WEBVTT

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Let's say I've got a triangle.

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There is my triangle
right there.

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And I only know the lengths of
the sides of the triangle.

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This side has length a, this
side has length b, and

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that side has length c.

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And I'm asked to find the
area of that triangle.

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So far all I'm equipped with is
the idea that the area, the

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area of a triangle is equal
to 1/2 times the base of

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the triangle times the
height of the triangle.

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So the way I've drawn this
triangle, the base of this

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triangle, would be side c, but
the height we don't know.

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The height would be that h
right there and we don't

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even know what that h is.

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So this would be the h.

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So the question is how do
we figure out the area

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of this triangle?

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If you watched the last
video you know that you

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use Heron's formula.

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But the idea here is to try
to prove Heron's formula.

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So let's just try to figure
out h from just using

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the Pythagorean theorem.

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And from there, once we know h,
we can apply this formula and

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figure out the area
of this triangle.

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So we already
labeled this as h.

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Let me define another
variable here.

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This is a trick you'll see
pretty often in geometry.

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Let me define this is x, and if
this is x in magenta, then in

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this bluish-purplish color,
that would be c minus x, right?

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This whole length is c
-- the whole base is c.

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So if this part is x, then
this part is c minus x.

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What I could do now, since
these are both right angles,

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and I know that because this is
the height, I can set up two

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Pythagorean theorem equations.

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First, I could do this left
hand side and I can write that

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x squared plus h squared
is equal to a squared.

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That's what I get from
this left hand triangle.

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Then from this right hand
triangle, I get c minus x

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squared plus h squared
is equal to b squared.

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So I'm assuming I know a, b and
c, so I have two equations

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with two unknowns.

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The unknowns are x and h.

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And remember, h is what
we're trying to figure out

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because we already know c.

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If we know h, we can
apply the area formula.

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So how can we do that?

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Well, let's substitute
for h to figure out x.

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When I say that I mean let's
solve for h squared here.

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If we solve for h squared
here we just subtract x

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squared from both sides.

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We can write that x squared --
sorry, we could write that

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h squared is equal to a
squared minus x squared.

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Then we could take this
information and substitute

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it over here for h squared.

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So this bottom equation
becomes c minus x

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squared plus h squared.

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h squared we know from this
left hand side equation.

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h squared is going to be equal
to -- so plus, I'll do it in

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that color -- a squared minus x
squared is equal to b squared.

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I just substituted the
value of that in here, the

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value of that in there.

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Now let's expand this
expression out.

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c minus x squared, that
is c squared minus

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2cx plus x squared.

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Then we have the minus --
sorry, we have the plus a

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squared minus x squared
equals b squared.

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We have an x squared and
a minus x squared there,

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so those cancel out.

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Let's add the 2cx to both
sides of this equation.

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So now our equation
would become c squared

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plus a squared.

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I'm adding 2cx to both sides.

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So you add 2cx to this,
you get 0 is equal to

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b squared plus 2cx.

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All I did here is I canceled
out the x squared and then I

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added 2cx to both sides
of this equation.

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My goal here is to solve for x.

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Once I solve for x, then
I can solve for h and

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apply that formula.

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Now to solve for x, let's
subtract b squared

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from both sides.

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So we'll get c squared
plus a squared minus b

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squared is equal to 2cx.

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Then if we divide both sides by
2c, we get c squared plus a

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squared minus b squared
over 2c is equal to x.

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We've just solved for x here.

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Now, our goal is to solve
for the height, so that

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we can apply 1/2 times
base times height.

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So to do that, we go back to
this equation right here

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and solve for our height.

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Let me scroll down
a little bit.

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We know that our height
squared is equal to a

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squared minus x squared.

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Instead of just writing x
squared let's substitute here.

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So it's minus x squared -- x
is this thing right here.

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So c squared plus a
squared minus b squared

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over 2c, squared.

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This is the same
thing as x squared.

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We just solved for that.

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So h is going to be equal to
the square root of all this

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business in there -- I'll
switch the colors -- of a

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squared minus c squared plus
a squared minus b squared

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-- all of that squared.

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Let me make it a little bit
neater than that because

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I don't want to--.

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The square root -- make sure I
have enough space -- of a

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squared minus all of this stuff
squared -- we have c squared

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plus a squared minus b
squared, all of that over 2c.

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That is the height
of our triangle.

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The triangle that we
started off with up here.

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Let me copy and paste that
just so that we can remember

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what we're dealing with.

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Copy it and then let me
paste it down here.

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So I've pasted it down here.

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So we know what the height
is -- it's this big

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convoluted formula.

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The height in terms of a, b
and c is this right here.

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So if we wanted to figure out
the area -- the area of our

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triangle -- let me
do it in pink.

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The area of our triangle is
going to be 1/2 times our base

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-- our base is this entire
length, c -- times c times our

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height, which is this
expression right here.

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Let me just copy and
paste this instead of--.

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So let me copy and paste.

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So times the height.

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So this now is our
expression for the area.

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Now you're immediately saying
gee, that doesn't look a lot

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like Heron's formula,
and you're right.

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It does not look a lot like
Heron's formula, but what I'm

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going to show you in the next
video is that this essentially

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is Heron's formula.

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This is a harder to remember
version of Heron's formula.

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I'm going to apply a lot of
algebra to essentially simplify

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this to Heron's formula.

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But this will work.

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If you could memorize this,
I think Heron's a lot

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easier to memorize.

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But if you can memorize this
and you just know a, b and

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c, you apply this formula
right here and you will get

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the area of a triangle.

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Well, actually let's just apply
this just to show that this at

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least gives the same
number as Heron's.

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So in the last video we had a
triangle that had sides 9, 11

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and 16, and its area using
Heron's was equal to 18

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times the square root of 7.

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Let's see what we get when we
applied this formula here.

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So we get the area is equal
to 1/2 times 16 times the

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square root of a squared.

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That is 81 minus -- let's see,
c squared is 16, so that's 256.

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256 plus a squared, that's
at 81 minus b squared,

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so minus 121.

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All of this stuff is squared.

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All of that over 2 times c
-- all of that over 32.

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So let's see if we can
simplify this a little bit.

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81 minus 121, that is minus 40.

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So this becomes 216 over 32.

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So area is equal to
1/2 times 8 is 8.

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Let me switch colors.

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1/2 times 16 is 8 times the
square root of 81 minus 256.

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81 minus 121, that's minus 40.

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256 minus 40 is 216.

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216 over 32 squared.

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Now, this is a lot of
math to do so let me

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get out a calculator.

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I'm really just trying to show
you that these two numbers

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should give us our same number.

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So if we turn on
our calculator--.

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First of all, let's just
figure out what 18

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square root of 7 are.

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18 times the square root
of 7 -- this is what

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we got using Heron's.

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We got 47.62.

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Let's see if this is 47.62.

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So we have 8 times the square
root of 81 minus 216 divided

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by 32 squared, and then we
close our square roots.

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And we get the
exact same number.

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I was worried -- I actually
didn't do this calculation

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ahead of time so I might have
made a careless mistake.

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But there you go, you get
the exact same number.

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So our formula just now gave
us the exact same value

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as Heron's formula.

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But what I'm going to do in the
next video is prove to you that

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this can actually be reduced
algebraically to Heron's.

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