 Hello and welcome to the session. In this session, we are going to discuss rotation. See the given figure. In this figure, the blue star is the initial position of the star with respect to point A. Now, we rotated star in clockwise direction, keeping O fixed. The new position of the star is red star. This red star is called image of blue star after rotation with respect to a fixed point O. 45 degrees is the angle of rotation and O is the center of rotation. The arrow on angle tells us the direction clockwise and anti-clockwise of rotation. Here in this figure, it represents the rotation in clockwise direction. And the other figure is of rotating blue star anti-clockwise at 120 degrees. So, we can say that a rotation is a transformation in which the object is rotated about a fixed point and the direction of rotation can be clockwise or anti-clockwise. Here we shall discuss 90 degrees, 180 degrees and 270 degrees rotation of a figure center origin that is 00. First we are going to discuss 90 degrees rotation and we see the two arrows. First arrow was horizontal, later it became vertical. It means it is rotated to 90 degrees angle. Now, we shall see how to find image and vertices of the figure on coordinate plane when rotated at 90 degrees. Now, we consider a line AB on the graph. Now, if we want to rotate it at 90 degrees clockwise, we rotate the graph paper at 90 degrees and we get the following graph. We see that this is the rotated image of the given graph. Now, let us remain the axis and the numbers to get the rotated image of the given line AB. Hence, we got the rotated image of the line AB as A prime B prime with A prime having coordinates 3 minus 1 and B prime with the coordinates 1 minus 2. Geometrically, we can say that if a coordinate XY is in any quadrant and you have to rotate it at 90 degrees in clockwise direction, then first switch the coordinates, that is, XY becomes YX and then multiply the new Y coordinate by minus 1 so it becomes Y minus X. It means the coordinates XY when rotated at 90 degrees in clockwise direction becomes Y minus X. Like, if you have the coordinates T4 on rotation at 90 degrees in clockwise direction changes to minus 2. Now, we will do the same for anti-clockwise direction. Again, if we consider the line AB and if we want to rotate this line at 90 degrees in anti-clockwise direction, then we first rotate this graph at 90 degrees in anti-clockwise direction. We get the following graph after the rotation of 90 degrees in anti-clockwise direction. Now, again, we remain the axis and the coordinates so as to get the required rotated image of the given line AB at 90 degrees in anti-clockwise direction. And hence, we have got the rotated image of the line AB as A prime B prime with A prime having coordinates minus 3 1 and B prime with the coordinates minus 1 2. So, we can say that for any coordinates XY in any quadrant, if we want to rotate this at 90 degrees in anti-clockwise direction, then first switch three coordinates and then multiply the new X coordinate by minus 1. That is, any coordinates XY on rotation at 90 degrees in anti-clockwise direction changes to minus Y X. Like any coordinates, say minus 2 4 on rotation at 90 degrees in anti-clockwise direction changes to minus 4 minus 2. Now, we are going to discuss 180 degrees rotation, rotating 180 degrees is same as rotating at 90 degrees. So, we rotate graph two times at 90 degrees here direction does not matter. Now, if we consider the given line AB, we get the line A prime B prime as the rotated image of the line AB at 90 degrees in clockwise direction. Now, if we again rotate this line at 90 degrees in clockwise direction, we get the rotated image of the line AB at 180 degrees. And we again rotated the line A prime B prime in the similar manner as we did earlier and we get A double prime B double prime as the rotated image of the line A prime B prime at 90 degrees in clockwise direction. Now, we see that we have rotated line AB at 90 degrees twice and therefore, we get line A double prime B double prime as the rotated image of the line AB at 180 degrees having A double prime with the coordinates minus 1 minus 1. So, we get line AB minus 3 and B double prime with the coordinates minus 2 minus 1. Geometrically, we can say that to rotate a point 180 degrees above the origin, we multiply each coordinate by minus 1. The image is the same whether you rotate the figure clockwise or counterclockwise to say that any coordinate xy on rotation of 180 degrees in any direction changes to minus x minus y. And here, we have seen that if any line AB is rotated at 180 degrees in the coordinate plane with center origin, then the point A with the coordinates 1, 3 gets transformed to A double prime with the coordinates minus 1 minus 3 and the point B with the coordinates 2, 1 gets transformed to B double prime with the coordinates minus 2 minus 1. So, the new image is A double prime B double prime line. And now, we will discuss 270 degrees rotation and we know that rotating 270 degrees as rotating twice at 90 degrees. So, we rotate the graph 3 times at 90 degrees and here, direction does matter. Now again, we consider the line AB, line AB when rotated first at 90 degrees in clockwise direction. We get this line A prime B prime and line A prime B prime when rotated at 90 degrees in clockwise direction, we get line AB double prime B double prime and again rotating this line at 90 degrees in clockwise direction, we get the line A triple prime B triple prime. So, we can say that line AB when rotated at 270 degrees in clockwise direction becomes the line A triple prime B triple prime with A triple prime having coordinates minus 3, 1 and B triple prime with the coordinates minus 1, 2. So, we say that if our coordinate xy is in any quadrant and we have to rotate it at 270 degrees in clockwise direction then first we interchange the coordinates that is xy changes to yx and then we multiply the new x coordinate by minus 1. It means, xy changes to minus y, like if we have the coordinates minus 2, 4 on rotation at 270 degrees in clockwise direction changes to minus 4, minus 2. Now, for anti-clockwise direction if we want to rotate any point say A at 270 degrees in anti-clockwise direction with the coordinates 3, 1 on rotation at 270 degrees in anti-clockwise direction changes to 1, minus 2 that is we switch the coordinates and then multiply the new y coordinate by minus 1 that is if we have any coordinates xy on rotation at 270 degrees in anti-clockwise direction changes to y, minus x that is we interchange the coordinates and then multiply the new y coordinate by minus 1. We are going to discuss rotational symmetry which means the figure looks same after a certain amount of rotation of figure has rotational symmetry a rotation of 180 degrees or less clockwise. Counter-clockwise about its center reduces an image that fits exactly on the original figure it means if we rotate a figure at some angle maybe 45 degrees or 90 degrees or any other angle say 180 degrees or less clockwise or anti-clockwise we get the same figure there is no change in the figure even after rotation if we rotate the given figure by 90 degrees clockwise or anti-clockwise we get the same figure this completes our session hope you enjoyed this session.