# Rotation

Rotation is a transformation in which a figure is turned about a given point.

## Definition

A rotation of an object around a point or an axis is a continuous transformation that does not change the distance of any of the points on the object from the point or the axis..

## Rotation About A Point

## Formula

Set the origin \(O(0,0)\) as the center of rotation. When a point \(P(x,y)\) is rotated counterclockwise by angle \(\theta\), the resulting point \(P'(x',y')\) can be calculated with the formula

\[\begin{pmatrix}x' \\ y'\end{pmatrix} = \begin{pmatrix} \cos \theta & -\sin\theta \\ \sin \theta & \cos \theta\end{pmatrix} \begin{pmatrix}x \\ y \end{pmatrix}\]

In other words,

\[\begin{aligned} x' &= x\cos\theta - y\sin\theta \\ y' &= x\sin\theta + y\cos\theta\end{aligned}\]

## Rotational Symmetry

*See also: Symmetry*

**Rotational Symmetry** occurs if an object can be rotated less than \(360^\circ\) and be unchanged. The center of the rotation is called the *point of symmetry*.

Take for example some of the letters of the alphabet, which have rotational symmetry as they can be rotated \(180^\circ\) and remain the same.

## Problem Solving

If point \((81,62)\) is rotated about point \((25,64)\) by \(30^{\circ}\) in each rotation, how many rotations will be required to get the final point again as \((81,62)\)?

We have to rotate the point by \(360^{\circ}\) to get it back. So the rotations required are \(\frac {360^{\circ}}{30^{\circ}}\)= \(12\) rotations

Let \(f (x)\) be the relation from \(N->\){\(5,10\)} equal to \(10^{\circ}\) when \(x\) is odd and \(5^{\circ}\) when \(x\) is even. Here \(x\) denotes \(x^\text{th}\) rotation of point \((2,5)\) and \(f (x)\) is angle rotated clockwise by this \(x^\text{th}\) rotation when rotated about the point \((8,15)\). If after \(a\) rotations, the final point is again \((2,5)\), find the minimum value of \(\frac {a}{2}\).