The Pythagorean Theorem and the Distance Formula (Explained)

Video Series: Functions and Their Graphs (Step-by-Step Tutorials for Precalculus)

(D4M) — Here is the video transcript for this video.

00:00
in this episode you’ll learn the pythagorean theorem and the distance
formula and how they’re related so what is the pythagorean theorem
so first off we’re going to need a triangle so let’s start off with triangle abc
and we got some links of the sides a and b
and the longest side so let’s go ahead and make this triangle right here
and let’s call this side right here this this vertex right here c
and so this will be side of link c and let’s call this here b and this here a
and so we have side of length of b inside of length a here
and so can we pick out the pythagorean theorem from a lineup
is number one the pythagorean theorem or is number two the pythagorean theorem
so look at statement one here it looks like the form right here if p then q and

00:01
statement number two looks like it has the form if q then p
so we’re going to call statement number one here the pythagorean theorem
and number two here the converse of the pythagorean theorem
and we’ll talk about the converse here in a little bit so but first
is the pythagorean theorem true why is it true so in order to talk about why the
pythagorean theorem is true notice this equation has lots of squares in it so
what i’m going to do is i’m going to start off by drawing a square here
so let’s make this a square here and i’m going to put this
right triangle here and this in the square here so i’m going to call this
here length b here and length a and then i’m going to do another b and another a
and another b and another a and then a b and an a
and what i’m going to do is i’m going to form these line segments right here

00:02
notice this this is a square in the sense that
this side is a plus b this side has length a plus b and so what we get here
um we’re under the assumption here that this is a right triangle so let’s put
these right angles in here we got this square once we know these
are right angles in here we know these have the same links here side c
or side of the hypotenuse link c now if we want to find the area in here
because that’s exactly what the pythagorean theorem here is saying is
the area c squared here so c squared is equal to what
it’s the area of the large square which is a plus b squared
but that’s too much let’s take away these four areas here
so minus four times and how much do these try right triangles here what’s
the area of them each one has area one half a times b

00:03
and so this right here will be a squared plus two a b
plus b squared minus two a b so in fact this is just a squared plus b squared
and so that gives us some intuition as to why the pythagorean theorem is true
all right so now the next question might be is the pythagorean theorem useful
and so let’s go see that so are there any numbers here that
satisfies this equation right here so after looking a little bit of time
you’ve realized 3 4 and 5 does so for example 9 plus 16 is 25
then you also notice 8 6 8 and and 10 does so 36 plus 64 is equal to 100
in fact there are many of these um and then you might realize um how

00:04
about 5 12 and 13 so 5 12 and 13 also works so there’s 25 plus 144 is 169.
in fact there are many of these these are called the pythagorean triples
they’ve been known for at least 4 000 years but in fact they’re actually lots of
real numbers that will satisfy this equation in fact that’s kind of the
usefulness of this equation is that we can go from right triangle to an equation
and now the thing nice about this equation is if any two are known then we
can go find the third so for example we could go solve for c
how would we solve for c we would take the square root of both sides
notice that each one of these is positive so we just need the positive
root here well we could go and solve this for also for a we would move the b
squared over and we would take the positive root
so that would be c squared minus b squared and similarly we can solve this for b

00:05
so knowing any two of these we can go and find the third okay so um well
why is the distance formula true so here’s the distance formula
and so what we have here is two points x1 and y x1 y1 and x2 y2 and let’s go
ahead and draw these two points let’s say here’s x1 and y1
and let’s say this is x2 and y2 and so what is the distance between
these two points so we claim it’s this given to us by
this right here but how do we know that so what we’re going to do is we’re going
to draw a vertical line down and a horizontal line across
and we’re going to get this right triangle right here and so what is the
length of this side right here so this side right here has length
x2 minus x1 it’s all of this take away this part right here so minus

00:06
x1 and that gives us this leftover distance
and then this one right here is y2 take away y1 so y2 minus y1
with this distance right here now i drew it in quadrant 1 but actually
you know if you’re looking for the distance between any two points we could
be in any quadrant right so i’m going to put in absolute value on these
that gives us the distance no matter what quadrant we’re in
so this is the distance here and this is the distance here and we can use
pythagorean theorem to find the length of the hypotenuse the pythagorean
theorem says that d squared is equal to uh this side squared [Music]
plus this height here y2 minus y1 squared and so then solving for d
we have the positive root because d is positive
now i don’t need the absolute value anymore because i’m squaring this so i
can just say here x two minus x one in fact uh to stay consistent with this

00:07
that’s x two and x one and that’ll be square and then plus y2 minus y1
and so that gives us our typical distance formula there okay so
now you may ask the question though which is first so if we look at
two points here and try to do an example so let’s look at the example say here a
is the point uh two minus two one and b is the point here 2 4
what would be the distance between these two points
so we can label this as x1 y1 and we can label this as x2 y2
and so what will be the distance between
these two points so the distance will be x2 so 2 minus the x1 which is already a

00:08
minus 2. plus and now y2 minus y1 so 4 minus 1 squared
and so this distance is just square root of what is this right here that’s 4 16
plus 9 so we’re looking at third thirty four um that’s three
this is uh sorry this is a three and this is a three here
so this will be five squared plus 16 yes so square root of 34. now what if we
switch these and relabel these so now let’s say here this is x2 and y2
and that’s our a and this is our b here 3 4 and now let’s say this is x 1 y 1.
so now what will the distance be so the distance here now will be
square root of so we’re always going to do x2 minus x1 so i’m going to be

