How can I rotate a figure 90 degrees clockwise about a specific point?

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Rotating a Figure 90 Degrees Clockwise About a Point

Rotating a figure around a point is a fundamental concept in geometry that can be applied in various scenarios such as graphics programming, art, and even physical modeling. Here’s how you can achieve this in a systematic way:

Step-by-Step Process

  1. Identify the Center of Rotation: Choose the point around which you want to rotate your figure. This could be the origin (0, 0) or any other point in the coordinate plane.
  2. Determine Coordinates: Write down the coordinates of the vertices of the figure you intend to rotate. For example, a triangle with vertices A(x1, y1), B(x2, y2), and C(x3, y3).
  3. Translate the Figure: Subtract the coordinates of the center of rotation from each vertex’s coordinates. This will help you to rotate around the origin. If (hx, hy) is the center of rotation, the new coordinates become:
    • A'(x1 – hx, y1 – hy)
    • B'(x2 – hx, y2 – hy)
    • C'(x3 – hx, y3 – hy)
  4. Apply the Rotation: To rotate each point 90 degrees clockwise, you can use the transformation rules:
    • For each point (x, y), the new coordinates after 90-degree clockwise rotation become (y, -x).
  5. Translate Back: After applying the rotation, translate the points back to their original position by adding the coordinates of the center of rotation (hx, hy). So your final coordinates will be:
    • A”(y1 – hx, -x1 + hy)
    • B”(y2 – hx, -x2 + hy)
    • C”(y3 – hx, -x3 + hy)

Example

Let’s say we want to rotate point A(2, 3) 90 degrees clockwise about point O(1, 1).

  1. Translate Point A to the Origin: (2 – 1, 3 – 1) = (1, 2)
  2. Rotate Point Around the Origin: (2, -1)
  3. Translate Back: (2 + 1, -1 + 1) = (3, 0)

So, the new coordinates of point A after a 90-degree clockwise rotation about the point O(1, 1) are (3, 0).

Conclusion

Rotating figures can seem complex at first, but by breaking it down into clear steps, you can master this technique. Whether you’re creating graphics or solving geometry problems, understanding how to rotate shapes is an invaluable skill!

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