Math Antics - The Pythagorean Theorem - Free Educational videos for Students in K-12

Math Antics - The Pythagorean Theorem - By Mathantics

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00:03	Uh huh . Hi , I'm rob . Welcome to
00:07	Math Antics . In this lesson , We're going to
00:09	learn about the Pythagorean theorem or Pythagoras theorem as it's
00:13	sometimes called . And you may be wondering what's the
00:15	theorem and who in the world is Pythagoras ? Well
00:20	, in math , a theorem is simply a statement
00:22	that has been proven to be true from other things
00:25	that are either known or accepted to be true .
00:27	And Pythagoras . Well , he was this really smart
00:30	dude who lived a long time ago in ancient Greece
00:32	and he proved the theorem . Well , historians aren't
00:35	completely sure it was actually Pythagoras who proved it .
00:38	It could have been one of his students or followers
00:41	, but he usually gets credit for it anyway .
00:46	The main thing that you need to know is that
00:47	the Pythagorean theorem describes an important geometric relationship between the
00:52	three sides of a right triangle . We're going to
00:54	learn what that relationship is in just a minute .
00:57	But first , there's several things that you need to
00:59	know before you can truly understand the Pythagorean theorem or
01:02	use it to solve problems . First of all ,
01:04	to understand the pythagorean theorem , you need to know
01:07	about angles and triangles , and you also need to
01:09	know a little bit about exponents and square roots .
01:12	So if those topics are new to you , be
01:14	sure to watch our videos about them . First second
01:17	, even though the pythagorean theorem is about geometry ,
01:20	you'll need to know some basic algebra to actually use
01:22	it specifically . You'll need to know about variables and
01:25	how to solve basic algebraic equations that involve exponents .
01:28	We cover a lot of those topics in the first
01:31	five videos of our algebra basic series . Okay ,
01:35	now that you've got all that background info covered ,
01:37	let's see what the Pythagorean theorem actually says . The
01:40	theorem can be stated in several different ways , but
01:42	the one we like best goes like this for a
01:45	right triangle with legs A and B . And hip
01:47	. Until you see a squared plus B squared equals
01:51	C . Squared . As you can see from this
01:53	definition , the pythagorean theorem doesn't apply to all triangles
01:57	, It only applies to right triangles , as you
02:00	know , right triangles always include one right angle that's
02:03	usually marked with a square right angle symbol . To
02:06	help you identify it , and you need to know
02:09	which angle is the right angle because it helps you
02:11	identify an important side of the triangle called the hypotenuse
02:15	. The hypotenuse is the longest side of a right
02:17	triangle , and it's always the side that's opposite of
02:20	the right angle . In other words , it's the
02:22	side that doesn't touch or help form the right angle
02:25	itself . In order to use the pythagorean theorem ,
02:29	you need to be able to identify the hypotenuse because
02:31	that's what the variable C . Stands for . In
02:33	the theorem , C . Is the length of the
02:35	hypotenuse side . The other two sides of the triangle
02:39	. The ones that do touch or form the right
02:41	angle are called its legs . Our pythagorean theorem definition
02:45	uses the variable names A and B . To represent
02:48	their lengths . Oh , and it doesn't matter which
02:50	leg is called A . And which leg is called
02:52	B . As long as you keep track of which
02:54	is which after you make your initial choice . Okay
02:58	. Now that we know the various parts of the
03:00	Pythagorean theorem , let's think about what the relationship or
03:03	equation A squared plus B squared equals C squared is
03:07	really telling us it's telling us that if we take
03:10	the lengths of the two legs sides A and B
03:13	. And square them , which means multiplying them by
03:16	themselves . A squared is eight times a and B
03:19	squared as B times B . And then if we
03:22	add those two squared amounts together , they will equal
03:26	the amount you'd get if you square the hypotenuse side
03:28	, which would be C squared or C . Time
03:31	. See that may sound a little confusing at first
03:34	. So let's take a look at a special example
03:36	of a right triangle . That will help the pythagorean
03:38	theorem make a little more sense . This right triangle
03:41	is called a 345 triangle because its sides have the
03:44	relative lengths of 34 and five and by relative lengths
03:49	, I mean that the units of length don't really
03:51	matter . Besides . Could be expressed in any units
03:54	inches , meters miles , whatever . So the triangle
03:58	could be of any size as long as it's lengths
04:01	would have the proportions 34 and five relative to each
04:04	other , starting with the side . That's three units
04:07	long . Which will call side A . What do
04:10	we get if we square that side ? Well ,
04:12	an arithmetic squaring three means multiplying three times three which
04:16	equals nine . And the geometric equivalent of squaring something
04:20	actually results in a square shape . As you can
04:23	see . This square contains nine unit squares , so
04:26	this red area represents the value a squared in the
04:29	Pythagorean theorem . Next , let's look at the side
04:32	that's four units long , which will call site B
04:35	. Squaring four means multiplying four times 4 which is
04:38	16 . Again , the geometric equivalent of that is
04:41	a literal square . That is four units on each
04:44	side and covers a total area of 16 units .
