COORDINATE GEOMETRY| CLASS 10| DISTANCE FORMULA|NCERT
COORDINATE GEOMETRY| CLASS 10| DISTANCE FORMULA|NCERT|EASY EXPLANATION
IF YOU LIKE THE VIDEO PLEASE LIKE,FOLLOW AND SHARE
IF YOU LIKE THE VIDEO PLEASE LIKE,FOLLOW AND SHARE
Category
📚
LearningTranscript
00:00Hello everyone. Welcome to insightful maths. Today's session is very special for me because
00:06this is the first video that I am making for the class 10 students. So I am picking up
00:11with the coordinate geometry that is of course very interesting and one of the easiest topic
00:16that I hope everyone can enjoy. From the coordinate geometry a particular topic distance formula
00:22will be learning today wherein what is the derivation of distance formula, how to use
00:28that and the related questions so that by the end of this video you will be able to
00:34solve all the questions and know from where has this distance formula come. Okay. So let's
00:40begin the session. As you can see here on the slide, there are certain coordinates of
00:48the points being given a point A point B and point C. Three points are given on the coordinate
00:55axis. Coordinates are basically wherein a point is represented as X, Y wherein this
01:03is the X coordinate and this is the Y coordinate. Correct. Now we can see here the coordinates
01:09of point A are X1, Y1. Coordinates of point B are X2, Y2 and for C how to find the coordinate
01:19of C, we can see here the X coordinate here, the distance of this point C from Y axis that
01:26is the X coordinate. It is X1 and distance from this X axis is the Y coordinate that
01:33is X2. Correct. Now we have already learned Pythagoras theorem since grade 7 we are being
01:40learning that. So we'll apply the same concept of Pythagoras property here to know the distance
01:47between the points A, B. It is a right angle triangle we can see here. Now we know that
01:54if we apply Pythagoras property in this right angle triangle A, B, C. Hypotenuse, the longest
02:01side is A, B. So A, B square should be equal to the sum of the square of other two sides.
02:12Other two sides are one is AC that means AC square plus another side is CB or BC that
02:21means BC square. If I add the square of these two I will get the square of A, B. Now distance
02:30of B and C. Have a look here if let's say this is my origin 0. Right. Let's say this
02:39is a point which is at distance 6. This is a point which is at a distance 3. This point
02:45is A and this point is B. If I need to know what is the distance between this point A,
02:53B. What is the distance. This is 4 and this is 5. 1 unit, 2 unit and 3 unit. The distance
03:02between A, B is 3 unit. How do we get that distance from here to here minus distance
03:11from origin to here. So we will be left with only this distance. So how do we get this
03:17distance between A, B. That is 6 minus 3. Answer is 3. We will apply the same logic
03:24here. The distance of point B from the origin it is X2 and distance of point C from the
03:33origin that is X1. So if I take the difference of these point I will get this distance which
03:40is BC. So I'll write here BC is equal to X2 that is the bigger one the longer one minus
03:50X1. OK. So what will be BC square. It will be the square of these two. Similarly now
03:57we need to find AC square that is the vertical one AC. So the greater height Y1 if from this
04:07complete distance Y1 that means from origin from here till the top. This is complete Y1.
04:16If I subtract this distance Y2. I will be left with this distance only and this is what
04:27we are looking for. So this distance AC is Y2 bigger one minus Y1 and I'm looking for
04:37the square. So Y2 minus Y1 whole square. We are supposed to find what is the square of
04:45the distance between AB that means what is the distance between AB. Let us put in this
04:51formula this one using the Pythagoras property. So AB square is equal to this is Y2 minus
05:01Y1 whole square plus X2 minus X1 whole square. But this is the square we are getting. We
05:11have to find the value of AB the distance only. So square when removed it go as under
05:17root. If I say 2 square is equal to 4. So what is 2 equal to that is under root of 4
05:25under root of 4 is 2 only we know that. So whenever we remove this square it goes as
05:30square root of another side. So AB is equal to under the root. We can interchange the
05:37position also it hardly matters. If I write 2 square plus 3 square or 3 square plus 2
05:44square they are exactly the same. Addition is commutative. OK. So I'm starting with the
05:49X coordinate first it hardly matters whether you write Y2 minus Y1 or X2 minus X1. So this
05:55is X2 minus X1 whole square plus Y2 minus Y1 whole square. This is your distance formula
06:07between the two points AB wherein the coordinates of the two points are X1 Y1 and X2 Y2. What
06:14you are going to do take the difference of X coordinate square it up plus take the difference
06:21of Y coordinate square it up add them and find the under root that is the distance between
06:28two points. I hope it is understood how do we derive the distance formula and which property
06:34we have used. We have used Pythagoras theorem. I hope everyone is well aware of this theorem.
