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CCSS.Math:

we're told that polygon Q is a scaled copy of polygon P using a scale factor of 1/2 polygons Q's area is what fraction of polygons P's area pause this video and see if you can figure that out alright my brain wants to make this a little bit tangible and once we get some practice you might be able to do it without drawing pictures but they're saying some arbitrary polygon Q and P so let's just make a simple one since we're talking about area I like to deal with rectangles since it's easy to think about areas of rectangles and so let's see polygon Q is a scaled copy of polygon P so let's start with polygon P and I will do this in red so polygon P let's just say I'm just gonna create an arbitrary polygon so let's say that this is and I'm gonna scale it by 1/2 so I'm gonna make it sides have nice even numbers so let's say this side right over here is 4 and this side right over here is equal to 8 this is polygon P right over here it's a quadrilateral it's in fact a rectangle and it's area area is just going to be 4 times 8 which is 32 now let's create polygon Q and remember polygons Q is a scaled copy of P using a scale factor of 1/2 so we're gonna scale it by 1/2 so instead of this side being 4 it's going to be 2 and instead of this side over here that's being 8 the corresponding side in the scaled version is going to be 4 so there you go we've scaled it by 1/2 and now what is our area going to be well our area and this is polygon Q and so our area is going to be 2 times 4 which is equal to 8 so notice the polygon Q's area is 1/4 of polygon P's area and that makes sense because when you scale the dimensions of the polygon by 1/2 the area is going to change by the square of that 1/2 squared is 1/4 and so the area has been changed by a factor of 1/4 or another way to ant this question polygon Q's area is what fraction of polygons P's area well it's going to be one-fourth of polygons P's area and the big takeaway here is if you scale something if you scale the sides of a figure by one half each then the area is going to be the square of that and so one-half squared is 1 over 4 it was scaled by 1/3 then the area would be scaled or the area would be 1/9 if it was if you scaled by a factor of 2 then our area would have grown by a factor of 4 let's do another example here we are told rectangle n has an area of 5 square units let me do this in a different color so rectangle and has an area of 5 square units James drew a scaled version of rectangle N and labeled it rectangle P so they have that right over here this is a scaled version of rectangle n what scale factor did James used to go from rectangle and to rectangle P so let's think about it we give us rectangle P right over here and let's think about its dimensions this height is 1 2 3 4 5 it's 5 high and it is 1 2 3 4 5 6 7 8 9 wide and so it's area it's area is equal to 45 now rectangle rectangle n had an area of 5 square units so our area let me write this down so n area to P area an area to P area we are multiplying by a factor of 9 for going from an area of 5 square units to 45 square units notice an area is 5 I'm gonna did in that color ends area is 5 square units P's area we just figured out is 45 square units and so we have it growing by a factor of 9 what would be the scale factor if our area grew by a factor of nine well we just talked about the idea that area will grow the factor with which area grows is the square of the scale factor so one way to think about it is scale factor scale factor squared is going to be equal to nine or another way to think about it our scale factor scale factor is going to be equal to three to go from n to P now let's verify that that we answered their question but I just wanted us to feel good about it let's draw a rectangle that is scaled down from P by a factor of three so or a rectangle if we were to scale it up by a factor of three we get rectangle P so it's bottom wouldn't have length three instead of nine so it'd be like this so that would be three and its height instead of being five it would be 5/3 5/3 is one and two thirds so it go about that high it would look something like that it would be five thirds and so our rectangle n would look like this and what is its area well five thirds times three is indeed five square units so notice when we have the area growing by a factor of nine the scale factor of the size to go from five thirds to five you multiply by three to go from three to nine you multiply by three