Class 9 Maths Chapter 1 | Distance Between Two Points in the 2-D Plane | NCERT Ganita Manjari

Added: 2026-05-31

[00:00:03]Hello students, this is Prachi. Welcome to our channel and today we are going to do a new topic of Class Nine Chapter First that is Distance Between Two Points in 2D Plane. So till now we have covered all the exercises till here, exercise 1.1 1.2.

[00:00:20]His introduction, Think and Reflect, has also been covered. And if you want to see it, you will find its link in the description box. And today we are going to start a new topic.

[00:00:31]Distance between two points in a 2D plane.

[00:00:34]Ok? What is a 2D plane? We have already seen that we call this a 2D plane. This is the positive X axis. This is the negative X axis. And if you go here then it is positive Y and if you come here then it is negative Y.

[00:00:49]Here +1 + 2 + 3, here minus works.

[00:00:53]Plus here and minus here. Ok? And this is our zero that is origin. We call it a two-dimensional plane because it has two dimensions.

[00:01:03]One horizontal and one vertical. Now, if I want to find the distance between two points, how do I find it? So let us look at that first.

[00:01:16]Like we have done graphs before.

[00:01:19]We are also given these maps here.

[00:01:23]Like let's talk about it. Let me zoom in here. You will see it easily. Yes. If we talk about this, suppose there are any two points here. Ok? For example, if someone tells you the distance from W3 to W4 is these two points. How much distance is it? We will count the units. Isn't it? 1 2 3 4 Units can be counted like this also.

[00:01:47]Or we can see what is the distance from the origin to here, isn't it, and what is the distance from the origin to here? Three. So let's subtract three from s. 7 - 3 became 4. So you can count like this 1 2 3 4 or you can subtract this from this. Isn't it? Even if you subtract this point from this point, you will still get its distance.

[00:02:12]So we took it out. Similarly, if you want to find out S2S S3 then you can count this also.

[00:02:18]1 2 3 units is a distance of 3 units.

[00:02:22]If not like this, then if you want to find out the total then see from here where is it going? It is going like this.

[00:02:29]From here to here it became eight. And it's over and where is s2 going? Five. So from here to here it's five. So five out of eight went. Ok? Five of these eight are gone. How much did it save? Three. So you can remove it like this also. Ok? And if you need to find out the distance and someone needs to find out from the point, we can do that too.

[00:02:50]Like from S2 to F point.

[00:02:54]So you can see there's a unit here.

[00:02:57]Ok? But suppose I have to find the distance from here to here, from W4 to S2, how would I do it [nasal sound]? That means, till now we were directly extracting the points, this is what we were extracting.

[00:03:15]This was parallel to the x axis. We were finding its distance. This was parallel to the y axis.

[00:03:20]So we took it out straight away.

[00:03:23]But there should not be any distance parallel to it.

[00:03:26]So how do we find the distance between those two points? So let us look at it and understand it. So to understand this type of distance, we bring a graph. Ok?

[00:03:38]Now look at this graph.

[00:03:42]Suppose now there are these three points. Ok? Now suppose I have to find the distance from A to D.

[00:03:47]Now there is no straightforwardness in this. Isn't it? This line X is neither parallel to the X axis nor parallel to the Y axis. So how do I find this point and this line? Okay.

[00:04:02]So first of all, understand how can I come here?

[00:04:07]Look up to here if I draw it up to here like this. This is your point till D, should I draw it till here like this. And I should join this with this.

[00:04:17]So I reached from A to D in straight lines. Reached from A to D in a straight line. Isn't it?

[00:04:28]So let's name it as C point. Ok? So from A to D, I have reached this straight path. Ok? Now look, this straight line is 90°. Isn't it?

[00:04:42]Because our axes are parallel. This is 90°. Now from here you might be getting an idea of what I am trying to say.

