Part 1 of Proof of Heron's Formula Part 1 of the proof of Heron's Formula
Part 1 of Proof of Heron's Formula
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- 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.
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