To rotate a point 180° counterclockwise around another point, you can follow these steps:
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Identify the point to rotate and the center of rotation.
Here, the point to rotate is \((−4, 6)\) and the center of rotation is also \((−4, 6)\). -
Since a 180° rotation around a point results in the same distance from the center but in the opposite direction, the resulting point will be located directly opposite on a straight line through the center.
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Calculate the resulting point:
For a 180° rotation, the coordinates of a point \((x, y)\) around a center \((cx, cy)\) can be found using the following formula:
\[ (x', y') = (2 \cdot cx - x, 2 \cdot cy - y) \] -
Substituting in the values: \[ (x', y') = (2 \cdot (-4) - (-4), 2 \cdot 6 - 6) \] \[ = (-8 + 4, 12 - 6) \] \[ = (-4, 6) \]
Thus, the resulting point after rotating \((−4, 6)\) by 180° counterclockwise around \((−4, 6)\) is \((-4, 6)\).
This means that rotating a point around itself will not change its position.