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AS Level Mathematics 9709 | Coordinate Geometry 2 | Ratio & Midpoint | Raja Shafqat — Transcript

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0:08Assalamu Alaikum, son. In our last
0:11class, we covered the first point of
0:13coordinate geometry, which was the
0:15distance formula and the ratio of
0:17collinear points. Points on the same
0:19line. To find their ratio, we use the
0:22difference of x-coordinates and the
0:24difference of y-coordinates. Today, we
0:26are finally starting the second point.
0:29This second point is named "Finding
0:31unknown point using ratio." What was it
0:33previously? Previously, we were given
0:36the points. Previously, we were given
0:39the points and we found the ratio.
0:41Alright, son? Previously, we were given
0:44the points and we found the ratio. Now,
0:46today will be the reverse approach,
0:48where the ratio will be given. And we
0:50will have to find the points. That is
0:52why its name is what it is. "Finding
0:54unknown point using ratio." Now, there
0:57are two or three options for this.
0:58First, let me explain it to you. Let's
1:00start with an example so the concept
1:02becomes clear first. Suppose you have
1:04this line here. I'll just randomly plot
1:08some points on the line. To explain the
1:10concept to you. Alright? Like this, I
1:13have marked three points on the line.
1:15Suppose you have this point here. Let's
1:18say this is point A, this is point B,
1:20and this is point P. Suppose the
1:22coordinates of point A are given as 2,
1:253 and the coordinates of point B are
1:27given as 12, 18. And point P is your
1:31unknown point. Meaning, all three of
1:33these points are on the line. Point A
1:35is already given. B is already given.
1:37And point P is unknown. And what is the
1:39examiner telling us? The examiner says
1:41that point P lies on line AB, given
1:44that point P lies on line AB. Point P
1:49lies on the line through A and B, or
1:53line AB, such that the ratio of AP and
1:59PB is 2: 3. This is the hint the
2:03examiner is giving you. He says that
2:05point A and B are given. Point P is
2:08unknown. Point P is unknown. The
2:11examiner says point P lies on the line
2:14through A and B such that the ratio of
2:17AP and PB is 2: 3, and the examiner
2:20says to find the coordinates of point
2:23P. So, what is our target? That we need
2:27to find the coordinates of this unknown
2:30or missing point. Clear, son? We need
2:33to find the x-coordinate and the
2:34y-coordinate. This is our target. We
2:36need to find the coordinates. Now,
2:39there are two or three options for this
2:41. It depends on the students. Let me
2:43tell you the first option. Let's start
2:46like O-Level Math. O-Level Math used to
2:48have a topic. It was called vectors. In
2:50vectors too, we used to study this
2:51concept. Let me guide you. Look, the
2:55very first thing, the very first thing
2:57you should note is what is the
2:59horizontal and vertical gap between
3:02points A and B? Meaning, if you notice,
3:05this x-coordinate is 2 and its
3:07x-coordinate is 12. What is the x-gap
3:09between 2 and 12? The x-gap between 2
3:11and 12 is 10. So the first thing I did
3:14was find the x-difference between A and
3:17B. What is the difference between the
3:19x-coordinates? And in the same way,
3:21find the difference between their
3:23y-coordinates. Now look, its
3:25y-coordinate is 3. Its y-coordinate is
3:2818. So what is the gap between these
3:30two? The vertical gap between these two
3:33is 15. Let's double-check, what is the
3:38gap between 2 and 12? 10, and what is
3:42the vertical gap between 3 and 18? The
3:46vertical gap is 15. So we have first
3:48found the x-difference and y-difference
3:50. Meaning these points A and B are
3:53forming a triangle. Between them, this
3:56x-gap is 10 and this y-gap is 15.
4:00Alright, now like O-Level Math, we know
4:02this point is on the line. What is your
4:05ratio? The ratio is 2: 3, so tell me
4:07what is the total ratio? If I ask you
4:10what the total ratio is? Then you would
4:13say the total ratio is five. The total
4:15ratio is five. Meaning the sum of the
4:17ratio is five. So it means I can say
4:19that point P is on the line. Point P is
4:22on the line. So if this total ratio is
4:24five. This is two and this is three.