04:47	So this blue area represents B squared in the Pythagorean
04:50	theorem . And finally , let's deal with the hypotenuse
04:54	or sight see which is the longest side . It's
04:56	five units long . Squaring five means multiplying five times
05:00	5 which is 25 . And the geometric equivalent is
05:04	a five x 5 square . That has an area
05:06	of 25 units . So , this green area represents
05:09	c squared in the Pythagorean theorem . Now that you
05:12	can see how the arithmetic parts of the Pythagorean theorem
05:15	are related to the geometric parts of this right triangle
05:18	. Let's check to see if the pythagorean theorem is
05:20	really true , at least in this special case on
05:23	the arithmetic side , if you add up the amounts
05:25	A squared and B squared , they really do equals
05:28	C squared because nine plus 16 equals 25 . And
05:32	with a little rearranging of our unit squares , you
05:34	can see that the area of the squares formed by
05:37	the two legs really does equal the area of the
05:39	square form by the hypotenuse , wow ! Those ancient
05:42	greek dudes really were smart . Okay . But I
05:46	know what some of you are thinking . That's cool
05:48	and all . But why should I even care about
05:50	the pythagorean theorem ? What's it good for ? Well
05:53	, as always , that's a good question . And
05:55	the answer is like many things in math . The
05:58	pythagorean theorem is a useful tool that can help you
06:01	use what you do know to figure out what you
06:03	don't know specifically if you have a right triangle ,
06:06	but you only know how long two of its sights
06:08	are . The Pythagorean theorem tells you how to figure
06:11	out the length of the third , unknown side .
06:13	For example , imagine that you have a right triangle
06:16	that's two cm long on this side and three cm
06:19	long on this side . But we don't know how
06:21	long the hypothesis . No problemo , the Pythagorean theorem
06:25	tells us the relationship between all three sides of any
06:28	right triangle . So we can figure it out .
06:31	We know that a squared plus B squared equals C
06:34	squared . So let's plug in what we do know
06:36	into that equation and then solve it for what we
06:38	don't know . Again it doesn't matter which of the
06:41	two legs is called A . Or B . So
06:43	let's just label them like this and then substitute to
06:45	for A . And three for B . And the
06:48	pythagorean theorem equation . That gives us an algebraic equation
06:52	that has just one unknown sea . If we solve
06:55	this equation for C . In other words , if
06:57	we rearrange the equation so that C . Is all
06:59	by itself on one side of the equal sign ,
07:02	then we'll know exactly what C is . Well ,
07:04	no , the length of that side of the triangle
07:07	First . We need to simplify the left side of
07:09	the equation since it contains the known numbers . And
07:12	according to the order of operations , we need to
07:14	simplify the exponents . 1st two squared is four and
07:18	three squared is nine . Then we add those results
07:21	four plus nine equals 13 and we have the equation
07:24	13 equals c squared which is the same as c
07:27	squared equals 13 . Then to get see all by
07:30	itself , we need to do the inverse of what's
07:32	being done to it since it's being squared , the
07:35	inverse operation is the square root . So we need
07:38	to take the square root of both sides . Taking
07:41	the square root of c squared just gives us C
07:44	. Which is what we want on this side of
07:45	the equation . But it gives us a little problem
07:48	on the other side because it's not easy to figure
07:50	out what the square root of 13 is . It's
07:52	not a perfect square . So it's going to be
07:54	a decimal and probably an international number but that's okay
07:58	because it's fine to just leave our answer as the
08:00	square root of 13 . Sure you could use a
08:03	calculator to get the decimal value if you really need
08:05	one . But in math it's very common to just
08:08	leave square roots alone . Unless they're easy to simplify
08:11	. So the sides of this right triangle are two
08:14	centimeters three centimeters and the square root of 13 centimeters
08:19	. Let's try another example for this right triangle .
