Hey guys! Today, we're diving into a fun problem involving coordinate geometry and transformations. We'll be looking at how Randy translates a triangle on the coordinate plane and figuring out the rule he used. This is a fantastic way to understand how shapes move around in the coordinate system, a core concept in math and computer graphics. So, let's get started and unravel this geometric puzzle!
The Initial Setup: Triangle ABC
First, let's break down the initial conditions. Randy's initial triangle, ABC, is defined by three vertices on the coordinate plane. These vertices are crucial as they dictate the shape and position of the triangle before any transformation occurs. The coordinates provided are:
- A (7, -4)
- B (10, 3)
- C (6, 1)
To truly grasp what's happening, imagine these points plotted on a graph. Point A is located at (7, -4), meaning 7 units along the x-axis and -4 units along the y-axis. Point B sits at (10, 3), so 10 units on the x-axis and 3 units on the y-axis. Lastly, point C is at (6, 1), or 6 units on the x-axis and 1 unit on the y-axis. Connecting these points gives us triangle ABC. Understanding these initial coordinates is the foundation for figuring out how Randy moved the triangle.
The coordinate plane is our playground here. It's the grid where we plot points and shapes, and understanding how things move on this grid is what coordinate geometry is all about. Visualizing the triangle in this plane helps us see the transformation more clearly. Before we dive into the translated triangle, let's solidify this initial setup. Think of these coordinates as the triangle's "before" picture. What comes next is the "after," and our mission is to figure out the rule that connects these two states.
Remember, guys, in coordinate geometry, each point's location is precisely defined by its coordinates. Any change in these coordinates means a change in the point's position, and consequently, the shape's position. This brings us to the next part: the translated triangle. We'll see how these coordinates have shifted and then deduce the rule that caused the shift. So, keep these initial coordinates in mind as we move forward – they're our starting point in this geometric journey.
The Translated Image: Triangle A'B'C'
Now, let's turn our attention to the transformed triangle. After Randy translates the figure, we have a new set of vertices, forming triangle A'B'C'. These new coordinates tell us where the triangle has moved in the coordinate plane. The coordinates of the image are:
- A' (5, 1)
- B' (8, 8)
- C' (4, 6)
Notice how each point has shifted from its original location. A, which was at (7, -4), is now at A' (5, 1). B, initially at (10, 3), has moved to B' (8, 8). And C, which started at (6, 1), is now at C' (4, 6). These changes in coordinates are the key to unlocking the translation rule. A translation in geometry means sliding a figure without rotating or resizing it. It's a pure movement, a shift from one place to another, and this shift is consistent across all points of the figure.
The challenge now is to identify the specific shift that occurred. How many units did the triangle move horizontally, and how many units did it move vertically? This is where the beauty of coordinate geometry shines. By comparing the original coordinates with the translated coordinates, we can pinpoint the exact nature of the translation. This involves looking at the difference in the x-coordinates and the difference in the y-coordinates for each point. These differences will reveal the rule Randy used to transform triangle ABC into triangle A'B'C'.
Guys, visualizing this translation is super helpful. Imagine the original triangle and then picture it sliding to its new position. The distances it slides horizontally and vertically are constant for all points. This constant shift is what we're trying to find. So, let's get ready to analyze the changes in coordinates and decode the translation rule. This is where the math becomes a detective game, and we're the detectives!
Decoding the Translation Rule: The Shift in Coordinates
Alright, let's put on our detective hats and figure out the rule Randy used to translate the triangle. To do this, we'll compare the original coordinates of triangle ABC with the translated coordinates of triangle A'B'C'. The goal is to find a pattern – a consistent change in the x and y values that applies to all three points. This consistent change is the translation rule. Let's break it down point by point:
- Point A (7, -4) to A' (5, 1): To go from x = 7 to x = 5, we subtract 2. To go from y = -4 to y = 1, we add 5. So, the shift for point A is (-2, 5).
- Point B (10, 3) to B' (8, 8): Similarly, from x = 10 to x = 8, we subtract 2. From y = 3 to y = 8, we add 5. The shift for point B is also (-2, 5).
