following is a video for 12.8 notes and this is uh title or the title of the section is called congruent and similar solids so we're going to look at the attributes of those um so the first thing I want to talk about real quick is similar solids okay so similar solids okay and we know if we have a similar solid if the corresponding measurements corresponding [Music] measurements measurements are proportional proportional okay and so what I would say here is that my height here of this cone if I took the height sub one and divided that by height sub2 this would be equivalent to me saying R sub one ided R sub 2 okay so if I take the radius of this and divided by the radius of that it would be equivalent to this height divided by this height okay and this is just kind of very similar uh to what we uh did when we had similar triangles and similar polygons okay the sides are proportional okay and we can set up this relationship now we have another one where we have congruent solids okay congruent solids um and what we would do here is uh we would do the same thing and that just means that your corresponding measurements so your corresponding measures measurements are congruent all right so if I took H sub one um this height and divided it by this height um then that would equal the um this slant height L sub one divided by this slant height L sub two uh but if I divided those they would not only equal each other but they would also equal one because they are the same okay so if I take something and divide it by another thing and if those are the same measurements then I'll just get one okay so uh the one thing I do want to talk about um on here and you'll need to write this kind of on the right side um I want to talk about similar solids similar similar solids okay now we have the we have a couple things that we can talk about we can talk about the ratio of side lengths ratio of sides and that would be just a simple ratio A over B okay um if we talk about the ratio of surface area then that's going to give us a^2 over b^ 2 which is the quantity A over B and I Square it now remember that the ratio of side lengths that's just a single measurement okay we raise that to the first Power and to the first Power we're talking about area that's a square unit okay so we can say if we have similar solids we can find the surface area and we take that ratio and we Square it okay but if we take the ratio of volume okay which is a cubic measurement that would be A over B A cubed over B Cubed which would be equivalent to me saying Just A over B that quantity cubed okay and I'll show you how this works here in a second um but we want to use this notion that if we're talking about um the ratio of surface areas and we take the ratio of sides that we can find about the side lengths and then we can just Square it okay so let's put this into practice a little bit uh in number one it tells us determine whether each pair of solids is similar congruent or neither if the solids are similar then State the scale factor so what I want to do is I want to look at the corresponding side so I'm going to look at two cm here and 6 cm here I'm going to divide them so I'm going to go 2 over 6 okay and I want to determine is that equal to this height divided by that height so I'm going to take four and divide it by 12 all right um so I take four and divide it by 12 now on this one I'm going to do the last one is 3 over9 equivalent to all these okay and so what we can find out is that yes in fact these are equivalent you could cross multiply and determine if they are proportional but I know that all of these simplify to 1/3 okay so this is what we would call a similar solid okay not aide solid all right so my scale factor all right would be one over three we could also write that as 1 two 3 okay one colum 3 all right let's take a look at number two all right so let's see if these are similar um now my radius here is equal to five not 10 radius is equal to 5 m uh here my diameter is equal to 10 m now if I get my diameter I know my radius is just half of that so if I divide that by two I get that my radius is 5 m okay so if I take the radius here which is 5 m and divided by the radius over here which is 5 m then I get that we have a value of one 5 ID 5 gives me one and so I know that these are congruent okay congruent solvid all right they are the same same size okay all right take a look at number three uh if two similar pyramids have Heights of 4 in and 7 in what is the ratio of the volume of the small pyramid to the volume of the large pyramid so I know I have two similar pyramids okay Heights of four and seven okay so I have a scale factor okay which is the same as the ratio of their side lengths all right and that would be four ID s okay so then if I wanted to find the ratio of their volume then I would simply just take 4 over 7 and I would Cube the whole thing now when I cube a fraction that means I'm going to take the numerator cubed and divide it by the denominator cubed all right and if I take four cubed and that's four * 4 * 4 uh then I'm going to get 64 and then if I take 7 cub in my calculator then I get 343 and this is my answer okay this is the ratio of our volume I take my ratio of side lengths and I Cube it and that's what I get all right let's take a look at number four now number four um is a word problem it says two stockpots are similar cylinders okay so I'm just going to draw a quick cylinder here um it's usually helpful to draw a picture um the smaller stock pot has a height of 10 in okay and a radius of 2.5 in so this is my radius R is equal to 2.5 in okay and my height is equal to 10 in um the larger stock pot has a height of 16 so I'm going to come down here I'm going to draw a bigger one pardon my drawing doesn't look very good but you get the idea my height is equal to 16 in and I don't know what my radius is okay but I'm just going to leave it here I don't know what my radius is okay and it says now what is the volume of this so what's my volume okay now what I can do this is kind of a tricky problem all right um I could take let me find my scale factor okay I know that these are similar tells me that they're similar so I have a scale factor and I'm going to take this height divided by this height I'm going to take 10 over 16 all right which is equivalent to me if I take both of those and divide it by two I get 58 okay so that's my scale factor okay this is also the ratio of my sides okay now I know this that if I take the ratio of sides 5/8 and if I Cube this thing then that's going to give me the ratio of volumes okay and that's going to be 5 cubed divided by 8 cubed and if I put those in the calculator then I get 125 divided by 512 okay that's just 5 cubed is 125 8 cubed is 512 this is the ratio of volumes okay so I want use this to find the volume of this I don't know what the volume of this is but I do need to have the volume of my smaller one now my volume of the smaller one is going to be um < * 2.5 2ar time my height which is 10 now what I'm going to do I'm going to leave it like this I'm going to take 2.5 SAR is 6.25 and then multiply that by 10 this gives me 62.5 * pi all right and that's my volume of my smaller one so if I set up a new proportion here if I say 125 / 512 then that's equal to the volume of my smaller one which is 62.5 * pi being divided by the volume of my big of my big cylinder so I could cross multiply and divide and solve for the volume of the big all right and if I cross multiply and divide that I actually end up getting a volume of 84.2 and that's Ines cubed okay so if I know that something is similar and I uh know the volume of one I can actually calculate the volume of the other provided I'm given a side length so I have to start off with my scale factor okay Cube everything get the ratio of volume and then set up a proportion with that ratio of volumes with the other volume and I'll I'm able to solve for the other volume okay I hope this helps please get started on your 12.8 homework and I hope you enjoyed this video as much as fun as much fun as I had making it