00:09
looking at now minus 2 minus 3 squared plus 1 minus 4 squared
and so what will this distance right here be that’s a minus 5 but it’s still
25 and this is minus 3 and then that’s a square so it’s minus 3 times -3 so you
still get square root of 34. so in terms of which is first
it doesn’t matter you can do them any you can label them any way you want the
distance is a geometric concept it’s unique no matter which way you
algebraically represent it okay so now but the question is but uh
how are they related how are the distance formula and the pythagorean
theorem related so when we try to answer this question
here um how are they related what we’re going to do is we’re going to say okay
um we’ve already taken the distance formula and said you know what the
distance formula is true and we we show that it’s true when we

00:10
use the pythagorean theorem to do that but what if you wanted to show the
pythagorean theorem is true using the distance formula instead and so if we
want to look at something like this what if we have a right triangle
so let’s take a right triangle right here and and since it’s a right triangle
we can move it to the origin right here and so let’s say this is a and this is b
and so this right here is point b and this is point a
and this is our origin right here at point c and so what can we say using the
distance formula here so what will this point right here be so this is length a
so this is the point here a0 and this is the point right here 0b
and so what will this distance right here be you know using the
distance formula so the distance right here will be
the distance will be square root of and then we can use a minus 0 and then

00:11
square it and then 0 minus b then square and so this tells us that d squared is
equal to a squared plus and then this is a b squared here so in other words the
length of the hypotenuse right here which is d in this case it satisfies the
you know pythagorean theorem here so we can label
this as c or d it doesn’t matter we’re going to get the pythagorean theorem
so in other words in order to show the distance formula we
use the pythagorean theorem but if you want if you want to show the pythagorean
theorem is true you can just use the distance formula so that’s exactly how
they’re related um now what about the converse though what about the converse
so let’s look at the converse here so for a triangle abc with longest side
if you know this equation is true then in fact the triangle must be a
right triangle so why is that true so we need to look at some triangles

00:12
first so let’s look at this triangle right here and this is triangle here acb
or abc and we don’t know this angle right here
but this is length a this is length b and this is link c
and what we do know is that if right so we know that a squared plus b squared
equals c squared now what i’m going to do is
we want to show this is 90. we want to show it’s a right angle a right angle so
it says then it’s a right it’s the right triangle so what i want to do is i want
to take this a and this b think think about them as sticks or
something like that that you can pick up and move and i want to use this a and
this b and and form a right triangle right here
and this is going to be potentially a different triangle right here so i’m
going to use different letters let’s say this is point p this is point q

00:13
and this is point r right here and i’m going to use a and i’m going to use b
and i’m putting them together as a right angle right here
now we already know the pythagorean theorem is true
so whatever this uh side of length here is we don’t know what that length is but
let’s call it a d but because this is a right angle here i formed it using a
right angle here so the a squared plus b squared is in fact equal to d squared
so here at this part we don’t know what the d is we can’t say it’s equal to c
but at this time we don’t know this angle and we don’t know these are equal
but we know this equation holds for this
one and this equation holds for this one but for very different reasons this
equation holds for this triangle by hypothesis
this equation holds by the pythagorean theorem now
we can make the connection between the two so
the c squared will be equal to a squared
plus b squared which in fact is equal to d squared

00:14
but because both of these are positive links in fact this says that c squared
equals d squared or in other words c is equal to d
so now we know in fact this d must be equal to the c
now that we know that they’re equal to each other what we can use is we can say
by the side side side congruence triangle uh theorem
that these triangles must be congruent this is b this is b and a and a and now
we know these are equal so we have all three sides so
by side side side triangle abc so triangle abc is congruent to p prq
and therefore we can therefore we know that the angle acb
or angle c right here is a right angle because this one is

00:15
and so that’s why the pythag the converse of the pythagorean theorem is
also a true statement okay so now the question is is it useful
so is the converse useful so let’s just look at a simple example
to understand how it is useful here we have three points and there’s
nothing special about these three points we can take any three points and ask
this question do these vertices form a right triangle
so in this case we’re going to show that it does form a right triangle
so what we’re going to do is we’re going
to label this this right here is point a and this right here is point b
and this right here is point c and i’m gonna find the distance between
these two and these two and these two and then i’m gonna see if our converse
of our pythagorean theorem uh gives us what we want so here we go let’s find

00:16
the distance between a and b so the distance here is between a and b
so we’re going gonna be looking at two minus four plus and then one minus zero
and so this right here comes out to be square root of five
and now let’s look at the distance between b and c
and so the distance between uh b and c so i’m going to be looking at minus one
minus two so minus one minus two and i’m going to be looking at here minus 5
minus 1. and so this right here is what um minus
3 so that’s 9 and then the 36 so we’re looking at square root of 45
and so what about the distance between a and c so the distance between a and c

00:17
would be so now we’re looking at minus one minus four [Music] and plus and then
minus five minus zero so minus five squared
and so this would be 25 plus 25 we’re looking at square root of 50. and so now
we’re going to ask the question square root of 5 plus square root of 45
does that equal to square root of 50 squared so this is 5 plus 45
equals 50. so yes so yes this is a right triangle
so what we can say is we can take any three points actually and ask the question
do these three points form a right triangle and we just check if the
longest side over here is that squared is equal to the square

00:18
the sum of the squares of the other two sides and so yes the converse is
is useful all right so at this point i want to say
thank you for watching and i’ll see you in the next episode
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About The Author
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David A. Smith (Dave)

Mathematics Educator

David A. Smith is the CEO and founder of Dave4Math. His background is in mathematics (B.S. & M.S. in Mathematics), computer science, and undergraduate teaching (15+ years). With extensive experience in higher education and a passion for learning, his professional and academic careers revolve around advancing knowledge for himself and others. His work helps others learn about subjects that can help them in their personal and professional lives.

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