06:44This is taught in grade 7. You must have done so many questions based on this. So let us
06:49now apply this distance formula for solving the questions for. First question is find
06:56the distance between the following pair of points. Right. First question accordingly
07:02second third fourth second third is also given to you. Let me do this first one for
07:07you. First one the number given here. Let me just put it as a coordinate X1 Y1. It is
07:14given to you as 2 3. Another one. Let's say it is X2 Y2 and it is given to you as 4 comma
07:231. We know that distance formula means I need to find the distance. If I take it as a point
07:31A just a coordinate X1 Y1 point B just a coordinate X2 Y2. I need to find the distance
07:39between A B. So what was the formula. First of all under root difference in the X coordinates
07:46X2 minus X1 whole square plus difference in the Y coordinates that is Y2 minus Y1 whole
07:55square. Now we just need to put the values and we have to be very very strong with the
08:01addition and subtraction of our integers. So X2 minus X1 X2 is what 4 and X1 is 2. So
08:10I will first input the values 4 minus 2 whole square plus Y2 is 1 minus Y1 is 3 1 minus
08:223 whole square and of course the under root of these two. 4 minus 2 is 2. So 2 square
08:31plus 1 minus 3 is minus 2. Then we have subtracting two numbers and the bigger number is negative
08:40answer is negative. So minus 2 square. Please mind the step very very carefully. 2 square
08:48means 2 into 2 that is 4 plus what is minus 2 square. When we have to square a negative
08:55number answer always comes positive. Minus 2 square means minus 2 into minus 2 minus
09:03minus cancel. So 2 into 2 it is again 4. So 4 plus 4 under root 4 plus 4 is what 8. So
09:14we have to find under root of 8. It is not a perfect square but let's see how best we
09:19can simplify. 8 can be written as 2 into 4 2 into 2 and 2 into 1. This is under the root
09:292 into 2 into 2. If I'm finding the square root I need to make a pair of 2 2 numbers
09:362 numbers here. So 1 will go out and rest of the number will remain inside the bracket.
09:43So what have we found the value of a b. The distance between these two points is 2 root
09:502. I hope this much is understood. Similarly you can do the second part. Third part I'm
09:57doing for you because it is dealing with variables. So you may get confused. If not it is very
10:04good but I still want to do this part for you to add the clarity. Third one now we have
10:11point a which is a b. That means this is x 1 y 1 point b here is minus a minus b. And
10:22what is that. This is our x 2 y 2. I have to find the distance between a b. So step
10:30number one it is necessary to write the distance formula which is x 2 minus x 1 whole square
10:37plus y 2 minus y 1 whole square. Start with an under root. Now x 2 is minus a minus x
10:511 is a. I will write a and complete square ticket x 2 minus x 1 minus a minus a whole
11:00square plus y 2 is minus b minus b. There is a minus and y 1 is b. So minus b minus
11:10b whole square and of course under root two negative numbers together to be added along
11:17with the minus sign. So minus a minus a is minus 2 a whole square plus minus b minus
11:27b is minus 2 b whole square. And of course it will go inside the under root.
11:35Now look at this carefully. Minus 2 a square means minus 2 a into minus 2 a. Negative gets
11:45cancelled. Minus minus is cancelled. 2 multiplied by 2 is 4. a multiplied by a is a square.
11:544 a square. In the similar manner right hand side will be this one. Minus 2 square is 4
12:02and b square is b square and of course it will go under the root. Now a and b are variable.
12:10We don't know its value. How to find the root? But we can observe one thing. 4 is common
12:16in both. If I take 4 as common, I can write 4 into a square plus b square is the root.
