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Dilations

Added Feb 22, 2019 •

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hello everybody let's talk about dilations yeah that's right about what happens when you go to the optometrist and they want to do an eye exam what time. Is it kind of dilations in geometry so I want to tell you a little bit about a friend of mine this is Kent and Kent as you can see got a really sweet family portrait taken and he wanted to put this family portrait in a frame that says you can see it doesn't really fit very good because the print that he got was a wallet-size and the frame was like a 5 by 7 frame so you know what he had to do right he had to take that wallet-size picture I had to go back to the photo studio and he had to get what we call an enlargement he had to take the picture and he had to get bigger version of it when you take something and in large it or reduce it when you make it big shrink it and it stays the same shape and proportions this is what we call in Geometry a dilation that dilation can mean your free image gets bigger or it could mean your preimage gets small can we say that it maintains the same proportions what we really mean is that the dilation takes a 2 and 1/2 inch by 3 and 1/2 in photo and turns it into a 5 by 7 photo and if you notice but that's proportional because the top doubled and the bottom doubled as well for a dilation to be a dilation it has to be proportional image in the perimeter the same shape which is different sized versions of that shape okay so here's what it looks like on a coordinate plane let's say you got triangle ABC and point a is at -1 and point B is at 1, 0 and point C is at 0, negative one and you can verify that just by looking at the graph and let's say you wanted to make triangle ABC twice as big what would you do if you wanted to double the size of triangle ABC well what you would do is just take all of the coordinates and double each of the numbers so a prime wouldn't be at -1, 2 it would be double that the negative one gets doubled and becomes a negative two and the two it's doubled and it becomes a 4 and we're going to do the same thing to make a b Prime and a c Prime and if you look at all those coordinates all the cornices got double and when we plot out those coordinates guess what happens triangle a prime B Prime C Prime is a dilated version of triangle ABC and it's twice as big so basically all of these x coordinates and Y coordinates became new X and Y Barnett's but when you do a dilation the x-coordinates and the y-coordinates both get doubled k this is what we call a scale factor the scale factor is what you multiply the preimage how to get the new image and in this case we multiply everything by 2 because we wanted the size to double scale factor all right so here's an example in the figure triangle X Prime y Prime at Z Prime is a dilation of triangle XYZ find the scale factor of the dilation and classified as an enlargement or reduction first can we classify it as an ally reduction before we do the scale factor yeah I don't know that I have to explain this but an enlargement is when something gets bigger and a reduction is when something gets smaller okay so how do we know if this is an enlargement or reduction well which triangle did we start with did we start with the bigger one or did we start with the smaller one we started with the smaller one how did I know that because that's triangle XYZ about the apostrophes and the one with the apostrophes is where we ended up and guess what just happened it got bigger so this is an enlargement not a reduction okay now we can find the scale factor the scale factor of this enlargement is what did you have to multiply everything by to make it grow did it get twice as big or two and a half times of signs or 3 or 10 times as big well here's what we're going to do find the scale factor we're going to figure out set point Y and point wide Prime are at the following coordinates why is it three, two endpoint why Prime is it 6, you can verify that just by looking at the coordinates on the graph and we're going to ask a question what number do you have to multiply pointwise coordinates by to turn them into the court set point why Prime will the three got doubled to turn into a six and the two got doubled to turn into a 4 so I think that means that the scale factor on this problem is but we doubled everything so the scale factor is to all right that wasn't too bad we were able to identify it as an enlargement with a scale factor of 2 all of the examples so far that we did have a scale factor of 2 but you can have a different scale. you can triple the size of something okay you can even quadruple the size of something or you can make something half the size if your scale factor is 1/2 what you start with is going to give you something smaller and what just happened really quickly on the screens on them for some reason was a little note telling you that when the scale factor is a action like that it's going to be a reduction not enlargement which kind of makes sense because the scale factor of 1/2 means it's half as big which means it's going to be smaller all right let's summarize dilation a dilation is when you enlarge or reduce something but it keeps the same proportions or in other words it keeps the same shape but a different size the scale factor of a dilation is what you multiply Dupree try to get the new image so that doubles the scale factor is too if it triples a scale factor is 3 if it's half as big the scale factor is half so if you wanted to write arrow notation for a dilation then what you do is whatever that number is in front of the exes that's your scale factor so if you doubled it you'll be 2x, 2y if you triple it would be 3x, 3 why if you made it half as maybe 1/2 x, 1/2 way and that is basically how dilations work all right thanks for paying attention thanks for listening and thanks for learning with me and I will see you tomorrow

Uploaded by Jeff Marbut.

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Dilations

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