 Hello and welcome to the session. In this session we will discuss that center of dilation remains unchanged and skin factor enlarges or reduces the size of original figure. Now in our earlier sessions we have discussed the concept of dilation when center was at the origin. Now let us recall that dilation is a transformation in which a figure is enlarged or reduced with respect to a fist point towards the center of dilation. And the skin factor of a dilation is the ratio of the side length of the image to the corresponding side length of the original figure. Now consider the given figure. In this figure a blue line A B is enlarged to form a different paint line A dash B dash with respect to center. Now here you can see that there is a change in size that shape remains same. Now length of original line segment that is length of A B is 2 units and length of enlarged figure that is the line segment A dash B dash is 4 units. And now if you see the ratio of the corresponding sides that is A dash B dash upon A B is equal to upon 2 which is equal to 2 therefore the skin factor is equal to 2. Skin factor is equal to image length upon pre-image length of the dilation. The corresponding sides of the image and the pre-image are proportional and equal to the skin factor. Now let us denote the skin factor by K. Now if the absolute value of the skin factor that is absolute value of K is greater than 1 the image is an enlargement and if absolute value of skin factor is between 0 and 1 the image is a reduction that is negative then the image is enlargement or reduction depending on its absolute value that it may generate on the of the center of the pre-image. Now let us discuss the relation between the center of dilation and skin factor. Now in this figure we have drawn rays from the center of dilation O the vertices of the line segment A B. Now here you can see that these rays through the vertices of the image after dilation here O is the origin with coordinates 0 0 and coordinates of O are 0 0. So let us find distance O A by using distance formula. So this will be equal to square root of minus 0 O which is equal to which is equal to 2 root 2 which will be equal to 0 whole square minus 0 whole square which is equal to square root of is equal to square root of 32 and this is equal to 4 root 2. Thus we have 4 root 2 upon 2 root 2 which is equal to O V dash upon O V will be equal to 2 and this implies O V dash is that is K. So we can write this equation as A dash is equal to K into O V. Now the figure we can see that line segment A B is equal to V dash the constant of the image in dilation will be parallel that center for lines and center for line segment A will be the image of center of dilation. Now dilation with a scale factor of 2 can be written as D2 thus D2 is equal to that image of A under dilation with scale factor with center C and scale factor K is across bed maps every point P in the plane to a point P dash the following properties of the image center say P has image P dash is equal to absolute value of K that is absolute value of scale C P and here K is not equal to 1 and the dilation is between 0 and 1 dilation is enlargement of K is greater than we should note that center of dilation can be original at any other point when center of the point P with coordinates X Y will be transformed to the point P dash with coordinates K X, K Y then to draw dilated image we draw rings directed from the fixed point of dilation center C and then we mark the image points using ruler or compass with measurement such that C P dash is equal to K times C P and then we join the image points to get the dilated figure that is dilated using here in the first step let us take and R see we have drawn a rays passing through the vertex P vertex Q and vertex R now here ray of the vertex P to the vertex R now by using the ruler measure length C P is equal to 1 by 2 now image of point P the image points using ruler or compass with measurement such is equal to K times C P similarly C Q dash is equal to K times C Q dash is equal to K times C R now this implies C P dash is equal to now here scale factor is 1 by 2 so C P dash is equal to 1 by 2 into C P similarly C P dash is equal to 1 by 2 into R dash is equal to 1 by 2 into C R now using ruler of the line segment C P make image as the point P dash now the point P dash is the image of the point P will mark the point Q dash respectively using now let us try the points P dash Q dash and hence we get a triangle point C as the center of dilation first Q dash R dash is the image of triangle P Q R is equal to 1 by 2 now there is a reduction of the original triangle because the scale factor which is equal to 1 by 2 lies between 0 and 1 now triangle and triangle P dash U dash R dash proportional here we have dilated the given triangle using scale factor 1 by 2 in dilation the size changes but scale remains thus the image and pre-image are similar to each other in dilation now here that is triangle P Q R triangle P dash U dash R dash and these two triangles P dash Q dash upon P Q is equal to P dash R dash upon P R is equal to upon Q R we have discussion remains unchanged reduces or enlarges the given figure hope you all have enjoyed the session