# How to find a point after rotation?

Initially the position of the shape was in (100, 100). I am rotating (say 30 degrees) the shape as shown in the image below. I have found the starting point of the rotated object. Is there a formula to find the point of the shape when the rotation was 0 degree? Any help?

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What exactly do you mean by "position of the shape was in (100,100)"? What are these coordinates with respect to? – matt Jun 11 '12 at 6:02
(100, 100) is the left top corner of the first rectangle. – Karthik Jun 11 '12 at 6:04
And where is (0,0)? – matt Jun 11 '12 at 6:06
And which point are you rotating about? – Mark Bennet Jun 11 '12 at 6:15
Rotating w.r.t the center of the rectangle. – Karthik Jun 11 '12 at 6:18

Rotating any point $(x,y)$ on a plane about the origin by $\theta$ degrees (counter-clockwise) is given by $$\left[ \begin{array}{c} x' \\ y' \\ \end{array} \right] = \left[ \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \\ \end{array} \right] \left[ \begin{array}{c} x \\ y \\ \end{array} \right],$$ where $(x',y')$ denotes your new position.