Area of parallelograms and triangles
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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 the area of a rhombus. You could take half the product of its diagonals. And a rhombus is a parallelogram. But you can't just generally take half the product of the diagonals of any parallelogram. It has to be a rhombus. And 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 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. These two triangles are congruent. But we can do it in a pretty straightforward way. We can look. Obviously, AD is equal to BC. We have DC is equal to AB. And then both of these triangles share this third side right over here. They both share AC. So we can say triangle-- let me write this in yellow-- we could say triangle ADC is congruent to triangle-- let me get this right. So it's going to be congruent to triangle-- so I said A, D, C. So I went along this double magenta slash first, then the pink, and then I went D, and then I went the last one. So I'm going to say CBA. Because I went the double magenta, then pink, then the last one. So CBA, triangle CBA. And this is by side, side, side congruency. 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 of ABCD, the whole parallelogram, it's going to be equal to the area of triangle-- let me just write it here-- it's equal to the area of ADC plus the area of CBA. But the area of CBA is just the same thing as the area of ADC 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. 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 is perpendicular. We could call that the height right over there. So if you want the total area of parallelogram ABCD, it is equal to two times one half times base times height. Well, two times one half is just one. And so you're just left with base times height. So we can call this b. So it's just b times this height over here, base times height. So that's a neat result. And you might have guessed that this would be the case. But if you want to find the area of any parallelogram, and if you can figure out the height, it is literally you just take one of the bases, because both bases are going to be the-- opposite sides are equal, so it could have been either that side or that side, times the height. So that's one way you could 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 like that, and stand it on this side, so this would be point-- let me draw the points-- this would be point A. Let me 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 1-- sorry, not 1/2. That would be for a triangle. The area of this would be base times height. So you could say it's h times DC. So you could say this is going to be equal to h times the length DC. That's one way to do it. That's this base times this height. Or you could say it's equal to AD times, I'll call this altitude right here, I'll call this height 2. Times h2. Maybe I'll call this h1. h1, h2. So you could take this base times this height, or you could take this base times this height right over here. This is h2. Either way. So if someone were to give you a parallelogram, just to make things clear, obviously, you'd have to be have some way to be able to figure out the height. So if someone were to give you a parallelogram like this, they would 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 is 6, then the area of this parallelogram would literally 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 30.