12:26Now what is square root of 4? Square root of 4 is 2. So this 4 will be converted. Root
12:33of 4 is 2. 2 outside and inside we are still having a square plus b square and that is
12:40the answer of a b. The distance between these two points is twice under root a square plus
12:48b square. I hope this much is understood. If not, pause the video, do it yourself once
12:54again and then go further to the next question. Next question now says determine. Determine
13:02means find out if the points this, this and this are collinear. Collinear means the points
13:11which are lying on the same line. Collinear means on the same line. Before doing the question,
13:19let's understand one thing. If point a, point b, point c, whichever order, if the points
13:26are collinear, how will we know that they are collinear? If I add a b and b c, a b plus
13:38b c, what should I get? a b plus b c. It should be equal to a c. If there is any other combination
13:46other than this, it is not necessary that b will always be in the middle. It may be a,
13:52c or b. So that means a c plus c b. It should be equal to a b. If the sum of two numbers
14:04is coming as a third number, they are collinear, otherwise not. Collinear means on the same
14:11line. So let me just erase it. First of all, we have to find the distance between these
14:19three points. Only then we will know if the sum of two numbers is equal to the third number
14:23or not. Answer, we will give it a name. Let a is the point 1 5, let b is the point 2 3,
14:34let c is the point minus 2 minus 11. Now let's find the distance between them one by one.
14:42Let us first find what is the distance between a and b. Again, these are the coordinates
14:48given. We can easily use the distance formula for finding the distance in these two points.
14:54Under the root, formula has to be written x2 minus x1 whole square plus y2 minus y1
15:03whole square. You can take anyone as x1 or x2. It hardly makes any difference. But you
15:10have to remember that you will take x1 as y1. So let me take first number as x1, second
15:19as x2 and I'll follow this pattern. 2 is x2, 1 is x1. So I will write 2 minus 1 whole square
15:28plus another down is 3, 3 minus 5 whole square and it has to go under the root. 2 minus 1
15:39square plus 3 minus 5 is minus 2 square and then it is going under the root. Under the root,
15:491 square is 1 plus minus 2 square means minus 2 into minus 2, that is 4. And under the root,
15:594 plus 1 is 5. So I have got the distance between a and b, that is root 5. I'll write
16:05this equation number 1. Let us now find the distance between b and c. We have to write this
16:14formula every time. If we are finding the distance between these two, that's 1. Now
16:21upper number, b is our x1, below 1 is the x2. Under the root, below number, minus 2,
16:30minus upper number, minus 2, another is also minus 2, whole square plus lower number is minus
16:4211 and upper number which is 3, whole square. Minus 2 minus 2 is minus 4, minus 4 square,
16:55minus 11 minus 3 is minus 14 whole square. And we have to find under the root.
17:03Minus 4 square is 16 plus 14 square is, have a look, jisko yaad nahi hai, 14 into 14,
17:124 into 4, 16, 4 ones are 4 and 5 and again 1 into 14 is 14. It is 6, 5, 4, 9 and 1.
17:24It is 196. Ab hum kya karenge, in dono ko add karenge, 16 here, 6 and 6 is 12,
17:339 plus 1, 10 and 1, 11 and this is 212. So under root this one which is under the root 212.
17:44Let me see if this can be simplified further. 212 prime factorization,
17:512 ones are 2, 0, 2 6s are 12, 2 5s are 10, 2 3s are 6 and what about 53, that is a prime number.
18:03So I can write this as 2 under root 53. We have got BC, ab humne AB nikala, BC nikala, AC is left
18:13out. Let us find what is the distance of A and C. Now we are finding the distance between these two.
18:22AC is equal to, under the root, C is minus 2, so minus 2, upper one, minus 1, whole square,
18:33plus minus 11, minus 5, whole square. Minus 2 minus 1 is minus 3, is the square,
18:42minus 11 minus 5 is minus 16, is the whole square and of course under root.
18:49Minus 3 square means minus 3 into minus 3 plus 9 plus, ab 16 ka square agar aapko yaad hai,
18:57well and good. Ek cheez yaad rakhenge, negative number ke square mein minus gets cancelled,
19:02toh humein 16 ka square hi chahiye. 16 into 16, 6 is a 36, 6 and 3, 9 and this is 16,
19:136, 9 and 6, 15, it is 256. Ab 256 mein aapko 9 add karna hai, it will be 265 and of course,
19:26it goes under the root. Humein AC mila under the root 265. Let me see if I can simplify this
19:34further. 265, 6 plus 5, 11, 12, 13. It is not a multiple of 3, let me divide by 5. 5 5's are 25,
19:475 3's are 15. Mere paas kya aaya? It is 5 5's are 25, 5 3's are 15 but 53 is what? It is a
19:56prime number, it cannot be factorized further, so it will remain the same. I cannot take it further.
