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# Area of triangle proof

## Video transcript

we now know how to find the area of rectangles what I want to do in this video is think about how we can find the areas of triangles so we're starting here with a right triangle has a 90 degree angle right over here right triangle ABC and let's think about how we can find its area well maybe we can construct a rectangle out of triangle ABC and if we can construct a rectangle out of it and then maybe we can somehow find our area of part of that rectangle and the best way to construct a rectangle is to really duplicate ABC and then flip it over and put it right on top of this and just to verify that that definitely will be a rectangle so we know that this is 90 degrees right over here let's say that this is X degrees right over there we have X plus 90 Plus this thing have to be equal to 180 so this thing and this thing have to add up to 90 so let's just call this 90 90 minus X now let's flip this thing and rotate it around so that it will look like this so then you would have another triangle that would look just like this we're now this right angle on the flipped version is that right angle right over there this angle right over here this X is now this angle right over there and this 90 minus X is now this angle right over there and you can see X plus 90 minus X that'll give you a right angle and then you have X plus 90 minus X that gives you a right angle and you have four sides and four right angles you are definitely dealing with a rectangle and this rectangle has two of our original triangles in it so we can write we can write that the area of triangle ABC the area of triangle ABC that's what the brackets mean area of triangle ABC is going to be equal to 1/2 times the area of our entire rectangle and our entire rectangle let me add another point here let me call this D it's going to be 1/2 the area of rectangle a B C D and we know how to find the area of rectangle ABCD it's going to be equal to it's going to be equal to the base of the rectangle so this is going to be equal to one half times and so this part right over here let me do this in a different color the area of ABCD is equal to the base or the width of the rectangle so that's just the length of BC I'm just putting this in parentheses it doesn't mean it just BC is just the length of this segment right over here and I'm just putting in parentheses so that we don't get the letters jumbled up so it's going to be this width or this base right here times the height of the rectangle so times ad times sorry times a B so it's this base times this height gives us the area of the entire rectangle and the area of our right triangle is half of that so there we have it it's 1/2 times times this base times this height is the area of a right triangle so in general if you ever have a right triangle if you ever have a right triangle and this is a right angle right over there which isn't it you need one right angle in order to be a right triangle and this base has length B and this side over here has length H you know that the area the area is going to be equal to 1/2 times the base times the base of the the of the triangle that's the base of the triangle BC times the height of the triangle so you could view it this way if you look at the actual letter points and if you just view these measures is the base times height it's just 1/2 base times height and we only know that this works right now for a right triangle now let's think about it for other types of triangles that aren't necessarily right triangles so here I have a kind of an arbitrary triangle ABC and to approach figuring out its area what I want to do is just split this up into two right triangles so what I'm going to do is drop a perpendicular from B so I'm going to just slowly this was this was an actual structure you just literally drop something straight down from here and that line is going to be perpendicular to this base right over here to AC and let me call that point let me call that point D and what's useful here is now we've constructed we've turned that one triangle into two right triangles so we can say that the area let me write it this way we can say the area of triangle a B see that's what we want to figure out it's equal to the area of this character right here so it's equal to the area of triangle abd a BD plus the area of triangle plus the area of this magenta triangle so plus the area of B C D of B C D and this is useful because we know how to find the area of right triangles now and obviously this is 90 degrees and this is also going to be 90 degrees the area of abd is one-half base times height so it's going to be 1/2 times the base which is the length ad so 1/2 times ad times the height which is the length of BD right over here assuming that we can figure that out so this times that length so BD so that's the area of the blue triangle and now let's find the area the magenta triangle well once again it's a right triangle it's going to be one half or do that in magenta let me do that in magenta so it's going to be one half times the length of this base right over here which is DC the length of segment DC times the length of BD again times the length of BD again now you can factor out a 1/2 we can factor out a 1/2 BD from both of these terms 1/2 BD so you get 1/2 BD 1/2 BD times ad we're left with ad and that's not the same shade of blue we're left with AD plus DC plus DC plus DC close the parentheses and what is AD plus DC ad is the length that length and then DC is this length so if you were add up those two lengths you get the length of AC so this whole thing this whole thing is the length AC so we're left with the area of ABC is equal to I'll do this in a new color it's equal to 1/2 and I'll just switch the orders right over here 1/2 times h1 right in that same color times AC AC times BD times BD now what is this again well this is now 1/2 times the base which is this AC times the height which is BD so that's pretty cool it worked from right triangles and it actually if we know the height of a triangle notice this isn't one of the sides now for a right triangle it was one of the sides now for this arbitrary triangle it isn't but if we know it the area of this triangle is still 1/2 times the base times the height now what about a triangle like this how can we figure out its area well let's try to do it the same thing let's see if we can somehow either construct this out of right triangles or maybe make add a right triangle of this to make it into another right triangle and the easiest way to do that is we can kind of just drop a rock from right over there and then this and then where it kind of hits the ground would form a right angle and let's call this point D and what we care about what we want to find in this we want to find the area if we want to find the area of triangle ABC so that's what we care about triangle ABC is what we want to find the area of but the area of triangle ABC is going to be the area of this larger right triangle that we've set up it's going to be this larger triangle so it's going to be the area of ad be a DB minus the area of the smaller triangle minus the area of this smaller one right over here so minus the area of ADC of a D C so I just this blue on a DB this is just the whole thing this is the whole thing just so we're clear about what we're talking about now what is the area of a DB well we know how to find the area of right triangles ADB area of a DB is going to be is going to be one half times our base which is DB DB the length of segment DB times our height which is the length of ad times ad and then from that we want to subtract the area of the smaller triangle so that's going to be one half times our base which is DC that's the length of our base DC times the length of our height which is ad ad so here we can factor out a 1/2 and we can factor out an ad a 1/2 and an ad so let's do that so we factor out a 1/2 times ad and what's left inside is a DB DB minus a DC - DC now what is DB - DC we put the parentheses in - why - again so we have DB - DC well if you take the length DB is the length of this whole thing you subtract from that the length DC you're going to be leg left with CB so this character right over here this is C this is CB and so the area of ABC is going to be equal to 1/2 times CB I'm just switching the order of multiplication instead of that yellow color 1/2 times CB times ad times ad now what is this once again it's 1/2 times our base times our base times our height and once again the height in this case because it's not a right triangle it's it's it's not one of the sides someone would have to give you that information you would have to figure out what this actual height is but once what's neat is in any form of triangle the area is 1/2 base times height the height is kind of if you have the sides of a right triangle it's it's easy to figure out for these others if it's not given you might have to do it in some tricky way figure out figure out the height somehow but anyway hopefully you found that useful