To rotate a point by 90 degrees clockwise about a specific point $C$, you'll follow these general steps:

1. Find the coordinates of the point to be rotated:

Let the coordinates of the point you're rotating be $P(x, y)$, and the point $C$ (the center of rotation) be $C(x_c, y_c)$.

2. Translate the point so that $C$ becomes the origin:

Subtract the coordinates of $C$ from the coordinates of $P$ to shift the point $P$ relative to $C$: $P' = (x - x_c, y - y_c)$

3. Apply the 90-degree clockwise rotation:

The formula for a 90-degree clockwise rotation of a point $(x', y')$ around the origin (0, 0) is: $(x', y') \rightarrow (y', -x')$ This means you swap the $x'$ and $y'$ coordinates and change the sign of the new $x'$ coordinate.

4. Translate the point back to the original position:

Once the rotation is applied, translate the point back by adding the coordinates of $C$: $P_{\text{rotated}} = (y' + x_c, -x' + y_c)$

Example:

Suppose you want to rotate a point $P(3, 4)$ 90 degrees clockwise about the point $C(1, 2)$.

1. Translate point $P$ relative to $C$: $P' = (3 - 1, 4 - 2) = (2, 2)$

2. Rotate 90 degrees clockwise: $(x', y') \rightarrow (y', -x') \Rightarrow (2, 2) \rightarrow (2, -2)$

3. Translate back: $P_{\text{rotated}} = (2 + 1, -2 + 2) = (3, 0)$

So the rotated point is $(3, 0)$.

Let me know if you need further clarification or have a specific example you'd like help with!

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Related Questions:

1. How do I perform a 180-degree clockwise rotation about a point?
2. What happens if I rotate a point counterclockwise instead of clockwise?
3. How can I rotate a point about the origin?
4. Can I rotate shapes like triangles or rectangles around a specific point?
5. How do translations and rotations work together in geometry?

Tip: When rotating about any point that isn't the origin, always perform the translation first, apply the rotation, and then translate back to get the final position.