[00:04:49]Because you are in ninth standard. So this is a right angle triangle. So inside which I can take out this unit. I can remove this unit. And if I get this and this then I can definitely find this with the help of yes Pythagoras.

[00:05:06]Ok? So see how to remove it. 1 2 3 This is of three units. You can understand it like this or you can also see it like this that from here to here, from here to here, A point which is four four. Isn't it? And how much is this C point? There is a forest. So one out of four became three. You can see it like this also. So this is 4 - 1 this is 3 units.

[00:05:32]This is 3 units. Ok? I don't write about this for 1 minute.

[00:05:36]We will write this separately. Ok?

[00:05:40]So this is 4 - 1, this is your 3 units.

[00:05:44]Ok? After that if I talk about CD. Let me talk about the CD. So watch the CD or count it like this. You can count like this 1 2 3 4.

[00:05:55]Or you can see how far D is from here to here?

[00:06:00]How far is C from here to here? Three. So what is 7 - 3? Four. Ok? Now I will create the same thing on a new page. So let me take a normal page and create it here and show it to you. Like it was. It was like this and like this.

[00:06:26]This was your A, C and D. So we saw that there were 3 units and how many were there? Four units. Ok? And it was like this.

[00:06:37]Ok? That was exactly it.

[00:06:42]I made it here. Now you have to remove this thing. Ok? So what does Pythagoras say? Hypotenuse Square = Perpendicular Square + Base Square. Ok? Hypotenuse is AD Perpendicular is AC and base is CD AD If you want to find out then what is AC AC?

[00:07:04]How much is 3² and CD? 4² okay? And what will happen to you? AD² = 9 + 16 that is 25 so if AD is square what does it take? Root. Ok? So how much is this yours? 5 minutes.

[00:07:22]Ok? 2 How much is it here? 5 units. That means how much is your AD? There are 5 units. So now you have think and reflect questions related to this figure, let's do that. So the first Think and Reflect comes to you in moving from A point 34 to D point 71. Ok?

[00:07:46]So moving from point A to point D, what distance has been covered along the X axis and what about the distance along the Y axis. So how much distance did you travel along the x axis and how much distance did you travel along the y axis? Along means along with it, together means parallel. Ok? So this is the distance you've traveled along the X axis. Isn't it? What is the distance you have traveled along the CD and Y axes? Is the AC okay? So first you travelled Y distance AC and then travelled CD. This is when you got AD. Ok? So what do we write here?

[00:08:32]Four units. Look, this is four units.

[00:08:35]1 2 3 4 How many units along the x axis and along the y axis? 1 2 and three. So three units along the y axis. Ok?

[00:08:54]Ok? How did they come? You can also write it like this. How did Four come about? By doing 7 - 3.

[00:09:01]Ok? 7 - 3 here you can write 7 - 3 and here you can write 4 - 1 okay? So this is 4 - 1.

[00:09:14]Ok? The next question is can these distances help you to find distance AD? So can these distances help you to find AD? So yes, this is the distance he has helped us with.

[00:09:26]What help have you got? Because of them, what have we been able to achieve? Pythagoras was able to apply. Isn't it? We found AC, we found CD and from their distances we were able to find AD.

[00:09:39]With its help we have been able to extract AD.

[00:09:41]So yes we can right. So yes, yes with the help of Pythagoras and these distances, what we will write here again like this is AD² = AC² + CD² which is not what we wrote earlier.

[00:10:13]We will rewrite the same here. So how much did the AC cost? 3 units. These were 4 units.

[00:10:21]Because 9 + 16 that is 25. So AD becomes 25 which is 5 units. So you can write the same thing by mentioning it. Ok? So you have done it.

[00:10:35]So now if we come back to our map, if we talk about this map, we have found AD. Similarly, if I want to find DM, what should I do? I will draw a straight line from here along the y axis. Isn't it? From here I will draw a straight line which will be along the X-axis.

[00:10:55]So I can major from here also. Isn't it? How can I do it? See.

