Area of triangle formula derivation

if I give you three points like this and if I ask you can you find the area of the triangle you'll get if you connect these three points how do you think about this now my first instinct was to check if it's easy to find the base and the height because if it is then I'll just find the length of the base and the height and then half and to be a center height I know how to do that but over here the sides are not like horizontal or vertical so it's not very easy to find the base and the height okay so that method is not possible directly so then you can think okay maybe you can find the lengths of each of these sides that was the second thing that struck me maybe I can find this length this length and this length and then use Kieran's formula I don't know if you remember it because it's that root of s into s minus a into s minus B into s minus C formula you can use that and then you can find the area of this triangle that'll definitely work I encourage you to try it I just did the calculations looked so big that I was just gonna I realize that is probably not what this questions trying to get us to do which is to find the area of this triangle in somewhat a simple way so if both of those are not possible then what can you think of the thing about coordinate geometry is that it gives you all the lengths either horizontally or vertically and the problem here is that we don't have any of these lines being horizontal over because that's what makes this question not directly easy so then what can we do I want you to stop right now and think can you express the area of this triangle as some figures some triangles or rectangles or something such that all of them have only vertical or horizontal sides can you think of that now when I can't think of something I'm just gonna start drawing some vertical and horizontal lines and see where it leads me maybe I'll get lucky let us see I'm gonna draw a vertical line from here and then maybe a horizontal line from here and I can begin to see some pattern that I can form here so if I connect these two I see a triangle here then maybe I draw some more here another one there another one here and now finally I'm able to see there it is can you see it there is a rectangle over here which covers this entire region and there are these three triangles so let me shade these triangles triangles I have these three triangles whose area if I subtract from the original rectangle this big one over here I will get what I want which is the area of my of my triangle so I actually have my path ready this should work because my rectangle has only horizontal and vertical lines in it and my triangles have both all of them have a base and a height that is either vertical and do that that's vertical and horizontal so my job is done now my job is to find the lengths of these so that I can proceed and finding the lengths when in coordinates geometry for horizontal and vertical is usually direct so let's do it so let me first find this big side of the rectangle so that's gonna be this entire side and what is that equal to you can see that the x-coordinate of this line is x3 I'm saying this line because all points on this line will have the x-coordinate x3 and all points on this line will have the x coordinate x1 in other words this length is x3 minus x1 x3 minus x1 and the next thing I'll need is the y-coordinate this big line over here so what is that going to be equal to similar story y2 minus y1 y2 minus y1 all the points over here I have the y-coordinate y2 all the points here of the y-coordinate y1 you can do something similar it's a similar story for all of these other sides we need what other sides do we need I need the this so that I can find the area of this triangle so what is that going to be that's going to be equal to x3 minus x2 x3 minus x2 this one here I will need that is x2 minus x1 x2 minus x1 I will also need this one this y-coordinate that is y 2 minus y 3 y 2 minus y 3 and I will finally need the one which is why three - y1 y3 - y1 the diagram looks a little bit cluttered but we have everything we need so what is our next step from here our next step from here is to write the area of the rectangle down first what's the area of the rectangle its length into breadth or x3 minus x1 x y2 minus y1 y2 minus y1 this is the area of the rectangle now we need to subtract the areas of each of these triangles let me start with this one so let's say I do minus minus half into base into height right so half into base let me just take I can take which over I want is the base so bases let me take x2 - x1 x2 - x1 x height in this case that's y2 - y1 2 minus y1 and now I will subtract say this triangle minus half into for this triangle the base is x3 minus x1 x3 minus x1 and the height is y 3 minus y1 y3 minus y1 and finally the last triangles over here so it's 1 by 2 and 2 minus half into base base x3 minus x2 3 minus x2 into height Y 2 minus y 3 y 2 minus y3 now our let me just move this little bit of urea so now our job actually is to simplify this and the thing here is that even though this doesn't look too beautiful of the final result we get when we simplify a lot of terms cancel so the thing you