Area of parallelogram proof

we know that quadrilateral ABCD over here is a parallelogram and what I want to discuss in this video is a general way of finding the area of a parallelogram in the last video we talked about a particular way of finding a area of a rhombus you can take half the product of its diagonals and a rhombus is a parallelogram but you can't just generally take the product of the half the product of the diagonals of any parallelogram it has to be a rhombus so now we're just going to talk about parallelograms so what do we know about parallelograms well we know the opposite sides are parallel so that that side is parallel to that side and this side is parallel to this side and we also know that opposite sides are congruent so this length is equal to this length and this length is equal to this length over here now if we draw a diagonal I'll draw a diagonal AC we can split our parallelogram into two triangles and we've proven this multiple times that these two triangles are congruent but we can do it pretty in a pretty straightforward way we can look obviously ad is equal to BC we have DC is equal to a B and then both of these triangles share this third side right over here they both share they both share AC so we can say triangle let me write this in yellow we could say triangle ADC a DC is congruent to triangle let me get this right so it's going to be congruent to triangle that's it a DC so I went along this double magenta slash first then the pink and then and then and then I went D and then I went the last one so I'm going to say C be a because I want the double magenta then pink then the last one so C be a triangle C be a and this is by side-side-side congruence all three sides they have three corresponding sides that are congruent to each other so the triangles are congruent to each other and what that tells us is that the areas of these two triangles are going to be the same so if I want to find the area the area of ABCD the whole parallelogram it's going to be the equal to the area of triangle let me just write it here it's equal to the area of a DC plus the area of C ba but the area of CBA is just the same thing as the area of a DC because they are congruent by side-side-side so this is just going to be two times the area of triangle ADC which is convenient for us because we know how to find the areas of triangles the area of triangles is literally just 1/2 times base times height so it's 1/2 times base times height of this triangle and we are given the base of ADC it is this length right over here it is DC you could view it as the base of the entire parallelogram and if we wanted to figure out the height we could draw an altitude down like this so this this is perpendicular we could call that the height right over there so if you want the total area if you want the total area of parallelogram ABCD it is equal to 2 times 1/2 times base times height well 2 times 1/2 is just 1 and so you're just left with base times height so we could call this B so it's just be B times this height over here base times height so that's the neat result in this comment you might have guessed that this would be the case but if you find want to find the area of any parallelogram and if you can figure out the height it is literally just take one of the bases because both bases are going to be the you know opposite sides are equal so it could have been either that side or that side times the height so that's one way you can have found the area or you could have multiplied the other way to think about it is you could have multiplied so if I were to turn this parallelogram over it would look something like this it would look something like this so if I were to rotate it if I were to rotate it like that and stand it on this side so this would be point let me draw the points this it would be point a I need to make sure I'm doing this right yeah this would be point a this would be point D this would be point C and then this would be point B you could also do it this way you could say it's one it's sorry not 1/2 that before triangle it would be the area of this would be base times side so you could say it's a times DC so you could say this is going to be equal to H times the length D C times D C that's one way to do it that's this base times this height or you could say or you could say it's equal to ad it's equal to ad times I'll call this I'll call this altitude right here I'll call this height 2 times H 2 maybe I'll call this H 1 H 1 H 2 so you could take this base times this height or you could take this base times x times this height right over here this is H 2 either way so if someone were to give you a parallelogram just to make things clear and obviously you would have to be able some way to be able to figure out the height so someone will give you a parallelogram like this they were to tell you this is a parallelogram if they were to tell you that this length right over here is 5 and if they were to tell you that this distance this distance is 6 then the area of this parallelogram would really be 5 times 6 I drew the altitude outside of the parallelogram I could have drawn it right over here as well that would also be 6 so the area of this parallelogram would be would be 30