[00:11:00]Suppose I give it some name.

[00:11:02]Suppose I name him P. Ok?

[00:11:04]So now I have got a triangle.

[00:11:08]Ok? Now I can also send DMs from here.

[00:11:10]Tell me how to do it? If I want this thing, then see how many points are coming from here to here? From here to here it is nine and from here to here it is 7 so 9 - 7 is 2 units. And anyway, if you count, two units are coming. Ok? And from here, if we notice, how much is this? From here to here there is six.

[00:11:32]And from here till here there is forest. So this is five units. And we can also count 1 2 3 4 5. So this is two units and this is five units. So in this way we can solve it by applying Pythagoras. Ok? So let's plug it in and solve it.

[00:11:51]Ok? So how much MP did I get from here? 5 units. So how did the five come about? 6 - 1 5 units. Ok? And how much DP DP did I get? From here to here 9 - 7 that is 2 units. Ok? Now we have to put my Pythagoras inside it. So this is going to be hypotenuse equal to perpendicular square plus base square. DM² = MP² + DP². Ok? So DM, we have to find that [nasal sound].

[00:12:22]How much MP has come? 5 and how much DP has come? Two. Ok? So this becomes 25 + 4, okay, so dm² = 29 and when we remove the square, what will be your route here, so 29 units, that is your length of dm, okay, and if I ask you how much distance have you travelled along the x axis and along the y axis, then along means along with it, so along the x-axis you have travelled 2 units and along the y axis you have travelled 5 minutes.

[00:13:00]Ok? Ok. Now if I tell you to calculate the distance of am, what will you still do? We will make triangles like this.

[00:13:10]So here the mathematicians felt that it is not possible that we keep making triangles again and again and keep solving them by applying Pythagoras. So what he said was that we should make a general formula.

[00:13:21]And that flower was named Distance Formula. What is distance formula? Let's read this. So what is the distance formula? So look at this thing. It's a simple thing. Like we have it now.

[00:13:33]Ok? Now we have to find out the distance between these two points.

[00:13:37]Ok? So what are these points, these coordinates? This is x and this is y. And what is this? This is x and this is y. We write these points like this. Isn't it?

[00:13:50]First there is the x coordinate then the y. Then first the x coordinate then the y coordinate. And there are two points here. So I consider this as one point. I consider this to be two points. Isn't it? First point and second point. So this is the x y coordinate. Isn't it?

[00:14:03]So let's take its coordinates as x1 y1 and let's take its coordinates as x2 y2, that is, the x and y coordinates of the second point.

[00:14:10]So let's create x2 y2 here.

[00:14:13]This is your y axis. Ok? This is your x axis. Ok? The points here [nasal sound] were your A's and the points here were your M's.

[00:14:27]Okay? What coordinates of A did you have? What coordinates did you have for 3 4 and M? 96. Ok? We will also join this.

[00:14:35]Ok? And we take these coordinates as this is x1 y1 and we take this as x2 y2. Ok?

[00:14:46]So what does your distance formula look like directly inside this? am = x2 - x1² + y2 - y1² ok? So this is your distance formula.

[00:15:07]So you don't need to make triangles again and again.

[00:15:09]You can directly calculate, make and apply this flower. Ok? Now you will say that ma'am you had said that we do not have to do the order. I will do it with understanding. So yes you should not get cramps. Now I will explain to you how this formula came about.

[00:15:24]Ok? So what have we been doing till now? Till now we were solving here by making triangles. Isn't it? So now look at this thing. What if I make a triangle out of it? What if I make a triangle from here? This is from here and this is from here.

[00:15:48]Right? Here it is from here and here it is from here.

[00:15:50]Ok. What would it have become now? A triangle would be formed.

[00:15:53]What was it for? This was hypotension.

[00:15:56]Ok? I will make the same thing here. Let's make a rough one here and one here. Ok? This became a triangle.

