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# Proof of Heron's formula (1 of 2)

## Video transcript

let's say I've got a triangle there is my triangle right there and I only know the lengths of the sides of the triangle the side has length a this side has length B and that side has length C and I'm asked to find the area of that triangle and so far all I'm equipped with is the idea that the area the area of a triangle is equal to 1/2 times the base of the triangle times the height of the triangle so the way I've drawn this triangle the base of this triangle would be side C but the height we don't know the height would be that H right there and we don't even know what that H is so this is would be the H so the question is how do we figure out the area of this triangle and I if you watched the last video you know that you use Heron's formula but the idea here is to try to prove Heron's formula so let's just try to figure out H from basic from just using the Pythagorean theorem and from there once we know H we can apply this formula and figure out the area of this triangle so we already labeled this as H let me define another variable here let me define let me define this is a trick you'll see pretty often in geometry let me define this is X and if this is X in magenta then in this bluish purplish color that would be C minus X right this whole length is C the whole base is C so if this part is X then this part is C minus X and what I could do now because since these are both right-angles and I know that because this is the height I can set up two Pythagorean theorem equations first I could you do this left hand side and I can write that x squared plus h squared plus h squared is equal to a squared that's what I get from this left hand triangle and then from this right hand triangle I get C minus x squared C minus x squared plus h squared plus h squared is equal to b so I'm assuming I know a B and C so I have two equations with two unknowns the unknowns are X and H and remember H is what we're trying to figure out because we already know C if we know H we can apply the area formula so how can we do that well let's substitute for H to figure out X now when I say that I mean let's solve for H squared here if we solve for H squared here we just subtract x squared from both sides we can write that x squared sorry we could write that H squared is equal to a squared minus x squared and then we could take this information and substitute it over here for H squared so this bottom equation becomes C minus x squared plus h squared H squared we know from this left-hand side equation H squared is going to be equal to so plus I'll do it in that color a squared minus x squared is equal to is equal to B squared I just substituted the value of that in here the value of that in there now let's expand this expression out C minus x squared that is C squared minus 2c X plus x squared and then we have the minus a sorry we have the plus a squared plus a squared minus x squared equals B squared is equal to V squared we have an x squared and a minus x squared there so those cancel out those cancel out and let's add the two C X to both sides of this equation so now our equation would become C squared plus a squared plus a squared I'm adding to C X to both sides so you add to C X to this you get 0 is equal to B squared plus 2 C X all I did here is I cancelled out the x squared and then I added to C X to both sides of this equation my goal here is to solve for X and once I solve for X then I can solve for H and apply that formula now to solve for X let's subtract b squared from both sides so we'll get we'll get C squared plus a squared minus B squared is equal to 2 C X and then if we divide both sides by 2 C we get C squared plus a squared minus B squared over 2 C is equal to X we've just solved for X here now our goal is to solve for the height so that we can apply 1/2 times base times height so to do that we go back to this equation right here and solve for our height so let me scroll down a little bit we know we know that our height squared that our height squared is equal to a squared a squared minus x squared well instead of X just writing x squared let's substitute here so it's minus x squared X is this thing right here so C squared plus a squared minus B squared over 2 C squared this is the same thing as x squared we just solved for that so H is going to be equal to H is going to be equal to the square root of all this business in there I'll switch the colors of a squared minus C squared plus a squared minus B squared all of that squared which let me make it a little bit neater than that because I don't want to the square root the square root make sure I have enough space of a squared minus all of this stuff squared we have C squared plus a squared minus B squared all of that over 2 C that is the height of our triangle the triangle that we started off with up here let me copy and paste that just so that we can remember what we're dealing with let me copy it then let me paste it down here so I'll paste it down here so we know what the height is is this big convoluted formula the height is in terms of a B and C is this right here so if we wanted to figure out the area the area of our triangle I'll do it in a let me do it in pink the area of our triangle is going to be one half times our base our base is this entire length C times C times our height times our height which is this expression right here let me just copy and paste it instead of so let me so copy and paste so times the height so this now is our expression for the area now you're immediately saying gee that doesn't look a lot like Heron's formula and you're right it does not look a lot like Heron's formula but what I'm going to show you in the next video is that this essentially is Heron's formula this is a harder to remember version of Heron's formula I'm going to apply a lot of algebra to essentially simplify this to her own formula but this will work if you could memorize this I think herons is a lot easier to memorize but if you can memorize this and you just know a B and C you apply this formula right here and you will get the area of a triangle now oh well actually let's just apply this just to show that this does at least gives the same number as herons so in the last video we had a triangle that had sides 9 11 and 16 and its area and its area using herons was equal to 18 times the square root square root of 7 let's see what we get when we apply this formula here so we get the area is equal to 1/2 times 16 times the square root the square root of a squared that is 81 minus 81 - let's see C squared is 16 so that's 256 250 six plus a squared that's 81 plus 81 minus B squared so minus 121 minus 121 all of this stuff is squared all of that over 2 times C all of that over 32 so let's see if we can simplify this a little bit 81 minus 121 that is minus 40 so this becomes 216 over 32 so area is equal to 1/2 times 8 is 8 let me switch colors it 1/2 times 16 is 8 times the square root of 81 the square root of 81 minus 256 let's see 81 minus 121 that's minus 40 to 50 6 minus 4 is 2 16 to 16 over 32 squared now this is a lot of math to do so let me get out a calculator I'm really just trying to show you that these two numbers are should give us our same number so if we turn on our calculator if we turn on a cow first let's figure out what 18 square roots of 7 are 18 times the square root of 7 this is what we got using herons we got 47.62 let's see if this is 47.62 so we have 8 times the square root of 81 minus 2 16 divided by 32 squared and then we close our square roots and we get the exact same number I was worried I actually didn't do this calculation ahead of time so I might have made a careless mistake but there you go you get the exact same number so our formula just now gave us the exact same value as Heron's formula but what I'm going to do in the next video is prove to you is prove to you that this can actually be reduced algebraically to herons