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The Pythagorean Theorem Made Easy! | Real-Life Word Problems & Right Triangle Proofs
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May 17, 2025
Welcome to today’ math lesson, where I break down the Pythagorean Theorem using real-world examples In this video, we explore: A volleyball court scenario where Nora serves the ball across the diagonal—how far did it really travel? A balloon problem involving right triangles and how to calculate height using limited information. Step-by-step instructions on how to apply the a² + b² = c² formula to find missing sides of right triangles. How to verify if three side lengths form a right triangle using the Pythagorean Theorem. Fun classroom interaction, sketches, and memorable math tips to help you visualize and retain the concept. Perfect for pre-algebra and algebra students learning how geometry and algebra connect through word problems and real-life applications. Whether you're preparing for a quiz, helping a student, or reviewing core math concepts—this is the Pythagorean video lesson you didn’t know you needed! Don’t forget to like, subscribe, and share if this helped clarify anything. Drop a comment below if you have any questions or want more lessons like this.
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0:01
all right So let's talk about the Pagan
0:03
theorem section 10.4 So I'm going to
0:05
start with a word problem
0:08
and then we're going to elaborate on
0:10
that Right So suppose that you guys that
0:12
play volleyball right A volleyball court
0:15
is 30 ft wide and 60 ft long That's a
0:18
that's the right measurement Is that
0:19
right That's what they claim Right Now a
0:22
player suppose Norah want to serve right
0:26
What's going on over there What's what's
0:28
going on there
0:30
What's going on between you two Let's
0:32
get ready All right So a volleyball
0:34
court is 30 ft wide and 60 ft long A
0:37
player Nora you say is want to serve the
0:40
ball from one corner of the court to the
0:42
opposite corner because she feels like
0:44
there's a weak player that I can't catch
0:46
a serve Right So if you serve it right
0:50
How far is this How did you figure it
0:53
out About like 60 ft long You don't know
0:57
You don't know that We know that in the
0:58
court right We can draw the court We can
1:01
draw it Serve from one corner to the
1:03
other You just stand at the corner You
1:04
want to serve and serve straight It's
1:06
not that hard You can't So let's say
1:09
this is here right Nor's here right This
1:13
is Nora and she has a volleyball She
1:15
want
1:22
right That's she woke up that morning
1:25
and she didn't feel like doing
1:26
[Music]
1:32
Please stop You're making it worse So
1:34
now she want to turn right She hits the
1:36
ball from there and she want to
1:39
serve the net She want to turn from this
1:42
corner right to this corner
1:46
This
1:47
is like
1:50
[Music]
1:52
she wants to serve from here to here
1:54
Right Now how do we know what the
1:57
distance is We can use do an estimation
2:00
We can use We want to know the exact
2:02
distance So we have to use the
2:07
right If you look at this this is a
2:10
right angle So it forms a right triangle
2:12
So therefore the application is we're
2:14
going to use theorem to get to that
2:16
Right So now before we do that I'm going
2:18
to explain this real quick So in any
2:22
given right triangle any given right
2:24
triangle any right the sum of the
2:28
squares of the length of the leg is
2:31
equal to the square of the length of the
2:34
hypotenuse Can you and yeah I'll draw
2:38
that And the formula is a square + b
2:40
square is equal to c squares Right So
2:43
this is the right triang that I drew
2:45
Right And then these sides are called
2:48
the
2:49
legs They're called the legs All right
2:51
These are called the legs and this is
2:54
called the
2:56
hypotenuse In any given triangle the
2:59
sides where you have the right angles
3:01
This is always going to be the leg You
3:02
can call it B and this is A And then the
3:04
hypotenuse always we use the letter C
3:06
for the hypotenuse The hypotenuse is the
3:09
side opposite the right angle Everyone
3:12
repeat that The hypotenuse is the side
3:15
opposite
3:17
the right angle Right So this is always
3:19
going to be the hypotenuse and all the
3:20
other sides are called the length This
3:23
you can call this a or b doesn't matter
3:25
But the hypotenuse is always going to be
3:27
the side that is opposite the right
3:29
angle Right So Peter Goras came up with
3:34
his formula and I just read you the
3:35
story and how he came up with a formula
3:37
He saw a blacksmith and he heard the
3:39
sound and he went ahead and put the
3:40
squares in a in a triangle and he he
3:43
realized that So so and that became like
3:45
something that that be used right So
3:48
according to Pythagoras and according to
3:50
what he
3:51
