Side lengths after dilation

the graph below contains triangle ABC and the point P draw the image of triangle ABC under dilation whose center is P and a scale factor of 2 so essentially we want to scale this so that every point is going to be twice as far away from P so for example B right over here it's the same y-coordinate as P but its x-coordinate is 3 times or is 3 more so we want to be twice as far so if this maps to point B we just want to go twice as far so we're at 3 away we want to go 6 away so point P is x-coordinate is at 3 now we're at 9 likewise Point C is 3 below P well we want to go twice as far so we'll go 3 more and point a is 4 above P well we want to go 4 more we're going to go twice as far 1 2 3 4 and we get right over there then they ask us what are the lengths of side a B and it's image a B a B right over here let's see we might have to apply the distance formula let's see it's the base right over here the change in the change in X between the two is 3 and the change in Y is 4 so this is actually a 3 4 5 right triangle 3 squared plus 4 squared is equal to 5 squared so a B is 5 units long essentially just using the Pythagorean theorem to figure that out and it's image well its images should be twice as long and let's see whether that actually is the case so this is a base right over here that's of length 6 this has a height or this change in Y I could say if we could I'm really just trying to figure out this length which is the hypotenuse of this right triangle that I don't have my drawing tool so I apologize but this height right here is 8 so 8 squared is 64 plus 6 squared is 36 that's 100 which is 10 squared just 10 squared and so notice our scale factor of 2 the corresponding side got twice as long each of these points got twice as far away from our center of dilation