get looks pretty beautiful actually I was surprised when I first saw it so pause this video right now and watch what happens when you expand all of these terms and look at what cancels and look at what the final result you get is I'm gonna do it now I actually don't need the diagram anymore I only have to simplify this I can look at the diagram later so one thing I'm gonna do first is I'm gonna take this 1/2 outside you know I don't want to keep dealing with Hobbes I really don't like that so I'm gonna take the 1/2 outside so I know my answer is going to have 1/2 outside which means that I can now start simplifying each of these I'm gonna do it in each of these in one one line and then let's look at what we get so I'm gonna put a two over here so that I have I've forgotten these halves and I added a two over here I know I'll have a half outside so what is this gonna be now the way I like to think about such things is I get very confused when I do signs and this too and all that so I'll first look at the terms then I'll add the signs so I'm going to have X 3 y 2 X 3 y 2 X 3 y 1 X 1 y 2 and x 1 y 1 I know these are going to be my four terms now I can figure out my signs so or then we're going to have a two so I know that all of them are going to have a two and I know that the middle two terms are going to be negative because it's a minus B right and these two are going to be positive so 2 X 3 y 2 minus - and finally a plus so then I'm gonna do this similarly I'm gonna say X 2 y 2 is going to be the first term X 2 y 1 is the second term X 1 Y 2 is the third term X 1 Y 1 is the last term now I'll figure out the signs there's a negative outside oh I hate it when that happens that that's the that's where I make most mistakes so what I'm gonna do is I'm going to add negative here this would have been negative so this is positive here positive here and negative here I'm going to keep the same pattern going so that I know it's going to be minus plus plus minus so I don't have to think about it at all think about it at all that's what I wanted to say so X 3 y 3 X 1 let let me X 3 y 1 X 3 y 1 x 1 y 3 and x 1 y 1 there it is and again it's going to be minus plus plus minus similar story here X 3 y 2 X 2 sorry X 3 y 3 X 3 y 3 X 2 y 2 and X 2 y 3 to move this a little bit so we can see that yeah X 2 y 3 and again it's minus plus plus minus now our job is to start canceling out because we know something's gonna cancel out this this expression is really large so we've seen that the final expression is actually quite smaller than this so let me first see X 3 y 2 can I find X 3 y 2z or i can find one here so this 2 goes away and this entire expression goes away X 3 y 1 I can see one over here so this goes away and this 2 goes away X 1 y 2 I can see u plus over here this is minus so this one goes away plus 2 X 1 y 1 I can find one x one y one over here and I can actually find one more x one y one over here so oh the center I think goes away I like it when things start getting canceled like this let me see if there's anything else x2 y2 x2 y2 ok I didn't see one here and a positive one over here x2 y1 I can't see anything x3 y3 I can see one over here this goes so finally it looks like nothing else will go so how many terms do I have one two three four five and six so these six terms are going to be there in my final expression but if you notice it's actually much simpler than the other then then what we started with so I'm gonna start writing this expression now so I'm gonna take the X is common because I can see that there is an X 3 y 2 here X 3 y 1 and so on so X 1 Y 3 is there X 1 Y 2 is there so I can take X 1 common and write it X 1 into so it's not like there's a big reason I'm doing it it's just that I find that the final expression looks more beautiful than we do it like this so X 1 into what Y 3 minus y 2 y 3 minus y 2 now what is the next one I'll take X 2 common out over here and then I'll have if I take X 2 common I'll get X 2 y 1 minus I 0 minus y 3 so Y 1 minus y 3 y 1 minus y 3 plus I'll take x3 common now and I will get Y 2 minus y 2 minus oh dear why do minus y 1 y 2 minus y 1 and there we have it now I know about you when I see this I'm a mist that what they're getting here is so much simpler compared to like what we get when we do the hurons formula or any of those things it's basically just the first coordinate one of the x coordinates the difference of the other two plus the second one 1 minus 3 the difference of the other 2 the third one 2 minus 1 so this is your expression and notice that you may sometimes get a negative value for this entire expression but the area of a triangle can't be here negative so what we do is we take the absolute value you'll calculate this entire thing and take the absolute value so why you get the negative is that the order in which we you choose the points may sometimes be different so you'll get a negative value so take the modulus of the absolute value of that to find the area and in the case where the 3 points that you've chosen don't even form a triangle they form a straight line the 3 points are collinear we say there you will get this value to be 0 so getting 0 for this means that the 3 points are collinear