[00:16:04]Ok?

[00:16:07]So this was hypotenuse. Let it be this point. We accept this. Let us assume a point q.

[00:16:11][nasal sound] Let's assume the q point here. Ok? Now what do we have to do with it? We have to check how many units this is and how many units this is.

[00:16:19]We have to check this. So, I told you that you can count like this. 1 2 3 4 5 6 units. Isn't it? You can count it like this or you can also see how far it was from here to here? It was nine. And how much is it from here to here? It is three.

[00:16:36]So what has come to you? He has come after scoring 9-3. Isn't it? So what is a nine? This is its x coordinate and this is its y coordinate sorry its x coordinate. What did you have? x2 What was it you had? x1 So let me rewrite it here. Look, this is what you had. Up to here you had 9 and here you had 3, right? So how did this come to you? By doing 9 - 3. Isn't it?

[00:17:07]By doing 9 - 3. So what is 9? x2 And what is this? x1 So now what we do here is 9 - 3 and you get this unit. Ok? I won't do much now. I'm just telling you how it came about. So we got this unit by doing 9 - 3 qm.

[00:17:24]Ok? 9 - 3 by that's six. Here you go qm. Ok? Ok. After that you have to remove this thing. Isn't it? Now check this. How much is this? This is six. Isn't it? Six. From here to here is six and how much is this? From here to here it's four.

[00:17:44]So it will come only then, if 4 out of 6 are gone then what is six? What is its y2 unit and four? It has y1 units, right? So what did we get next?

[00:17:54]6 - 4 So we got two. That is qa you got it. Isn't it? So now look, if I applied Pythagoras here, what would I get?

[00:18:06]am² am² = qm² and q ah qm² + aq² is what you get. Ok? So this is am² = qm qm How did you calculate it? Six, right? And how did Six come about? He has come after scoring 9-3.

[00:18:28]What is 9 - 3² + aq? Your aq has come or your qa has come, 2 that is 6 - 6 - 4 has been squared. Ok? So what was your 9? x2 and three what's yours? x1 plus six what 's yours? What is y2 four? Hole square of y1.

[00:18:54]This is done am and square. If you remove the square, what do you put here? Root. So look, your same flower has arrived.

[00:19:01]Ok? So if we do it from Pythagoras, the same thing comes out.

[00:19:04]Ok? And that's why the mathematicians thought, what's the need to put such a big flower again and again? We will directly tell you this formula. Ok? So let's solve this quickly and see what is the distance in between. Ok?

[00:19:23]If we talk about triangle, then first of all if we talk about triangle then it will become am² = qm² how much was qm? 6² then qa² that is 2² so how much am² do you have? 36 + 4 am² is 40 and how much am will you get?

[00:19:45]40 units. Ok? Here you go in the form of a triangle. Same thing if I do it here with distance formula then check it with distance formula.

[00:19:55]x2 - x1 means 9 - 3² + 6 - 4² am = 6² + 2² so this becomes 36 + 4 that is 4 units. So this is your triangle method. This is your distance flower.

[00:20:19]Ok? I hope now the distance full is clear to you. Ok? So how did this distance formula come into existence and how did we understand it? Ok? Now the question arises to you that if I have considered the first point here. This is considered to be second. So I could consider this one first and this one second. So, would there be a difference in some questions? Is there a difference in some questions? So I don't come.

[00:20:48]You could think of this as x1 y1. This could also be considered x2 y x2 y2. Why so?

[00:20:55]Because look, understand this thing. I will also show you how to solve it.

[00:21:01]If I take a here as 3 and 4 as x2 y2 and here I take it as x1 y1 let's try the formula on this now am = x2 - x1 means 3 - 9² + 4 - 6² okay? So look what's coming.

[00:21:23]This is -6² This is -2² So -6² 36 - 2 squared four two. So look, it's still coming. That means it doesn't matter to us. Consider this as the first point or this as the first point, whichever you wish. Even if there is a minus, it will become square. If it comes, you will get only simple length.

