 Hello and welcome to the session. In this session first we will discuss about point reflection, the reflection of a point, say a point P. In the fixed point O is the point P dash such that the point O is the immediate point of the segment P P dash. Consider a fixed point O let this point be P. So the reflection of this point P would be a point P dash which is the image of point P and this point O is the immediate point of P P dash. So we can say that the reflection of the point P in the point O is the point P dash such that O is the immediate point of P P dash. So this is the point reflection of the point P and this fixed point O is called the center of reflection. Now the reflection of the center of reflection is the center itself. So we can say the reflection of the point O is the point O itself and we can write this as reflection of the point O in point O is the point O itself and symbolically we can write the reflection of point P in the point O is P dash in this way. This can also be written as reflection of the point P dash in point O is point P. Next we have reflection in the x-axis. A reflection in the x-axis could be denoted by rx. It maps a point with coordinates xy onto the point with coordinates x-y that is the sign of the ordinate is changed under reflection in the x-axis. So if we have a point say a point A with coordinates xy then the reflection of this point A in x-axis would be a point A dash with coordinates x-y that is the sign of the x-axis is unchanged and the sign of the ordinate is changed. Consider a point P with coordinates x0 then this point lies on the x-axis as the ordinate is 0. So we can say that the reflection of such points like point P with coordinates x0 is the point itself. So such points with coordinates x0 that is in which the ordinate is 0 are their own reflections as they lie on the x-axis. Consider a point A with coordinates 12 the reflection of this point A in the x-axis would be a point A dash with coordinates 1 minus 2 that is we change the sign of the ordinate. Next we discuss the reflection in the y-axis the reflection of a point in y-axis is denoted by ry. It maps the point with coordinates xy on to a point with coordinates minus xy. So this shows that under the reflection of a point in the y-axis the ordinate of a point remains unchanged while the sign of the x-axis is changed. For any point P with coordinates xy the reflection under y-axis would be a point P dash with coordinates minus xy. Consider this point A with coordinates 12. Let us now find out the reflection of this point under y-axis. For this we only need to change the sign of the fc sa that is 1 would become minus 1 and the sign of the ordinate remains the same that is 2 remains as it is. So the coordinate of the image of the point A under reflection y-axis would be minus 12 and this point A dash with coordinates minus 12 is the reflection of the point A under y-axis. The points like point P with coordinates 0 y such points lie on the y-axis as the x-coordinate is 0 that is fc sa is 0. So the reflection of such points that is point P with coordinates 0 y is the point P itself such points are their own reflections because they lie on the y-axis. Next we discuss the reflection in the origin reflection in the origin denoted by this ro maps the points with coordinates xy onto the point with coordinates minus x minus y that is under reflection of a point in the origin the sign of both fc sa and ordinate changes. For a point say a point A with coordinates xy the reflection of this point A in the origin is denoted by the point A dash with coordinates minus x minus y that is the sign of both fc sa and the ordinate changes. For example if we have a point P with coordinates minus 12 then the reflection of this point P in the origin would be a point with coordinates in which the sign of the fc sa and the ordinate is changed that is it would become 1 and minus 2. So 1 and minus 2 is this point so this is the point P dash which is the reflection of the point P in the origin. Next we have invariant point points that remain unaltered under a reflection transformation in points and invariant point on image for example any point say point P in which the coordinates are of kind x0 that is the fc sa is x and the ordinate is always 0 such points invariant the reflection in the x axis since these points lie on the x axis so when they are reflected in the x axis the coordinates of their images are also of the form in which the coordinates are x0. Famously case with the points of the form 0 y such points are invariant under reflection in y axis as these points lie on the y axis and when they are reflected in the y axis the coordinates of its image would be the points with coordinates 