Triangle Translation: Finding The New Y-Value

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Triangle Translation: Finding the New Y-Value

Hey guys! Let's dive into a fun geometry problem involving triangle translation. We're given a triangle, and we need to figure out where one of its points ends up after it's been shifted around the coordinate plane. Don't worry, it's easier than it sounds! We'll break it down step-by-step, and you'll be a translation pro in no time. So grab your pencils, and let's get started. We have a triangle with vertices P(-2, 6), Q(-8, 4), and R(1, -2). The question is, what happens to the point P after the triangle undergoes a translation described by the rule (x, y) → (x - 2, y - 16)? What is the new y-value of the point P' after the translation?

Understanding the Problem: Triangle Translation and Coordinates

First things first, what exactly does triangle translation mean? Think of it like sliding the triangle across a sheet of paper without rotating or changing its shape. Every point in the triangle moves the same distance and in the same direction. In our case, the translation rule (x, y) → (x - 2, y - 16) tells us exactly how each point moves. The rule specifies that we subtract 2 from the x-coordinate and subtract 16 from the y-coordinate. So, the original point P(-2, 6) will be moved 2 units to the left (because of x - 2) and 16 units down (because of y - 16). This new location is P', which is what we need to find.

Now, let's talk about coordinates. Remember that in a coordinate plane, every point is defined by an (x, y) pair. The x-coordinate tells us how far to the left or right the point is from the origin (0, 0), and the y-coordinate tells us how far up or down the point is. In our problem, we only need to focus on the y-coordinate of point P after the translation. This means we're only interested in the vertical movement of the point.

Let's apply this to the problem. We start with the point P(-2, 6). The translation rule tells us to subtract 2 from the x-coordinate and 16 from the y-coordinate. So we'll take the y-coordinate of P, which is 6, and subtract 16. That gives us 6 - 16 = -10. This is the y-coordinate of P' after the translation. Therefore, the y-value of P' is -10.

To make sure you've got it, let's consider a different point. Let's imagine we wanted to find the new coordinates of Q after the translation. Q starts at (-8, 4). Applying the translation rule, we'd do (-8 - 2, 4 - 16), which equals (-10, -12). Notice how we applied the same rule to Q as we did to P, just with different starting numbers. Pretty cool, right? The translation rule consistently moves every point in the triangle by the same amount, maintaining the triangle's shape and size. Now you can find the new coordinates for all of the points.

Step-by-Step Solution to Find the New Y-Value

Let's break down the problem step by step to find the y-value of P' after the translation. We'll start with the original coordinates of point P, which are (-2, 6). Then, we have the translation rule (x, y) → (x - 2, y - 16). This rule tells us that for any point (x, y), its new location after the translation will be (x - 2, y - 16). To find the new y-value of P', we need to focus on the y-coordinate of the original point P, which is 6. We will then apply the translation rule to this y-coordinate.

The translation rule tells us to subtract 16 from the y-coordinate. So, we'll take the original y-coordinate of P (which is 6) and subtract 16: 6 - 16 = -10. Therefore, the new y-coordinate of P' is -10. This is the y-value we were looking for. The x-coordinate of P' is -2 - 2 = -4, but the question only asks for the y-value, and we have found it. We can say that P' has the coordinates (-4, -10).

This might seem like a lot of steps, but it's really straightforward. The key is to understand the translation rule and how it affects the coordinates of each point. In this case, the y-value changes by subtracting 16. Once you get the hang of it, you'll be able to solve these problems quickly and easily. Remember, every point on the triangle moves according to the same rule, making the transformation consistent throughout the shape. After the translation, the triangle still exists, only in a different location on the coordinate plane.

Selecting the Correct Answer and Key Takeaways

Now that we've found the y-value of P', let's look at the multiple-choice options and select the correct answer. We determined that the y-value of P' is -10. Now, let's look at the given options:

A. -18 B. -16 C. -12 D. -10

The correct answer is D. -10, as it matches the y-value we calculated. Congratulations, you've solved the problem!

Here are the key takeaways from this problem:

  • Triangle Translation: Understand that this is a sliding motion where every point moves the same way.
  • Translation Rule: This rule dictates how much the x and y coordinates change.
  • Coordinate Changes: Apply the rule to find the new coordinates of each vertex.
  • Focus: Concentrate on the y-value when asked for the new y-coordinate.

By following these steps, you can confidently solve any translation problem. Remember that the x and y coordinates move independently based on the translation rule. Also, make sure to read the question carefully to identify what is being asked of you. Sometimes, you only need to determine the new x-coordinate, or sometimes only the new y-coordinate. Keep practicing, and you'll get the hang of it in no time. This problem illustrates a fundamental concept in geometry, demonstrating how shapes can be manipulated on a coordinate plane. These concepts are a foundation for more complex topics later on. So, understanding triangle translations is critical.

Visualizing the Translation and Tips for Success

To make things even clearer, let's visualize this translation. Imagine the triangle PQR on a graph. Point P is at (-2, 6). The translation rule (x, y) → (x - 2, y - 16) moves P 2 units to the left and 16 units down. If you were to draw this out, you would see that the new point P' is located at (-4, -10). Visually, the entire triangle has simply slid down and to the left without changing its orientation or shape. This is what a translation does. The triangle doesn't rotate or change its size; it just moves to a new position. Try sketching this out on graph paper to help you visualize it. This will make the concept even easier to grasp.

Here are some extra tips to help you with translation problems:

  • Draw a Diagram: Sketching the triangle and the translated position can help you visualize the changes.
  • Label Coordinates: Write down the coordinates of each vertex clearly to avoid confusion.
  • Break it Down: Focus on one coordinate (x or y) at a time, especially if you're just looking for the change in one of them.
  • Practice: The more problems you solve, the more comfortable you'll become with translations.

Remember, the core concept here is understanding how the translation rule affects each coordinate. It's a straightforward process once you grasp the basics. Don't be afraid to draw diagrams or write out the steps to help you visualize and solve the problem. Practice with different translation rules and triangle orientations to build your confidence. You can even create your own problems with different points and translation rules. This will help reinforce your understanding and make you even better at solving these types of problems. With practice, you'll become a master of triangle translations! Geometry can be fun, and with the right approach, you can master these concepts. Keep practicing and keep up the great work!