Triangle Transformation: Finding Coordinates And Translation

Jan 10, 2026 by Tim Redaksi 61 views

Hey guys! Let's dive into a fun geometry problem involving triangle transformations. We're going to explore how a triangle gets moved around the coordinate plane. Specifically, we're going to figure out the coordinates of some missing points and the exact movement that happens during a translation. This is super helpful for understanding how shapes change position and how the coordinate system helps us keep track of it all. So, grab your pencils, and let's get started!

Understanding the Problem

Alright, so here's the deal: We have a triangle called FGH that gets translated (or slid) across the plane to become triangle PQR. We're given the coordinates of a couple of the original points (F and G) and two of the translated points (P and R). Our mission? To find the missing coordinates of H and Q and to identify the translation that takes FGH to PQR. Sounds like a fun puzzle, right? This kind of problem is a classic in geometry because it really tests your understanding of how transformations work and how to apply them using coordinates. Understanding translations is fundamental to more advanced concepts in geometry and computer graphics. It forms the basis of how we understand movement and positioning in space.

To break it down, a translation is a transformation that moves every point of a shape the same distance and in the same direction. Think of it like sliding the triangle across the table without rotating or changing its size. The cool thing about translations is that we can describe them using a simple rule that tells us how much to shift the x and y coordinates. So, for every point (x, y) in the original triangle, the translated point (x', y') will be given by a rule like (x + a, y + b), where 'a' is the horizontal shift and 'b' is the vertical shift. This concept is applicable in various fields, from video game design where objects are constantly translated to engineering where structures undergo precise movements. Therefore, mastering the ability to calculate these translations is a valuable skill to have.

Finding the Translation Vector

Okay, let's figure out how FGH got to PQR. We know that point F (6, 18) moved to become point P (8, 4). Since a translation moves every point the same amount, we can use the coordinates of F and P to figure out the translation rule. Specifically, let's analyze how the coordinates of F changed to get to P. To get from F to P, we need to see how the x-coordinate and the y-coordinate changed individually. The x-coordinate went from 6 to 8, meaning it increased by 2 (8 - 6 = 2). The y-coordinate went from 18 to 4, meaning it decreased by 14 (4 - 18 = -14). Therefore, the translation rule is (x, y) -> (x + 2, y - 14). This rule means we're adding 2 to the x-coordinate (moving to the right) and subtracting 14 from the y-coordinate (moving down). This process is crucial because it allows us to generalize the transformation to any point on the original triangle. Thus, understanding this allows us to move any point by the same amount, ensuring the shape's integrity is maintained.

The translation vector can be written as <2, -14>. This vector summarizes the overall shift in the x and y directions. We can use this vector to predict where any point in the original triangle will end up. This step is pivotal because it gives us a clear understanding of the motion involved. The translation vector is a very important concept in geometry and is widely used in various applications. Being able to determine the translation vector accurately is fundamental to many types of geometric problems.

Finding the Coordinates of H

Now that we know the translation, we can use it to find the coordinates of point H. We know the coordinates of G (-2, 8). Since G is translated to Q, and FGH translates to PQR using the same rule, we can apply the translation rule we found to point G to find Q. We will apply the translation rule (x, y) -> (x + 2, y - 14) to the coordinates of G. So, if G is (-2, 8), then Q will be (-2 + 2, 8 - 14), which simplifies to (0, -6). Therefore, the coordinates of Q are (0, -6). Using this approach is consistent, as we can apply the rule to all the points of the triangle. The goal is to perform the same translation on all points, hence, maintaining the integrity of the triangle. Furthermore, this also helps to check our work. If the new points do not yield the same kind of triangle, then there is a mistake somewhere in our calculation. However, applying the same translation vector to H would yield its coordinates after translation to form R. We can deduce the coordinates of H by working backward. We know that R is the result of translating H. We know that R is (12, -6). Now, since the translation rule is (x, y) -> (x + 2, y - 14), the inverse is (x, y) -> (x - 2, y + 14). So, to get back to H, we apply the inverse translation to R. Thus, H = (12 - 2, -6 + 14) = (10, 8).

This shows how understanding the translation rule allows us to easily find the new coordinates. It's like having a map that tells us exactly where each point will end up after the move. This is also how we can verify the consistency of the translation and that all points move correctly.

Finding the Coordinates of Q

We've already found the coordinates of Q! Remember, we used the translation rule on the coordinates of G (-2, 8) to find Q. Applying the rule (x, y) -> (x + 2, y - 14), we get Q as (0, -6). This process emphasizes that every point in the original triangle is subject to the same transformation. This is what makes a translation a rigid transformation, meaning it doesn't change the size or shape of the figure – it only changes its position. So, the relative position of the points in FGH is preserved in PQR. Moreover, this property is vital in many fields such as computer graphics and image processing, where preserving the original shape is essential during translation.

By finding the coordinates of Q and H, we've successfully solved the problem. The translation helps us understand how the points in a shape move when undergoing transformation. Now you've got a grasp of how to handle triangle translations – finding missing coordinates and all. Awesome work!

Summary of Results

Translation Vector: <2, -14>
Coordinates of H: (10, 8)
Coordinates of Q: (0, -6)

Conclusion

So there you have it, guys! We've successfully navigated a triangle transformation problem. We found the missing coordinates of H and Q and figured out the translation that moved our triangle from FGH to PQR. Remember, translations are all about sliding shapes without changing their size or shape. And the translation vector is your best friend when it comes to understanding and calculating these shifts. Understanding translations is an essential stepping stone to grasping more complex geometric transformations such as rotations, reflections, and dilations. By mastering this concept, you build a solid foundation for further exploration in geometry and related fields. Keep practicing, and you'll become a pro at these problems in no time! Keep exploring and have fun with math!