Finding The Translation Rule For Triangle LMN To L' In Coordinate Plane

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Hey guys! Let's dive into a fun math problem involving coordinate geometry and translations. We've got a right triangle LMN, and it's been moved on the coordinate plane. Our mission is to figure out exactly how it was moved – what's the translation rule that was used? Let's break it down step by step.

Understanding the Problem

Okay, so we're given a right triangle called LMN. We know the coordinates of its vertices: L is at (7, -3), M is at (7, -8), and N is at (10, -8). This triangle has been translated, which means it's been moved without being rotated or resized. After the translation, the new coordinates of point L, which we call L', are (-1, 8). Our task is to find the rule that describes this translation. In other words, we need to figure out how many units the triangle moved horizontally and vertically.

Initial Setup and Coordinates

Before we jump into calculations, let’s make sure we’re clear on the given information:

  • Original Triangle Vertices:
    • L: (7, -3)
    • M: (7, -8)
    • N: (10, -8)
  • Translated Point:
    • L': (-1, 8)

The key here is to focus on how point L has moved to L'. This will give us the translation rule that applies to the entire triangle since translations shift every point by the same amount in the same direction.

Determining the Translation Rule

To find the translation rule, we need to see how the x and y coordinates have changed from L to L'. The translation rule will be in the form (x, y) → (x + a, y + b), where 'a' is the horizontal shift and 'b' is the vertical shift.

Calculating the Horizontal Shift

The original x-coordinate of L is 7, and the new x-coordinate of L' is -1. To find the horizontal shift, we calculate the difference:

Horizontal Shift (a) = New x-coordinate - Original x-coordinate a = -1 - 7 a = -8

This means the triangle has been shifted 8 units to the left along the x-axis.

Calculating the Vertical Shift

Similarly, the original y-coordinate of L is -3, and the new y-coordinate of L' is 8. To find the vertical shift, we calculate the difference:

Vertical Shift (b) = New y-coordinate - Original y-coordinate b = 8 - (-3) b = 8 + 3 b = 11

This means the triangle has been shifted 11 units upwards along the y-axis.

The Translation Rule

Now that we have both the horizontal and vertical shifts, we can write the translation rule: (x, y) → (x - 8, y + 11). This rule tells us that to get from the original triangle to the translated triangle, we subtract 8 from the x-coordinate and add 11 to the y-coordinate.

Applying the Rule to Other Vertices

To double-check our translation rule, let's apply it to the other vertices, M and N, and see if we get reasonable results. This will help ensure that our rule is consistent and correct.

Translating Point M

Original coordinates of M: (7, -8) Applying the rule (x, y) → (x - 8, y + 11):

New x-coordinate = 7 - 8 = -1 New y-coordinate = -8 + 11 = 3 So, the translated point M' would be (-1, 3).

Translating Point N

Original coordinates of N: (10, -8) Applying the rule (x, y) → (x - 8, y + 11):

New x-coordinate = 10 - 8 = 2 New y-coordinate = -8 + 11 = 3 So, the translated point N' would be (2, 3).

Verifying the Results

By applying the translation rule to points M and N, we get M'(-1, 3) and N'(2, 3). While we don't have these points explicitly given in the problem, translating these points helps us confirm that our translation rule makes sense within the context of the coordinate plane. If we were to plot these points, we'd see they maintain the same relative position to L' as M and N did to L.

Why This Matters

Understanding translations in coordinate geometry is super useful for several reasons:

  1. Geometry Basics: It builds a strong foundation for more complex geometric transformations like rotations, reflections, and dilations.
  2. Real-World Applications: Translations are used in computer graphics, game development, and engineering to move objects around a screen or a design space without changing their shape or size.
  3. Problem-Solving Skills: Working through problems like this enhances your analytical and problem-solving skills, which are valuable in many areas of life.

Common Mistakes to Avoid

When working with translations, it's easy to make a few common mistakes. Here are some things to watch out for:

  • Incorrect Order of Subtraction: Make sure you subtract the original coordinates from the new coordinates to find the shift. Subtracting in the wrong order will give you the opposite sign and an incorrect translation rule.
  • Mixing Up x and y: It’s crucial to keep the x and y coordinates separate. Ensure you're applying the horizontal shift to the x-coordinate and the vertical shift to the y-coordinate.
  • Not Applying the Rule Consistently: A translation rule must apply uniformly to all points in the figure. If you find a rule that works for one point but not another, there's likely an error in your calculations.
  • Forgetting the Sign: Pay close attention to the signs (positive or negative) of the shifts. A negative shift in the x-direction means moving left, and a negative shift in the y-direction means moving down.

Conclusion

So, in this problem, we successfully found the translation rule that moved triangle LMN to its new position. By carefully calculating the horizontal and vertical shifts, we determined that the rule is (x, y) → (x - 8, y + 11). Remember, guys, the key to these problems is to break them down step by step and double-check your work! Understanding translations is a fundamental concept in geometry, and mastering it will help you tackle more advanced topics with confidence. Keep practicing, and you’ll become a pro at coordinate geometry in no time!