found if you square the legs of any of
3:54
these triangle you add them up is always
3:56
going to be equal to the square of the
3:58
hypotenuse Which means a
4:02
squ is always going to be equal to what
4:04
C^ square That's what he found out And
4:06
I'm going to prove it to you right Let's
4:08
assume that we have a triangle with
4:11
measurement eight right And then we also
4:15
have this equal to 15
4:19
Right So if you do if you find if you
4:22
want to find the hypotenuse all you have
4:25
to do is just do what 8 squar + what 15
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squar right So if you do 8 squar + 15
4:33
squar you're going to get that this is
4:34
64 right 8 square + what 15 squar is 225
4:39
Okay 225 And that gives you 289
4:45
And 289 is equal to what
4:48
C² right C square
4:52
C square Not just C C square Now the
4:54
question is are we looking for C square
4:56
or are we looking for C C So how do we
4:59
find C Square We take the square root
5:03
right We take the square root of that
5:06
What's the square root of 289 Who knows
5:08
it without thinking
5:11
17 right 10 Is it going to be plus or
5:15
minus 17 or just 17
5:17
Are you sure it's plus
5:20
We can't have
5:21
it Right So it's just going to be what
5:24
Regular 17 Right So C is equal to 17 So
5:28
if you do this 8 square + 15 square
5:30
you're always going to get the same
5:31
thing 220 289 right So in any given
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right triangle the square of both legs
5:40
added together is always going to give
5:42
you the square of the
5:44
hypotenuse This formula is going to
5:47
follow you in physics It's going to
5:49
follow you in algebra 2 It's going to
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follow you in algebra 1 pre-calculus
5:53
There's a lot of problems that we deal
5:54
with that have right triangles right And
5:56
even in calculus 2 we did a lot of thing
5:59
with um you can see a word problem So we
6:02
we're going to do all of that All right
6:04
So now let's go back to our problem and
6:06
let's try to solve this Right now let's
6:08
try to solve this I want to know how far
6:11
Nora was able to hit that ball Right
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Okay That's Nora All right
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Whatever the stick figure Nora Let's
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call it No we call Okay we call it Ma
6:22
That's fine You guys don't know We just
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call her right Let's call her Mora So
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Mora just hit the ball Right So how do
6:32
we find what is this side going to be
6:34
called What am I going to call this side
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How much is this here
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How much is this 60 60 So how am I going
6:44
to find that I'm going to do what 30
6:46
squared 60 equals to equals to C² right
6:52
And then 30 square is what 900 Okay And
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60 square is 3,600 3,600 So if you add
7:00
them together you get
7:03
4500 right C² Now how do I find C You
7:08
square root of 4500 What's the square of
7:11
4500 Alina what's the square root of
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4500
7:15
67.082039
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So we can write off to
7:19
67.1 right Is equal to C So she hit the
7:22
ball 67.1 ft right If you ever find a
7:27
hypot
7:30
I'm like so if you ever find a
7:33
hypotenuse that is less than the legs
7:35
you know you did something wrong because
7:36
the thought is always going to be the
7:38
longest side That's just truth Right now
7:41
let's do this problem here That's a
7:43
little different Right Now a balloon
7:45
advertising the opening of a store is
7:49
how do you say this word She or tethered
7:52
Tether Thank you for your lesson Right
7:55
See women are good that
7:59
uh my game this morning What were you
8:03
going to say She just taught me how to
8:06
spell So we appreciate you ladies right
8:08
And advertising the opening of his show
8:11
of the store this to the ground as shown
8:15
I'm going to draw a picture Right So see
8:19
now you have to come up with a picture
8:20
This is why this is important right So
8:22
I'm not saying you have to be an artist
8:24
but you do have to come up with pictures
8:26
So let's try and figure this out right
8:31
So let's try to figure this out So this
8:32
is the balloon right
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Okay
8:43
Right This is the ground All right And
8:45
they're saying about how many feet above
8:48
the ground is the balloon Zero Right So
8:52
here's what we're going to do What are
8:54
they looking for This is the balloon
8:55
here right So this is directly ground So
8:58