20:02This is AC. So if these points are collinear, the sum of the two sides, you know the sum of
20:12the two distance should be equal to the third distance. Ab is root 5 and BC is 2 root 53.
20:21Agar mein root 5 aur 2 root 53 ko add karo, will I be getting root 265? No. If I add AC and BC,
20:30will I get AB? It is again no. So we'll write here, since AB plus BC is not equal to AC,
20:43these points are not collinear. These points are not collinear. Once again, collinear means the
20:55points which are lying on the same line, the straight line. Agar ek point A hai, ye point B
21:01hai, ye point C hai, ye collinear kab hoga? Jab the sum of AB and BC, agar mein dono ko add karo
21:09and I get the longer distance, then it will be collinear, otherwise not. Let's move to another
21:16question. Another question now says, these three vertices of the triangle are given to you and you
21:24need to check if these are the vertices of an isosceles triangle. What is the criteria for a
21:31triangle to be called isosceles when its two sides are equal? Again, like we have done in the previous
21:39question, we will find the distance between these three points and check if any two distances coming
21:45equal, then it will be isosceles, otherwise not. So we'll write, let A is equal to 5, minus 2,
21:54coordinate of A, coordinate of B, 6, 4, coordinate of C is 7, minus 2. We are finding first the
22:04distance between AB. Har baar aapko distance formula likna hai. I am just writing it once,
22:10but in your exam, as and when you get this question, every time you have to write
22:15that I am finding the distance between these two, so I am using this formula.
22:19There are marks for writing the formula. So this is x2 minus x1 whole square, plus y2 minus y1
22:28whole square. Ab x2, aapko yaad rakhna hai agar aap pehle one ko x1, y1 lere ho, aur doosra ko x2, y2
22:36lere ho. Every time we have to follow this sequence. So 6 minus 5, I'll write, 6 minus 5,
22:44whole square, plus, here it is 4, upper one is minus 2. Ab yaha se dhyan do, 4 minus, kya minus
22:53karna hai? 4 me se, minus 2. And minus of minus 2 means plus 2. So mera question kya bana? I will
23:01still write here, y2 is 4, isme se kya minus karna hai? y1, jiska coordinate hai minus 2.
23:10So ye hume aise likna hai. And is poore ka square, and it has to go under the root. 6 minus 5 is 1,
23:191 square, plus, now 4 minus minus 2, ye dono minus cancel hua, 4 plus 2 is 6, that means 6
23:28square and under the root. 1 square is 1, 6 square is 36, under the root. 1 plus 36 means
23:36under root of 37. 37 is a prime number, it is not a perfect square, hum iska root nahi nikal sakte,
23:44we'll stop here. A bhi nikla, let us now find BC. Second point C ko ab hum x2 lenge,
23:52so second point, starting from 7. 7 minus 6, ka whole square, plus, C ka doosra point is minus
24:022, so we have to write, minus 2, minus 4, whole square. We have to be very good with integers
24:10before doing this question. 7 minus 6 is 1, so it is 1 square. Minus 2 minus 4 kitna hota hai?
24:182 negative numbers together to be added along with the minus sign, minus 6 ka whole square.
24:251 square is 1, minus 6 ka square lo, ya 6 ka square lo, that is the same, which is again 36.
24:331 plus 36 is 37, and of course under the root. This is equation number 2. Humne A bhi nikala,
24:41ek baar AC aur nikalte hain. First point, second point. 7 minus 5, under the root,
24:507 minus 5 whole square, plus, minus 2 neeche, ab dhyan do. Neeche it is minus 2. Minus 2 me
24:59se kya minus karna hai? Minus 2 once again. Minus 2 minus minus 2, it will be like this.
25:07Lower number minus 2 likha, upar wala number minus 2 hi minus karna hai. So minus 2,
25:14minus, kya minus karna hai? Minus 2. We know that minus and minus becomes plus,
25:20so this is minus 2 plus 2, which ultimately gets vanished. Hum yaha pe abhi likhte hain,
25:26then we will see, minus 2, minus minus 2, and iska whole square. 7 minus 5 is 2, it is 2 square,
25:35plus, minus 2, minus and minus cancel, plus 2, iska whole square under the root.
