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Have you ever wondered how a simple rotation can entirely change the coordinates of a geometric shape? In this article, we'll delve into the fascinating world of rotational transformations, specifically a 90-degree clockwise rotation about the origin. Get ready to unlock the secrets behind this transformation and see how it impacts the coordinates of a triangle.
Imagine you're given a triangle ABC, and your task is to rotate it 90 degrees clockwise around the origin. What does this mean? Well, it's like taking your triangle for a spin, but in the realm of mathematics, this spin has a specific rule that dictates how each point in the original triangle shifts to its new position in the rotated triangle, A'B'C'.
Before we jump into the rule, let's pause for a moment and think about what this rotation does to each point in the triangle. What happens to the coordinates of point A, B, and C when they are rotated to become A', B', and C'?
To understand this, let's set up a little table. We'll list the coordinates of the original points (pre-image) and the coordinates of the points after rotation (image).
After the rotation, the points transform into: - A': (5, -4) - B': (3, -4) - C': (5, -7)
Now, let's analyze these transformations. What pattern do you notice? For point A, the original x-coordinate (4) becomes the negative of that (negative 4) in the new y-coordinate of A'. Similarly, the original y-coordinate (5) becomes the new x-coordinate in A'.
Could this be a coincidence? Let's check the other points. For point B, the original x-coordinate (4) becomes negative 4 in the new y-coordinate of B'. And the original y-coordinate (3) becomes the new x-coordinate. The same pattern holds true for point C.
So, what's the rule? It appears that the new x-coordinate in the image is the negative of the original y-coordinate, and the new y-coordinate is the original x-coordinate. In mathematical terms, if we have a point (x, y) in the pre-image, the image point after a 90-degree clockwise rotation will be (y, -x).
This rule isn't just a lucky guess; it holds up for every point in the triangle. It's a powerful tool that allows us to predict the new coordinates of any point after a 90-degree clockwise rotation.
In conclusion, rotational transformations are more than just spinning shapes on a graph. They have a precise mathematical foundation that can be applied to any point in a shape. By understanding the rule behind a 90-degree clockwise rotation, we can unlock the secrets of how points shift and change in this intriguing mathematical dance.
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