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

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


Math Antics - The Pythagorean Theorem - By Mathantics



Transcript
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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