they want to know how far above the
9:00
ground is the balloon They want to know
9:02
the height
9:04
That's what they want to find Okay They
9:07
want to find his height Okay And now we
9:10
are given some some things in the in the
9:12
book We are told that the distance
9:14
between uh the distance between the rope
9:19
the between the this and where this is
9:21
this is here is 25 ft We told that this
9:24
is 25 ft
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right and this is 45 ft
9:30
So now what am I looking for I'm looking
9:33
for here the height right So how how far
9:36
is it above the ground We don't know
9:38
that 5^2 + h2 = 4 No What's the hypoten
9:48
So which one do we know now We know the
9:50
hypotenus and one of the legs right So
9:52
we know that a square we always going to
9:54
start with this formula A square + b
9:55
square is equal to c square Right Now
9:57
let's replace what we know What do we
9:59
know
10:01
We know a and c or b and c Doesn't
10:04
matter Right So we going to know let's
10:05
say we know a and c So we going to
10:06
replace a by what
10:09
20 5 + b² = to what 45 square That's
10:15
what I said That's what you said And
10:17
then you said no Okay My apologies
10:21
I don't think I heard that I don't think
10:24
I was 25 squared
10:29
625
10:31
My mind works like this Why are you
10:32
smiling That's not a good sign It's 625
10:35
right You're like you're probably like
10:36
big as you could say that Somebody help
10:39
me 45 No
10:42
Somebody help me Please please From the
10:46
goodness of her
10:47
heart 45
10:52
It's on the ground
10:55
All right 1,65
10:58
Exactly Is that right 45 * 45 Yeah
11:04
How much 2025 2025 All right Hey that's
11:09
the year Can I help solve this What am I
11:11
going to do
11:13
Algebra What am I going to do
11:20
Right So B² is equal to what What's 2025
11:32
What
11:33
100 All right And how do I find
11:48
415 So we can write it off from 42 right
11:52
So basically we
11:54
we're going to conclude that the balloon
11:56
is what 37 ft and
11:59
42 42 ft above the ground All right
12:04
So this is how
12:07
this is
12:09
how this is how
12:12
[Music]
12:16
this is how you figure it out Okay This
12:19
is how you do this So the pagan theorem
12:21
is really really easy to use and it's
12:23
really fun when you understand Now the
12:25
problem with pagan theorem is you have
12:27
to figure out how to draw those pictures
12:28
Sometimes it's done as obvious You have
12:30
to be able to like picture out what are
12:32
you being given because if the book does
12:34
not give you any picture now you have to
12:36
come up with what they're saying and
12:38
interpret that and then you're going to
12:39
apply the pyagon theorem So what I'm
12:42
going to do now is I'm going to give you
12:44
some work to do Okay And then um also
12:49
all right so there's there's something
12:51
else that you also need to know
12:53
Basically how do you how do you how can
12:56
you tell if if you're giving some
12:58
dimensions right How would you know if
13:01
the dimension that you're given are
13:02
forming a right triangle So example they
13:05
want you to verify that right So
13:07
basically let's say I give you three
13:09
three lengths I give you three lengths
13:12
Suppose I give you 6 8 and 10 6
13:16
cm 8 cm and then 10 cm And I want to say
13:21
verify that these three dimensions here
13:23
So the 10 will be C The tank is going to
13:25
be C We want to verify that this
13:27
actually we want to see We don't know
13:29
But does this form a right triangle No
13:31
We don't know yet We have to prove it
13:33
Right So let's assume that this is a
13:36
this is b and this is c So according to
13:40
pagoras a square + b square is equal to
13:43
what C square C square So now let's see
13:46
if this holds for this value Right So
13:48
we're going to go
13:49
6² + 8² is equal to 10 square Now we're
13:54
going to put a question mark here Why am
13:55
I putting a question mark Cuz you don't
13:57
know if it's true I don't know if it's
13:58
true I have to prove it Right If this is
14:00
true then yes I can conclude that this
14:02
forms a right triangle But if it's not
14:05
then it's not a triangle right Right
14:06
triangle So six square is what
14:10
64 right Is equal to 100 100 true or
14:14
false
14:16
Look at that
14:19
True right So because this is true my
14:22
conclusion is yes we have a don't
14:26
triangle right we do have a right
14:29
triangle because
14:32
okay well you don't need to guess you
14:35
just have to do what you just have to
14:36
verify to see that it's holes what if I
14:39
bring a tape measure to school and
14:41
measure out oh yeah you want to do that
14:43
that's going to take you a whole day
14:44
this is why we we just not going to do
14:46
that we just going to apply
14:48
u and then figure out like a split
14:50
second We don't need a measuring tape We
14:52
just apply the formula
14:56
All right