Unraveling Rotations Finding Original Triangle Vertices

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Hey guys! Ever wondered how shapes change when you spin them around? In geometry, this is called rotation, and it's a pretty cool transformation. Today, we're diving into a specific problem where a triangle has been rotated, and our mission is to find out where it originally was. We're given the coordinates of the rotated triangle's vertices, and we need to work backward to find the coordinates of the original triangle. Let's put on our detective hats and get started!

The Challenge: Decoding the Rotation

Our challenge revolves around a triangle that has undergone a 90-degree rotation about the origin. The coordinates of the rotated vertices, now labeled as A', B', and C', are given as follows: A'(6, 3), B'(-2, 1), and C'(1, 7). The core question we need to answer is: What were the coordinates of the original vertices A, B, and C before the rotation occurred? This involves understanding how rotations affect coordinates and applying the reverse transformation to find the original points. This problem isn't just a mathematical exercise; it highlights the importance of geometric transformations in various fields, from computer graphics to engineering. By solving it, we not only sharpen our math skills but also gain a deeper appreciation for the practical applications of geometry.

To tackle this, we need to understand how a 90-degree rotation affects the coordinates of a point. When a point (x, y) is rotated 90 degrees counterclockwise about the origin, its new coordinates become (-y, x). This is a fundamental rule of rotational transformations that we'll use to reverse the process. Think of it like this: the x and y coordinates switch places, and the original y-coordinate becomes negative. This transformation rule is the key to unlocking the original coordinates of our triangle.

To visualize this better, imagine a point in the first quadrant. After a 90-degree counterclockwise rotation, it moves to the second quadrant. The x-coordinate, which represented the horizontal distance from the y-axis, now becomes the negative y-coordinate, representing the vertical distance below the x-axis (but with the sign flipped). Similarly, the y-coordinate, which represented the vertical distance from the x-axis, now becomes the x-coordinate, representing the horizontal distance from the y-axis. This mental picture helps solidify the understanding of the coordinate transformation during rotation. We'll be using this principle to reverse the rotation and pinpoint the original positions of the triangle's vertices. So, let's dive deeper into the mechanics of reversing the rotation!

Unveiling the Rotation Rule

To crack this coordinate conundrum, we need to understand the rule that governs a 90-degree rotation about the origin. When a point (x, y) is rotated 90 degrees counterclockwise, its image becomes (-y, x). This transformation is the key to reversing the rotation and finding the original coordinates. This rule essentially swaps the x and y coordinates and negates the original y-coordinate.

But why does this happen? Let's break it down. Imagine a point plotted on a graph. Rotating it 90 degrees counterclockwise means we're essentially swinging it around the origin, changing its position relative to the x and y axes. The original y-coordinate, which represents the vertical distance from the x-axis, now becomes the horizontal distance from the y-axis in the new position. However, since we're rotating counterclockwise, this new horizontal distance is on the negative side of the x-axis, hence the negative sign. The original x-coordinate, representing the horizontal distance from the y-axis, becomes the new vertical distance from the x-axis.

This understanding is crucial because we're not just looking at a simple switch; the negation of the original y-coordinate is vital. It reflects the change in the point's position as it moves from one quadrant to another during the rotation. Think of it like this: a point in the first quadrant (where both x and y are positive) moves to the second quadrant (where x is negative and y is positive) after a 90-degree counterclockwise rotation. This change in quadrant is precisely what the negation captures. Now, armed with this knowledge, we can confidently apply the reverse transformation to find our original vertices.

This rule is our golden ticket to solving the problem. But how do we use it in reverse? If a point (x, y) becomes (-y, x) after a 90-degree rotation, then to go back, we need to apply the opposite transformation. This means if we have a point (a, b) that's the result of a 90-degree rotation, the original point was (b, -a). We're essentially reversing the swap and the negation. Let's see how this works in practice with our triangle's vertices.

Reversing the Rotation: Finding the Original Vertices

Now, let's roll up our sleeves and apply this reversed rule to the coordinates of A', B', and C' to find the original coordinates of A, B, and C. Remember, if A' is (6, 3), B' is (-2, 1), and C' is (1, 7), we need to swap the coordinates and negate the new y-coordinate to find the original points.

  • Finding A: A' is (6, 3). Applying the reverse transformation (b, -a), we get A as (3, -6). We've swapped the 6 and 3, and then negated the 6 to get -6. So, the original vertex A was located at (3, -6).
  • Finding B: B' is (-2, 1). Applying the same rule, we swap -2 and 1 to get (1, -(-2)), which simplifies to (1, 2). Therefore, the original vertex B was at (1, 2).
  • Finding C: C' is (1, 7). Swapping 1 and 7 and negating the 1 gives us (7, -1). Thus, the original vertex C was located at (7, -1).

So, there you have it! By applying the reverse transformation, we've successfully found the coordinates of the original vertices of the triangle. A was at (3, -6), B was at (1, 2), and C was at (7, -1). This demonstrates the power of understanding geometric transformations and how we can use them to solve problems by working backward. This process is not just about finding the coordinates; it's about understanding the underlying principles of how shapes move and change in space. Now, let's recap what we've learned and see the bigger picture of this problem.

Wrapping Up: The Original Triangle Revealed

Mission accomplished! We've successfully determined the original coordinates of the triangle's vertices before the 90-degree rotation. By understanding the rule of rotation and applying its reverse, we were able to unravel the mystery. The original vertices were:

  • A (3, -6)
  • B (1, 2)
  • C (7, -1)

This exercise not only reinforces our understanding of geometric transformations but also highlights the importance of working backward to solve problems. In many real-world scenarios, we're presented with the result of a process and need to figure out the starting conditions. This problem mirrors that situation perfectly.

The beauty of this problem lies in its simplicity and the clarity with which it demonstrates the principles of rotation. It's a great example of how a single, well-understood rule can be used to solve a seemingly complex problem. By breaking down the rotation into its component parts – the swapping of coordinates and the negation – we were able to reverse the process and find the original points. This approach of breaking down problems into smaller, manageable steps is a valuable skill not just in mathematics but in any field.

Furthermore, this problem showcases the interconnectedness of different mathematical concepts. Geometry, algebra, and coordinate systems all come together to provide a solution. This holistic approach to problem-solving is what makes mathematics so powerful and applicable in diverse areas. So, the next time you encounter a problem involving transformations, remember the principles we've discussed here, and you'll be well on your way to finding the solution. Keep exploring, keep questioning, and keep unraveling the mysteries of mathematics!

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Unraveling Rotations Find Original Triangle Vertices

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After a 90-degree rotation about the origin, the vertices of a triangle's image are A'(6,3), B'(-2,1), and C'(1,7). What were the coordinates of the original vertices?