PYTHAGORAS THEOREM| EASY EXPLANATION| CLASS 7| INSIGHTFUL MATHS

INSIGHTFUL MATHS

Transcript

00:00hey everyone welcome to insightful maths I hope you all are enjoying the videos

00:06I am uploading on different topics and for different grades so if you have not

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00:16informative videos. Topic for today is Pythagoras theorem it is basically the

00:21topic which is being introduced in grade 7 so we are going to discuss this topic

00:26I request please see the video till end to understand this topic and the

00:31application so let's begin you can see on the slide this theorem basically is

00:38named after a Greek mathematician called Pythagoras so you should know if we are

00:44talking about Pythagoras theorem it is based on the name of a Greek

00:49mathematician whose name was Pythagoras this you should know another thing is

00:55when we are talking about Pythagoras theorem this is applicable only in the

01:01case of a right angle triangle I hope everyone is aware what is a right angle

01:06triangle where one angle of the triangle is 90 degree the right angle okay so

01:13what this theorem does it gives us a relationship between the different sides

01:18of a right angle triangle let's take it further the Pythagoras theorem says that

01:25the sum of the square of the sides of a right angle triangle is equal to the

01:31square of the hypotenuse now what is this such a big statement it sounds

01:36little confusing but it is really very easy have a look at this picture is it a

01:41right angle triangle and it is right angle here in between the sides A and B

01:47let us first understand how do we name these sides in a right angle triangle

01:54the length which is just opposite to the right angle triangle this name is

01:59known as hypotenuse please have a look at the spelling this longest side of a

02:06right angle triangle which is just opposite to your right angle is called

02:11hypotenuse another two sides are known as the arms or one is height another is

02:18base whichever manner longest side is hypotenuse other two are arms generally

02:24this resting one this is which is joined to the base that is the base of a right

02:30angle triangle and this side A is the height this much is understood so what

02:36does the theorem says the square of the longest side which is the longest side

02:42here that is C so longest side square hypotenuse square is equal to the sum

02:49sum means I am adding two number which two number the squares of the other two

02:56side one side is of length a so I will take a square another side is B I will

03:03take B square so how do I frame the statement in a right angle triangle it

03:09is important to note down here in a right angle triangle the square of the

03:14hypotenuse is equal to the sum of square of other two sides you can frame it in

03:22this manner and who has given this property or the theorem it is named

03:27after the Greek mathematician Pythagoras taking this further let us

03:33understand why does this happen this is the proof of Pythagoras theorem let us

03:39consider that we have a right angle triangle you can see this yellow one it

03:43is a right angle triangle three different squares are attached to the

03:49three different sides of this right angle triangle this is the square you

03:53can see here this is the square attached to A square attached to B and square

03:59attached to C how do we find the area of a square area of a square is side into

04:06side so what is the area of C have a look how many what is the length of the

04:11side 1 2 3 4 5 it's a square if one side is 5 so 5 into 5 25 is the square the

04:21area of this one in the similar manner 1 2 3 and 1 2 3 3 into 3 is 9 area of this

04:30block is 9 in the similar manner 1 2 3 4 and 1 2 3 4 the area of this block is

04:384 into 4 16 do we and what is the area how do we find area if the side is C C

04:47into C area is C square here B into B area is B square and A into A area is A

04:56square here let us see if actually there is a relationship between these three

05:00numbers as per Pythagoras the longest side square that means 25 I'm not taking

05:0725 square the side is 5 and 5 square is 25 I have already taken the square okay

05:15so 25 should be equal to the sum of the other two what is A square here 16 and

05:23what is the B square given it is equal to 9 16 plus 9 wow that is coming 25

05:31both the sides are coming 25 only that means it is true so this is the proof of

05:36Pythagoras theorem we should understand the square of the longest side that is

05:42hypotenuse is equal to the sum of the square of other two sides now let us

05:49understand why we are actually learning this what is the use of learning

05:54Pythagoras theorem it determining the length of the side for example we have a

06:00right angle triangle if any two sides are given we can easily find the length

06:05of the third side for example A and B is known I can find C if C and B is known I

06:12can find A if C and A is known I can find B so in a right angle triangle if

06:18any two sides are given you can easily find the length of the third side to

06:24calculate the distance between two points in a coordinate system what is a

06:29coordinate system it looks I'll make it sure it looks something like this when

06:34we draw a bar graph on a graph paper what do we generally do we make some X's

06:39and then the bar this is basically the X 1 and the Y 1 these are the coordinate

06:46axis so if I have any two points here on the coordinate axis I can easily find

06:53the distance between two points using the Pythagoras property you are going to

06:58learn this in grade 10 so I am not touching it as of now but definitely in

07:04the next video I am going to cover this topic if anyone is interested to watch

07:08that please watch another video okay third one is it is basically used in

07:14construction and engineering ensuring that the structures are built correctly

07:19especially when dealing with the right angles and the diagonal bracing okay so

07:24basically in the construction work engineers generally use this property to

07:29ensure if the calculation is going correct navigation and surveying also

07:35uses this property to calculate the shortest path or the mapping distances

07:40in the navigation also it is being used I'll give you a practical application of

