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Proof of Heron's formula (1 of 2)
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. This 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. 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 would be the h. So the question is how do we figure out the area of this triangle? 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 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. 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. What I could do now, 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 do this left hand side and I can write that x squared plus h squared is equal to a squared. That's what I get from this left hand triangle. Then from this right hand triangle, I get c minus x squared plus h squared is equal to b squared. 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. 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. 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 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 2cx plus x squared. Then we have the minus -- sorry, we have the plus a squared minus x squared equals b squared. We have an x squared and a minus x squared there, so those cancel out. Let's add the 2cx to both sides of this equation. So now our equation would become c squared plus a squared. I'm adding 2cx to both sides. So you add 2cx to this, you get 0 is equal to b squared plus 2cx. All I did here is I canceled out the x squared and then I added 2cx to both sides of this equation. My goal here is to solve for x. 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 c squared plus a squared minus b squared is equal to 2cx. Then if we divide both sides by 2c, we get c squared plus a squared minus b squared over 2c 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. Let me scroll down a little bit. We know that our height squared is equal to a squared minus x squared. Instead of 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 2c, squared. This is the same thing as x squared. We just solved for that. So 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. Let me make it a little bit neater than that because I don't want to--. 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 2c. 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. Copy it and then let me paste it down here. So I've pasted it down here. So we know what the height is -- it's this big convoluted formula. The height 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 -- let me do it in pink. The area of our triangle is going to be 1/2 times our base -- our base is this entire length, c -- times c times our height, which is this expression right here. Let me just copy and paste this instead of--. So let me 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 Heron's formula. But this will work. If you could memorize this, I think Heron's 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. Well, actually let's just apply this just to show that this at least gives the same number as Heron's. So in the last video we had a triangle that had sides 9, 11 and 16, and its area using Heron's was equal to 18 times the square root of 7. Let's see what we get when we applied this formula here. So we get the area is equal to 1/2 times 16 times the square root of a squared. That is 81 minus -- let's see, c squared is 16, so that's 256. 256 plus a squared, that's at 81 minus b squared, so 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. 1/2 times 16 is 8 times the square root of 81 minus 256. 81 minus 121, that's minus 40. 256 minus 40 is 216. 216 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 should give us our same number. So if we turn on our calculator--. First of all, let's just figure out what 18 square root of 7 are. 18 times the square root of 7 -- this is what we got using Heron's. We got 47.62. Let's see if this is 47.62. So we have 8 times the square root of 81 minus 216 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 that this can actually be reduced algebraically to Heron's.