinate of L changes from -3 to 8. What do we add to -3 to get 8? The answer is 11. Therefore, b = 11. This indicates a vertical shift of 11 units upwards. Putting it all together, the translation rule is (x, y) -> (x - 8, y + 11). This rule tells us that every point on the triangle is shifted 8 units to the left and 11 units upwards.

Decoding the Translation Rule

The big question is: how do we figure out the magic formula that dictates this shift? Think of it like this: we're trying to find the difference between the original coordinates of point L and its new coordinates, L'. This difference will tell us exactly how much the triangle moved horizontally and vertically. Let's break it down:

  • Horizontal Shift: To find out how much the triangle moved horizontally, we compare the x-coordinates of L and L'. L's x-coordinate is -1, and L's x-coordinate is 7. The difference is -1 - 7 = -8. This means the triangle shifted 8 units to the left along the x-axis. Remember, a negative shift means moving left!
  • Vertical Shift: Now, let's tackle the vertical movement. We compare the y-coordinates of L' and L. L's y-coordinate is 8, and L's y-coordinate is -3. The difference is 8 - (-3) = 11. This tells us the triangle moved 11 units upwards along the y-axis. A positive shift indicates upward movement.

So, we've cracked the code! The translation rule is: (x, y) → (x - 8, y + 11). This nifty little formula tells us that every point on the triangle was shifted 8 units to the left and 11 units upwards. We can even test this rule on the other vertices, M and N, to confirm our findings.

Verifying the Rule: Let's Test It Out!

To be absolutely sure we've got the correct translation rule, let's put it to the test. We'll apply the rule (x, y) → (x - 8, y + 11) to the coordinates of points M and N and see if the resulting coordinates make sense. This is a crucial step in problem-solving – always double-check your work!

  • Point M: Original coordinates (7, -8). Applying the rule, we get (7 - 8, -8 + 11) = (-1, 3). So, M' should be located at (-1, 3).
  • Point N: Original coordinates (10, -9). Applying the rule, we get (10 - 8, -9 + 11) = (2, 2). Therefore, N' should be at (2, 2).

By applying the translation rule to points M and N, we've successfully calculated their new positions, M' and N'. If we were to plot these new points on the coordinate plane, we'd see that they form a triangle identical in size and shape to the original triangle LMN, just shifted to a new location. This confirms that our translation rule is indeed correct!

Why Translations Matter: Real-World Applications

Okay, so we've learned how to translate a triangle on a coordinate plane. But you might be wondering,