0 y only so they are invariant. This point O that is the origin with coordinates 00 it is invariant the reflection in x axis and y axis both that is this point does not change when it is reflected in the x axis as well as in the y axis. Also every point in a line is invariant the reflection in the same line. Suppose if we have a point on a line AB then this point is reflected in the line AB itself and so we can say that the point is not altered that is it is invariant. Next we have the reflection of a point in the line x equal to A x is equal to A is this line AB which is parallel to the y axis and it is at a distance A from the y axis that is we have OP as A. We take a point C with coordinates x1, y1 and let this point C be reflected in the line x equal to A that is in the line AB in C dash with coordinates y1 reflection of the point C in the line x equal to A. Now since C has coordinates x1, y1 so this means OL is equal to x1 and CL is equal to y1. Now as C dash has coordinates x2, y1 so this means that ON is equal to x2 and C dash N is equal to y1. Now we have OP is equal to A this means that OL plus LP is equal to A from the figure now OL is x1 so this means LP is equal to A minus x1 and also this LP would be equal to PN as we know that the reflection of a point is at the same distance from the line of reflection as the point itself and so LP would be equal to PN and so we have PN is also equal to A minus x1. Now ON is equal to x2 and from the figure we have OP plus PN is equal to x2. Now OP is equal to A and PN is equal to A minus x1 so here we have A plus A minus x1 is equal to x2 this means we get x2 is equal to 2A minus x1 so therefore as we had the reflection of the point C with coordinates x1, y1 is a point C dash with coordinates x2, y1 means that the reflection of the point C with coordinates x1, y1 is a point C dash with coordinates 2A minus x1, y1. So in general we can say that the reflection of a point in the line x equal to A maps any point with coordinates xy onto a point with coordinates 2A minus xy. Next we shall discuss the reflection of a point in the line y equal to B. Consider this line CD this is the line y equal to B that is it is at a distance B from the x axis so if we have this point as P so we can say that OP is equal to B. Consider this point A with coordinates x1, y1 and let this point A be reflected y equal to B that is in the line CD and let this point say A dash with coordinates x1, y2 be the reflection of a point A in the line y equal to B. Now as this OP is equal to B so this means ST would also be equal to ST the point A has coordinates x1, y1 so OT is equal to x1 and AT is equal to y1. From the figure we have ST is equal to SA plus AT so this means SA is equal to ST which is B minus AT that is y1 so SA is equal to B minus y1. Now A dash S would be equal to SA as A dash is the reflection of the point A and so it would be at the same distance from the line y equal to B as the point A so A dash S would be equal to SA. Now as SA is equal to B minus y1 so A dash S is also equal to B minus y1. Point A dash has coordinates x1, y2 so A dash T is equal to y2. Now from the figure we have A dash T is equal to A dash S plus ST and so this is equal to y2. Now A dash S is B minus y1 and ST is B so here we have B minus y1 plus B is equal to y2 or you can say that y2 is equal to 2B minus y1. The reflection of the point A with coordinates x1, y1 was the point A dash with coordinates x1, y2 so this means that the reflection of a point A with coordinates x1, y1 is the point A dash with coordinates x1, 2B minus y1 that is in place of y2 we put 2B minus y1. In general we can say that under reflection of a point in the line y equal to B the point with coordinates xy maps onto a point with coordinates x2B minus y. So if you consider the example the reflection of a point 2, 4 in the line x equal to 3 would be the point with coordinates 2A minus xy so 2 multiplied by A now A in this case is 3 minus the x coordinate of the point which is 2 comma the y coordinate which is 4 so this is equal to 6 minus 2, 4, 4 so this is the reflection of the point 2, 4 in the line x equal to 3. In the same way the reflection of the point 2, 4 in the line y equal to 3 if the point with coordinates x2B minus y so x in this case is 2 now the y coordinate of this image would be 2 into B which is in this case 3 so 2 into 3 minus the y coordinate of the point which is 4 so this is equal to 2 comma 2 this is the reflection of the point 2, 4 in the line y equal to 3. This completes the session hope you have understood the concept of point reflection reflection in the x axis y axis origin in variant points and the reflection of a point in the lines x equal to A and y equal to B.