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In general I know that if we rotate $(x, y)$ about origin through $180^\circ$ we will get new image $(-x, -y)$, but suppose that we make rotation not about origin but some other point $(a, b)$ does your result be rotation around origin + or - $(a, b)$? Suppose we have point $A(3, 27)$ and we want turn it by $180$ around the point $(2, -1)$, if we rotate $(3, 27)$ about origin by $180$ we get $(-3, -27)$ but how to connect $(2, -1)$ to this result?

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The idea is to translate to the origin, rotate, and then undo the translation... – J. M. Jul 23 '11 at 11:41
The general idea even has a name: Transform, Solve, Transform Back. – André Nicolas Jul 23 '11 at 13:12
up vote 2 down vote accepted

You can make a translation of axes so that $(2,-1)$ becomes the new origin. The new axes are $X=x-2,Y=y+1$. Then you compute the new coordinates of $A(3-2,27+1)=(1,28)$. The symmetric point $A'$ with respect to $(X,Y)=(0,0)$ is thus $A'(-1,-28)$ in the $XY-$coordinate system or $A'(-1+2,-28-1)=(1,-29)$ in the original $xy-$coordinate system.

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great answer thanks @Américo Tavares – dato datuashvili Jul 23 '11 at 11:54
@user3196: Thanks! – Américo Tavares Jul 23 '11 at 12:25

You can first move the point $(2,-1)$ to the origin, by adding $(-2,1)$ to all the points of the plane. Now the point $A$ goes to $(1,28)$. Now rotate: You get $(-1,-28)$. Now you have to return back: the image would be $(-1,-28)+(2,-1)=(1,-29)$. That's your result.

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