08:21	We know the length of the hypotenuse 6m and one
08:25	of the legs 4m but the length of the other
08:27	leg is unknown . So let's use the Pythagorean theorem
08:30	to find that unknown length . As usual we call
08:34	the hypotenuse side C . And let's call the leg
08:36	, we know site A and the leg we don't
08:38	know side B . Then we can substitute the known
08:42	values into the Pythagorean theorem and solve for the unknown
08:44	value , replacing the C with six . And the
08:48	A . With four gives us the equation four squared
08:51	plus B squared equals six squared , which we need
08:55	to simplify and solve for B first let's simplify the
08:58	exponents four squared is 16 and six squared is 36
09:03	. Now we need to isolate the B squared and
09:06	we can do that by subtracting 16 from both sides
09:09	of the equation On this side the Plus 16 and
09:12	the -16 . Leave us with just be squared .
09:15	And on the other side we have 36 -16 which
09:18	is 20 . We can now solve the simplified equation
09:22	for B by taking the square root of both sides
09:24	, which gives us B equals the square root of
09:27	20 . Again it's fine to leave your answer as
09:30	a square root like this and some of you may
09:32	know that the square root of 20 can be simplified
09:35	to two times the square root of five . We're
09:37	not going to worry about simplifying roots in this video
09:40	, but if you know how to do it awesome
09:42	. If you don't know , just leave the answer
09:44	as the square root of 20 m . Here's another
09:47	interesting one . What if you have a unit square
09:49	that's cut in half along a diagonal . Each side
09:52	of the square is one unit long , But how
09:55	far is it from one corner of the square to
09:57	the other along the diagonal ? Well , since the
10:00	diagonal divides the square into two right triangles , we
10:03	can use the pythagorean theorem to tell us that unknown
10:06	distance . We labeled the legs of the right triangle
10:09	A . And B . And the hypotenuse C .
10:12	And since we know that A and B are both
10:13	one , we can plug those values into the pythagorean
10:16	theorem equation , which gives us one squared plus one
10:19	squared equals c squared . Now we solve for C
10:23	. One squared is just one . So the left
10:25	side of this equation simplifies to one plus one ,
10:28	which is just two . That means c squared equals
10:31	two . And if we take the square root of
10:33	both sides , we get C equals the square root
10:36	of two . So that's how far it is across
10:38	the diagonal of the unit square . Okay , so
10:42	that's how you use the pythagorean theorem to find the
10:44	length of an unknown side of a right triangle ,
10:46	which is its most common use . But there's another
10:49	way that you can use the Pythagorean theorem that I
10:51	want to mention . You can also use the Pythagorean
10:54	theorem to test a triangle to see if it truly
10:57	is a right triangle . You know , in case
10:59	you're not already sure for example what if someone shows
11:03	you this triangle ? and ask , is this a
11:05	right triangle ? Well , it looks a lot like
11:07	a right triangle , but it doesn't have a right
11:09	angle symbol . And it would be kind of hard
11:11	to tell if this angle is exactly 90 degrees just
11:14	by looking at it . Maybe it's really close to
11:16	90 like 89.5 degrees . No worries . That the
11:20	factory and theorem can tell us for sure . If
11:22	we know the lengths of all three sides of the
11:24	triangle , if we know those lengths A , B
11:27	and C , then we can just plug them into
11:29	the pythagorean theorem equation to see if it holds true
11:32	in this particular case , since the two shorter sides
11:35	are each three centimeters and the longest side is four
11:37	centimeters . We can plug those values in for A
11:40	. B and C . And simplify to see what
11:42	we get three squared is nine . So on this
11:45	side of the equation , we get nine plus nine
11:47	which is 18 . And on the other side we
11:50	have four squared which is 16 . Oh , that
11:53	doesn't look right . Our equations simplified to 18 equals
11:57	16 , which is definitely not a true statement .
12:01	That means that the three sides of this triangle do
12:03	not work in the pythagorean theorem . They don't fit
12:05	the relationship A squared plus B squared equals c squared
12:09	. And since the pythagorean theorem tells us that all
12:12	right triangles , that that relationship , this triangle must
12:15	not be a right triangle . All right . So
12:18	now , you know what the Pythagorean theorem is ,
12:20	and you know how to use it , you can
12:22	use it to find a missing side of any right
12:24	triangle . And you can also use it to test
12:27	the triangle to see if it qualifies as a right
12:29	triangle . But as you can see , it takes
12:32	a lot of other mass skills to be able to
12:33	use the Pythagorean theorem effectively . So you may need
12:36	to brush up on some of those skills before you're
12:39	ready to try using it on your own . And
12:41	remember you can't get good at math just by watching
12:43	videos about it . You actually need to practice solving
12:46	real math problems , as always . Thanks for watching
12:48	Math Antics and I'll see you next time learn more
12:52	at Math antics dot com .

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