07:46this one here you can see that it is a wall right I hope you are able to see

07:50this white wall and there is a ladder here and there this ladder is lean

07:55against the wall okay are we able to make out that this is the vertical line

08:01and this is the ground the angle formed between the wall and the ground is it 90

08:10degree you can have a look around in your house and you can easily make out

08:14if this is the wall and this is the ground always the angle between these

08:18two is 90 degree so if this is 90 degree and the ladder is lean like this so

08:26which is the hypotenuse here if I need to know the length of the ladder what do

08:32I do besides making the right angle triangle one will be height one will be

08:38base so which is the longest side here hypotenuse or the longest side is the

08:44ladder here okay so all these ladder questions you need to understand this

08:50that this one is your hypotenuse ground is your base the distance between the

08:57base of the wall and the base of the ladder this distance is your base and

09:03this one is your vertical height let us do some questions to make you

09:08understand this concept well but a prerequisite the minimum requirement for

09:14doing this is you should know very well how to find the squares and the square

09:19root of the number which I am not covering right now I have already made

09:25the separate video on square and square root you can go to my channel and have a

09:29look there so you can easily understand how to find the square and square root

09:33of any number that will be applicable your hundred percent so have a look at

09:39question number one a right angle triangle has legs of length 6

09:43centimeter and 8 centimeter you need to tell what is the length of the

09:48hypotenuse so let us make it is always advisable to make a figure okay it gives

09:56more of clarity to you only you will find it much easier to solve the

10:01questions when the figure is there in your mind okay the legs are 6 centimeter

10:07and 8 which ever you can take if you wish to take this vertical as 6 you can

10:12if you wish to take base as 6 you can answer will be same let's say this is 6

10:18centimeter another is 8 centimeter question says what is the length of the

10:25hypotenuse I need to find the longest side so how will you start you will

10:30write here using Pythagoras theorem you have to first write Pythagoras theorem

10:37you have to mention here using Pythagoras theorem what do we know the

10:46square of the hypotenuse what is the hypotenuse here AC so you have to

10:51mention here longest side square is equal to the sum of which two side

10:57square a b a to a b square and another side is b c square I hope everyone is

11:07able to understood till here now just input the values you need to find a c

11:12square a b is what it is 6 so that is 6 square b c is 8 it is 8 square

11:20the square means the number multiplied by itself so what is 6 square it is 6

11:26into 6 that is 36 so this number is 36 8 square is 8 into 8 that is 64 if I add

11:36these two numbers that will give me 100 so a c square is equal to 100 if there

11:43is a square on one side and I need to remove this square it goes as under root

11:50on another side so the value of a c is under the root or square root of 100 how

11:57do we find square root if two similar numbers are there inside the root under

12:03root cancel one number cancel and one will come out I request please go and

12:08see that video made on square and square root you can easily solve this question

12:13so this is under the root 10 into 10 two similar numbers under root cancel

12:1910 cancel and only 10 will be left out so the value of a c is 10 don't forget

12:26to write the unit units mentioned here are centimeter so answer is 10 centimeter

12:32easy enough let's move to another question in a right angle triangle now

12:39the hypotenuse is given to you in a right angle triangle the hypotenuse is

12:4410 meters long let me just make it for you so that you can easily relate what

12:50all values are being given this value is 10 meter one leg is 6 meter up kissing

12:57for the 6 list if you hardly makes any difference let's say it is 6 meter you

13:02have to find the length of the other leg this one is missing that you need to

13:06find first of all give it the name ABC this is a right angle triangle right

13:13angle at B why because the longest side is just opposite to this right angle

13:19triangle and it is given to you as 10 again you write using Pythagoras

13:26property what does it says a c square hypotenuse square is equal to a b square

13:35plus BC square whatever values you know put the values here what is AC it is 10

13:43so you have to write 10 square is equal to I do not know a B keep it like this

13:50plus BC is 6 so 6 square find the value 10 square in the previous question we

13:58have done 10 square is for 10 into 10 hundred which is a b square plus 6

14:04square is 36 now if a number is being added on one other side and I take it to

14:12another side transposing it plus become minus minus become plus it is plus 36 so

14:20if I take plus 36 on another side it will be hundred minus 36 which is a b

14:27square the difference of these two numbers is 64 which is a b square I told

14:35you if we have to remove this square it goes as square root of this number so

14:42this is square root of 64 which is a b that means two similar numbers inside

14:50the root 64 is what 8 into 8 is 64 two similar numbers so one number cancel

14:58under root cancel we are left with just 8 and the unit here is meters so our

15:04answer is 8 meter so we have got this value this answer is 8 meter let's move

15:12to now the different kind of the question a triangle has sides of length

15:175 12 and 13 it is not given what is the hypotenuse what is the base nothing is

15:23being given just the length of three sides are given question is asking is it

15:28a right angle triangle we know that Pythagoras properties applicable only in

15:34the case of right angle triangle one more thing we know very well that always

15:39the longest side is the hypotenuse so out of these three which side should I

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