[00:21:42]Ok? For further clarity on this matter, we have been told extra things in NCERT.

[00:21:47]He studies. So the same thing is mentioned here in NCERT, one more extra thing is mentioned that look this is your y axis. This is the x axis. The three points a, dm, and adm are created here in the first quadrant and these are created in the second quadrant. We Know That This Is First Quadrant, This Is Second, This Is Third And This Is Fourth Quadrant.

[00:22:09][nasal sound] If we were to extend this further, the y axis of minus would start here. What's up now? Plus k is the y axis. Sorry plus the x axis is here. Here, the y axis is minus the x axis. Ok? So if you look at the coordinates, a4 is exactly 3 4, its exactly opposite here -3 and 4 are taken. So a was a' took. d was 71 so here I took -7 1 m was 96. Here m' is taken -9 and 6. Now from here, as I just told you, you can consider any point. Isn't it? So you can consider any point. 1 2 or you can consider 1 2 here also.

[00:22:51]Now we had calculated the units of these three and how much each one had brought. Isn't it? Whose length was how much? We had found that.

[00:22:58]I will write it down once.

[00:23:01]How much did AD earn? 5 units of AD had arrived. Ok? And how many AMs have come?

[00:23:08]40 units have arrived. Ok? And how much did the DM get?

[00:23:13]29 had come. Ok? It came three. Now the same thing has come in minus here. So now let's check if what's between them has still changed.

[00:23:25]Let us quickly check whether the distance between them has changed or is it the same. Let's do a quick calculation. We will do it only through distance formula.

[00:23:35]Ok? Is. So first of all let's take a' d'. Ok? So plug in the formula x2 - x1² + y2 - y1², okay? So you can consider anyone as a point. You can consider this as x1 y1, this as x2 y2, whatever you want. Ok?

[00:23:51]So let me take this as x1 y1 let me take this as x2 y2. Ok? There you have it -7² and -3².

[00:24:01]Ok? Understand it again here. This is x2, y2, x1, y1 so x2 - x1 so - x1 is -3 we will put. This is minus the full and instead of x1, it's -3, okay? Then what is y2 - y1 - y1? 4² okay? So how much is this? -7 - +3² This becomes -3² So - 7 + 3 - 4² and - 3² This becomes 16 + 9 25 okay? And how much is 25? 5 units. So a' t' came to us as much as we had before ad was.

[00:24:49]That is 5. Okay? If you check it like this and keep checking it, it will be exactly the same. Let me take out one more and show it to you.

[00:24:57]Like d'. Ok? And here it is, this is m'. I will take it out and show you.

[00:25:04]You will notice the third one yourself, it will be the same.

[00:25:06]Let's say x1 y1 x2 y2 okay? Put the flower. You can tell anyone.

[00:25:14]First point, second point, I have told you this again and again. Ok?

[00:25:18]Put the flower in it.

[00:25:20]x2 - x1 is correct? So minus this formula and x1 is -7 so I wrote it like this plus y2 - y1 okay? So this is minus 9 minus minus plus 7 and this is 5 squared 6 - 1 5 squared, okay? This is the square of 3 you have and this is 25 so this is 9 + 25 okay sorry this is to minus 2 so -2 is the square of -2 that is 4 okay so 4 + 25 that is 29 so see minus 2 came out and square of plus came out and 4 came out and you got 29 so see this is what we had 29 so that means what have we noticed here? If you do the same thing here, you will get the same 40.

[00:26:23]Meaning, what did you not change because of the change in axis? There was no change in length.

[00:26:31]Ok? So we have a question related to this, let us do that.

[00:26:34]Think and reflect. So right here you have the first question of Think and Reflect.

[00:26:38]What has remained the same and what has changed with the reflection? What has remained the same and what has changed after reflection?

[00:26:47]What does reflection mean? Look, in the centre it is working as a mirror.

