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CCSS.Math:

so quadrilateral ABCD they're telling us it is a rhombus and what we need to do we need to prove that the area of this rhombus is equal to 1/2 times AC times BD so we're essentially proving that the area of a rhombus is 1/2 times the product of the lengths of its diagonals so let's see what we can do over here so there's a bunch of things we know about rhombi and all rhombi our parallelogram so we there's tons of things that we know about parallelograms first of all if it's a rhombus we know that all of the sides are congruent so that side length is equal to that side length is equal to that side length is equal to that side length because it's a parallelogram we know that diagonals bisect each other so we know that this length let me call this point over here B let's call this e we know that b e is going to be equal to e d so that's b e we know that's going to be equal to e d and we know that a e we know that AE is equal to EC we also know because this is a rhombus and we proved this in the last video that the diagonals not only do they bisect each other but they are also perpendicular so we know that this is a right angle this is a right angle that is a right angle and then this is a right angle so the easiest way to think about it is if we can show that this triangle ADC is congruent to triangle ABC and if we can figure out the area of one of them we can just double it so the first part is pretty straight forward so we can see the triangle ADC we know that triangle a DC is going to be congruent to triangle ABC to triangle a b c and we know that by side-side-side congruence this side is congruent to that side this side is congruent to that side and they both share a C right over here so this is by side-side-side and so we can say that the area so because of that because of that we know that the area of ABCD the area of ABCD is just going to be equal to 2 times 2 times the area of we could pick either one of these we could say 2 times the area of ABC 2 times the area of ABC because the area of ABCD actually let me write it this way the area of ABCD is equal to the area of a DC plus the area of ABC but since they're congruent these two are going to be the same thing so it's just going to be two times the area of a b c now what is the area of a b c well area of a triangle is just 1/2 base times height so area of a b c is just equal to 1/2 times the base of that triangle times its height which is equal to 1/2 what is the length of the base well the length of the base is AC so it's 1/2 I'll color code it the base is AC and then what is the height what is the height here well we know that this this this diagonal right over here there's a perpendicular bisector so the height the height is just the distance from be e so it's AC times ve b e that is the height this is an altitude this is it intersects this base at a 90-degree angle or we could say B E is the same thing as 1/2 times BD so this is let me write this is equal to so it's equal to 1/2 times AC AC that's our base and then our height is B E which we're saying is the same thing as 1/2 times BD which is 1/2 times BD so that's it that's the area of just ABC that's just the area of this broader triangle right up there or that larger triangle right up there that half of the rhombus but we just said that the area of the whole thing is 2 times that so the area so if we go back if we use both this information and this information right over here we have the area of ABCD is going to be equal to 2 times the area of ABC well the area of ABC is this thing right over here it is so 2 times the area of ABC or baby see is that right over there so 1/2 times 1/2 is 1/4 times a C times B D and then you see where this is going 2 times 1/4 is 1/2 times AC times BD fairly straightforward which is a neat result and what actually I haven't done this in a video yet I'll do in the next video there are other ways of finding the areas of parallelograms generally it's essentially base times height but for a rhombus we could do that because it is a parallelogram but we also have this other neat little result that we proved in this video that if we know the lengths of the diagonals the area of the rhombus is 1/2 times the lengths the products of the lengths of the diagonals which is kind of a neat result