25:44Under the root, 2 square, ab minus 2 plus 2 toh cancel ho gaya 0, 0 ka square is 0 only.
25:50So under the root, 2 square means root 4 which is just equal to 2. Ye AC aaya. Since we are
25:57getting 2 sides, AB and BC as equal, so hence we have proved that it is an isosceles triangle.
26:05So you will say that since AB is equal to BC, given triangle is an isosceles triangle,
26:15and hence proved. This is how we can prove it is an isosceles triangle. A similar question comes
26:22for the quadrilaterals as well. For example, have a look at this question. Name the type of
26:30quadrilateral form, if any, by the following points and give reason for your answer.
26:52A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F
27:22, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F,
27:42G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F,
28:11G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F,
28:40G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F,
29:09G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F,
29:38G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F,
30:06G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F,
30:36G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F, G, A, B, C, D, E, F,
31:05√2 is equal to √2 if I square both the sides.
31:11We know that equality is maintained if we add, subtract, multiply, divide or square both the sides.
31:21You will see here I am putting both the sides as equal.
31:28A, B is equal to A, C.
31:30If A, B and A, C are equal, then A, B square and A, C square should also be equal.
31:36Now A, B is this one.
31:38So how much is A, B square?
31:40A, B square means the square of this whole root.
31:44If we do square of root.
31:46If I say square of under root 2.
31:49It means under root 2 into under root 2.
31:52Root, root cancel and it is just 2.
31:55When we do square of root.
31:57Root gets cancel and only the inside value remains.
32:01So A, B square is 2 minus x whole square plus 5 square.
32:09Similarly, how much is A, C square?
32:12Square of this whole root.
32:14Root will be cut and you will be left with 2 plus x whole square plus 9 square.
32:21I keep this value here.
32:23A, B square is equal to A, C square.
32:25Equal to 2 plus x whole square plus 9 square.
32:30Now we have to find the value of x.
32:33It is quite easy.
32:35Whole square of A minus B.
32:37We have to use the identity.
32:39Whole square of A minus B and A plus B.
32:42It is A square minus 2AB plus B square.
32:48Similarly, how much is whole square of A plus B?
33:13Now open whole square of A plus B.
33:17Plus twice of 2.
33:192 into 2 is 4.
33:214 into x is 4x.
33:23Plus second number of square plus 9 square.
33:27Now you see x square on left and x square on right.
33:31It is cancel.
33:33Plus 4 on right hand side and plus 4 on left hand side.
33:37Cancel.
33:39What is left with you?
33:414x plus 25 is equal to 4x plus 9.
33:479 is 81.
33:49If I take minus 4x to another side.
33:51Transpose.
33:53So minus 4x will become plus 4x.
33:55So 4x plus 4x.
33:57And on the left hand side.
33:5925 is already there.
34:01If I transpose 81.
34:03It will be minus 81.
34:054 plus 4 is 8.
34:07Right hand side is 8x.
34:09Now what is 25 minus 81?
34:111 positive, 1 negative.
34:13We have to take the difference.
34:15Bigger number negative.
34:17Answer has to be negative.
34:19Please do not do any oral calculation.
34:21It may cause an error.
34:2381 minus 25.
34:2511 minus 5 is 6.
34:27We have taken a carry over.
34:29It is 7.
34:317 minus 2 is 5.
34:33So your left hand side is 56.
34:35So x is equal to.
34:37Minus 56 by 8.
34:39And that is equal to minus 7.
34:41The value of x is minus 7.
34:43So what is your answer?
34:45We were looking for the coordinate of A.
34:47Which is x0.
34:49x is minus 7 and y is already 0.
34:51This is your answer.
34:53So I hope you are able to understand this topic.
34:57You are able to apply this formula.
34:59Do the NCERT questions.
35:01Thoroughly.
35:03And there is nothing more in this formula.
35:05It is all about practicing integers.
35:07Learning the formulas.
35:09And being accurate with your calculation.
35:11If you have liked this video.
35:13Please do not forget to like and subscribe.
35:15For the upcoming videos.
35:17Thank you so much for watching.
35:33Learn English for free www.engvid.com