- Point C (6, 1) to C' (4, 6): Again, from x = 6 to x = 4, we subtract 2. And from y = 1 to y = 6, we add 5. The shift for point C is (-2, 5).
Do you see the pattern, guys? For each point, the x-coordinate decreases by 2, and the y-coordinate increases by 5. This consistent change tells us that the translation rule is (x, y) → (x - 2, y + 5). This means every point on the triangle has been shifted 2 units to the left (because of the -2 in the x-coordinate) and 5 units upwards (because of the +5 in the y-coordinate).
The translation rule (x, y) → (x - 2, y + 5) is the key to understanding how Randy moved the triangle. It's like a set of instructions that tells us exactly how each point has been repositioned. This rule applies uniformly across the entire triangle, preserving its shape and size while changing its location. This is the essence of a translation in geometry. We've cracked the code! By analyzing the coordinates, we've identified the transformation rule. This exercise highlights how coordinate geometry allows us to describe movements and transformations mathematically.
Formalizing the Rule: (x, y) → (x - 2, y + 5)
So, let's formalize what we've discovered. The translation rule that Randy used can be expressed as (x, y) → (x - 2, y + 5). This notation is a concise way to describe the transformation applied to any point (x, y) on the plane. It tells us that the new x-coordinate is obtained by subtracting 2 from the original x-coordinate, and the new y-coordinate is obtained by adding 5 to the original y-coordinate. This rule encapsulates the entire translation: the horizontal shift of 2 units to the left and the vertical shift of 5 units upwards.
The beauty of this rule is its universality. It doesn't just apply to the vertices of the triangle; it applies to any point on the plane. If Randy had drawn a square or a circle, the same rule would apply to every point on that shape, shifting it in the exact same way. This is a fundamental concept in transformations: a single rule can describe the movement of an entire figure.
Guys, understanding this notation is crucial for working with transformations in geometry. It provides a clear and unambiguous way to communicate how a figure has been moved or changed. It's like a mathematical shorthand for describing complex movements. When you see a rule like (x, y) → (x - 2, y + 5), you should immediately be able to visualize the transformation: a slide 2 units to the left and 5 units up. This ability to translate between mathematical notation and geometric visualization is a key skill in coordinate geometry.
To recap, the rule (x, y) → (x - 2, y + 5) is more than just a formula; it's a complete description of the translation. It tells us the direction and magnitude of the shift in both the horizontal and vertical directions. This formal representation is the final piece of the puzzle, solidifying our understanding of how Randy transformed triangle ABC.
Conclusion: Geometry and Transformations Unveiled
In conclusion, we've successfully deciphered the translation rule Randy used to transform triangle ABC into triangle A'B'C'. By carefully analyzing the changes in coordinates, we identified the rule as (x, y) → (x - 2, y + 5), which represents a shift of 2 units to the left and 5 units upwards. This exercise showcases the power of coordinate geometry in describing and understanding geometric transformations.
We've seen how a translation, a fundamental geometric operation, can be precisely defined using a simple rule. This rule applies uniformly to all points on a figure, preserving its shape and size while changing its position. Understanding these transformations is crucial in various fields, from computer graphics and animation to engineering and design. The ability to mathematically represent movements and changes in shapes is a powerful tool.
Guys, this problem illustrates the importance of breaking down complex problems into smaller, manageable steps. We started with the initial coordinates, then looked at the translated coordinates, and finally, compared the two to find the pattern. This step-by-step approach is key to success in mathematics and problem-solving in general. By understanding the underlying concepts and applying them methodically, we can unravel even the most challenging puzzles.
So, next time you encounter a transformation problem, remember the steps we took here. Look for the pattern, formalize the rule, and visualize the transformation. With practice, you'll become a pro at decoding geometric movements. And remember, geometry isn't just about shapes and figures; it's about understanding the relationships between them and the transformations they undergo. Keep exploring, keep questioning, and keep having fun with math!