To rotate a point \(180^\circ\) counterclockwise around another point, you can follow these steps:
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Let the point to be rotated be \(P(x,y)\) and the center of rotation be \(C(a,b)\). In this case, \(P\) is the point that is being rotated, and \(C\) is the point \((-4, 6)\).
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The formula for rotating a point \(P(x,y)\) around the center \(C(a,b)\) by \(180^\circ\) can be expressed as: \[ P' = (a - (x - a), b - (y - b)) = (2a - x, 2b - y) \]
Let's say we want to rotate the point \( (x, y) \) around the center \( C(-4, 6) \). The resulting coordinates \(P'\) after the rotation will be calculated as follows:
- Let \( P = (x, y) \) (the coordinates of the point you want to rotate).
- The rotated point \( P' = (2a - x, 2b - y) \).
For instance, if you have a specific point to rotate (let's say \( P(0,0) \)), we will substitute:
- \( a = -4 \)
- \( b = 6 \)
- \( x = 0 \)
- \( y = 0 \)
Now plugging in the values, we calculate:
\[ P' = (2 \cdot -4 - 0, 2 \cdot 6 - 0) = (-8, 12) \]
If you specify the point you want to rotate, I can provide you with the exact coordinates of the newly rotated point after performing the \(180^\circ\) counterclockwise rotation around \((-4, 6)\).