Dilating shapes: expanding

the graph below contains the rectangle ABC P draw the image of a BCP under a dilation whose centre is at P and a scale factor is what is 1 and 2/3 what are the lengths of the side a B and it's image so we're going to do a dilation centered at P so for centering a dilation at P that means that every other point is going and a scale factor is 1 and 2/3 that means once we scale once we perform the dilute once we perform the dilation every point is going to be 1 in 2/3 times as far away from P well P is 0 away from P so P is still going to be its image is still going to be at P so let's put that point right over there now point C is going to be 1 in 2/3 times far as far as it is right now so let's see right now it is 6 away it's at negative 3 and P is or it's why its x-coordinate is the same but in the y-direction P is at 3 C is at negative 3 so it's 6 less we want to be 1 in 2/3 times as far away so what's one in 2/3 of 6 well 2/3 of 6 is 4 so it's going to be 6 plus 4 it's going to be you're going to be 10 away so 3 minus 10 that gets us to negative 7 so that gets us right over there now point a right now it is 3 more in the horizontal direction then point P's x coordinate so we want to go 1 and 2/3 as far so what is 1 and 2/3 times 3 well that's going to be 3 plus 2/3 of 3 which is another 2 so that's going to be 5 so we're going to get right over there that we could complete the rectangle and notice point B is now 1 in 2/3 times as far in the horizontal direction it was three away in the horizontal direction now it is 5 away from p's x-coordinate and in the vertical direction it in the vertical direction in the Y direction it was 6 below pease y-coordinate now it is one in 2/3 times as far it is 10 below it is 10 below p's y-coordinate so to answer these questions this little length of segment a B a B well we already saw that that is we're going from three to negative 3 that is 6 units long that is 6 units long and it's image well it's 1 1 in 2/3 as long we see it over here going from 3 to negative 7 3 minus negative 7 is 10 it is 10 units long we got it right