Dilations: scale factor

we are told that Pentagon a prime B prime C prime D prime e prime which is in red right over here is the image of pentagon ABCDE under a dilation so that's ABCDE what is the scale factor of the dilation so they don't even tell us the center of the dilation but in order to figure out the scale factor you just have to realize when you do a dilation the distance between corresponding points will change according to the scale factor so for example we could look at the distance between point a and point B right over here what is our change in Y our change in or even what is our distance our change in Y is our distance because we don't have a change in X well this is one two three four five six so this length right over here is equal to six now what about the corresponding side from a prime to be Brian well this length right over here is equal to two and so you could see we went from having a length of six to a length of two so you would have to multiply by 1/3 so our scale factor right over here is 1/3 now you might be saying okay that was pretty straight forward because we had a very clear you could just see the distance between a and B how would you do it if you didn't have a vertical or a horizontal line well one way to think about it is the changes in Y and the changes in X would scale accordingly so if you looked at the distance between point a and Point E our change in Y is negative 3 right over here and our change in X is positive 3 right over here and you can see over here between a prime and E Prime our change in Y is negative 1 which is 1/3 of negative 3 and our change in X is 1 which is 1/3 of 3 so once again you see our scale factor being 1/3 let's do another example so we are told that Pentagon a prime B prime C prime D prime V prime is the image and they don't they haven't drawn that here is the image of pentagon ABCDE under a dilation with a scale factor of 5 half so they're giving us our scale factor what is the length of segment a prime e prime so as I was mentioning while I read it they didn't they didn't actually draw this one out so how do we figure out the length of a segment well I encourage you to pause the video and try to think about it well they give us a scale factor and so what it tells us if the scale factor is five halves that means that the corresponding lengths will change by a factor of five halves so to figure out the length of segment a prime e prime this is going to be you could think of it as the image of segment AE and so you can see that the length of a e is equal to two and so the length of a prime e prime is going to be equal to AE which is two times the scale factor times five halves this is our scale factor right over here and of course what's two times five halves well it is going to be equal to five five of these units right over here so in this case we didn't even have to draw a prime V prime C prime D prime u prime in fact they haven't even given us enough information I could draw the scale of that but I actually don't know where to put it because we didn't even give us our center of dilation but we know that corresponding sides or the lengths between corresponding points are going to be scaled by the scale factor now with that in mind let's do another example so we are told that triangle a prime B prime C prime which they depicted right over here is the image of triangle ABC which they did not depict under a dilation with a scale factor of two what is the length of segment a B once again they haven't drawn a B here how do we figure it out well it's gonna be a similar way as the last example but here they've given us the image and they didn't give us the original so how do we do it well the key and pause the video again and try to do it on your own well the key realization here is that if you take the length of segment a B and you were to multiply by the scale factor so you multiply it by two then you're going to get the length of segment a I'm b-prime the image is length is equal to the scale factor times the corresponding length on our original our original triangle so what is the length of a prime B prime well this is straightforward to figure out it is one two three four five six seven eight so this right over here is eight so we have two times the length of segment a B is equal to eight and then you get the sag length of segment a B just divide both sides by two is equal to four and we're done