[00:26:51]If we have placed an object here, what do we see? His image is visible. Isn't it?

[00:26:57]You can see the back side inside the mirror. Ok? So what has been the case? It is being beans.

[00:27:02]Their shape has remained the same. Their length is the same. Isn't it? So we will write the same thing.

[00:27:13]Shape size.

[00:27:18]and the length of the sides of the triangle.

[00:27:27]Ok? The shape is the same, the size is the same, the length of the side of the triangle is the same.

[00:27:34]What has changed? Changed.

[00:27:38]Only its orientation has changed. Do you understand the orientation? It looks like this at first.

[00:27:43]Look, think for yourself. If I had a mirror here and I placed my hand here like this, then my hand would have come here like this. So, that means if the small finger is this close to it here, then this one is also this close to it. Isn't it? So what has changed? The orientation has changed.

[00:27:58]Ok? It is like this. It is like this.

[00:28:01]Look, the angle has changed like this. Ok? So what has changed here?

[00:28:06]Orientation.

[00:28:08]So we will right orientation of the figure. The orientation of the figure changed.

[00:28:18]Ok? The next question is asking would these observations be the same if triangle ADM is reflected in the X axis instead of the Y axis. Now, as we have reflected it along the y axis.

[00:28:30]Now if we leave this. Let us make this triangle here.

[00:28:35]We will make it downwards.

[00:28:38]So what will change? Will things remain the same? Will there be any difference? So think about what will happen?

[00:28:46]So I try to make it right here in front of you so that it can be easy. Look, this is zero. This is -1, -2, -3, -4, -5 and -6. Ok? Let's try to reach this level.

[00:29:03]So like this is the first point 3 4, now if I make this point below then where will 3 and 4 come? Minus will give 4. Isn't it?

[00:29:11]This will be the point. So this will be a dash this will be 3 - 4 okay? This is one point.

[00:29:19]What will be the point for d? Done for d is 7 and 1 one, right, of y, so one here will become -1.

[00:29:27]So this will be 7 and -1. So this is point d' 7 and -1 so the y coordinates are all coming in minus. Ok? And look, 9 and -6 will come out, m' m' is 9 and -6, which means x will remain the same. The y coordinate will change. Now we mix all these and make a triangle.

[00:29:52]it became.

[00:29:57]it became.

[00:30:04]Ok? So, this triangle has been formed.

[00:30:05]Absolutely. Its same reflection has been formed here.

[00:30:07]Now what will change and what will remain the same? What will remain of the beans? Again, if you apply the formula inside this, the length will remain exactly the same in a' t', inside this and inside this.

[00:30:18]The length will remain the same. It will remain the same size. Ok? It is the same shape.

[00:30:26]What has changed?

[00:30:30]Again, there is a difference in its orientation.

[00:30:34]Ok? If this is like this then this is like this.

[00:30:37]Plus, if we talk about other differences, what observations have you made? Have you observed what was happening here? Here the coordinates of the x axis were becoming negative. The y coordinates here are becoming negative. Isn't it? Look here, whatever coordinates were there, y was the same for all of them. Isn't it? All y's were the same but all x's became negative. What happened here? x are the same for everyone. Everyone's results became negative. Ok? So what would this observation be if it is reflected? Yes, it will remain the same. Also in the first case the x coordinates become negative and in the second case the y coordinates become negative.

[00:31:42]Ok? So within seconds it has become negative.

[00:31:44]Ok? So, this is what we have observed inside it.

[00:31:49]So this was your think and reflect and so thank you. That's all for this video.

[00:31:54]In this video we have covered Distance Between Two Points. And in the next video, we'll cover end of chapter exercises.

[00:32:02]Ok? So we will cover the last exercise of the chapter. 16 questions will cover it completely.

[00:32:10]So thank you. That's all this. That's all for this video. Stay tune for the next video. Thank you.

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