4:26Then I can also say what is this ratio?
4:28What is AP? AP is two out of five of
4:31the whole line. So we used to read it
4:33like this in O-Level Math too. I can
4:35say AP is two-fifths of the whole line.
4:39AP is two-fifths of the line segment AB
4:41. Meaning if this ratio was two. This
4:44ratio was three, so it is 2 out of 5. 2
4:46of AB. Now what is our target? Our
4:50target is to get the x-gap and y-gap
4:52between A and B to find P. What is the
4:54X gap between A and P? And what is the
4:57Y gap? So, note one thing. This gap,
4:59this point was 2, 3. How would this
5:01point have come about? This point was 2
5:03, 3. So that means if you move 10
5:05forward from 2, it becomes 12. And if
5:08you move 15 upward from 3, it becomes
5:1018. Meaning, you are standing at this
5:13location, at 2, 3. If you move 10
5:16horizontally and 15 vertically, add 10
5:18to 2. You reached 12, and add 15 to 3,
5:23so this point of yours became 12, 18,
5:27meaning A to A2. From 2 you moved 10 to
5:30the right, it became 12, and from 3 you
5:33moved 15 up, it became 18. That is what
5:35this means, right? So now, what is the
5:37first thing you will do? This AP of
5:39yours, I have found out what AP is. AP
5:43is 2/5th of the whole line. So first, I
5:46will look at what the X difference of A
5:48and P is. What is the X gap between
5:51them? Between A and P, and likewise,
5:53what is the Y gap between A and P? What
5:57is the Y difference? Okay, if you want
6:00the X difference between A and P, you
6:02can say 2/5th of the whole X gap. So
6:04the gap that came between them was a
6:06difference of 4 units. Meaning, if this
6:09entire difference between A and B is 10
6:11, then what is this gap between A and P
6:13? It is 4. Similarly, if I say Y
6:16difference, then what will be the Y
6:18difference of A and B? 2/5th of the
6:20whole Y gap. 2/5th of the whole Y gap.
6:23Meaning, this entire gap is 15. So what
6:25is this? What is the gap? This gap is 2
6:27/5th of the whole Y gap. 2/5th of the
6:31whole Y gap. So what is the Y gap? It
6:33is 15. So that means what is the
6:35vertical gap? The vertical gap is 6
6:38units. What is the vertical gap? The
6:40vertical gap is 6 units. So, if I
6:43summarize this and explain it to you
6:45again, tell me, what were the
6:47coordinates of point A? The coordinates
6:50of point A were 2, 3; point P was
6:52unknown. Point P is unknown. Your
6:55target is basically P. So from here I
6:57found out that this gap, the gap
7:00between A and P, is 4 units. And the
7:02vertical gap between A and P is the
7:04vertical gap I found out, which is six
7:07units. So, we used basic math. We used
7:10the concept of ratio. Meaning, this gap
7:13here, the horizontal gap, was a gap of
7:15four units. This gap was 4 units. And
7:19the vertical gap was 6 units. So that
7:24means if you want to find P, you move 4
7:27to the right from two, and 6 up from
7:29three, so what coordinates for P did
7:31you find? 6. 9, these are the
7:36coordinates of point P we found, so
7:39today we used a reverse approach here.
7:42Clear, kid? Today we used a reverse
7:44approach here. We were already given A
7:46and B. Point P was unknown. Point P was
7:49unknown. Clear, kid? Suppose a student
7:52tells me, "Sir, I have a doubt about
7:54this method of yours," or "I want to
7:56verify P's coordinates." To check it.
7:59Suppose a student has a doubt about
8:01what I said, then how can you verify it
8:03? You already had the A coordinate.
8:05What were the coordinates of A? The
8:07coordinates of A were 2, 3. What
8:09coordinates do you have for P? You have
8:12the coordinates for P as 6, 9; this is
8:14what we just found. And what are your B
8:17coordinates? The coordinates for B are
8:2012, 18, so in the last class, I told
8:22you the ratio trick. Suppose you have a
8:24doubt about my working. Then what is
8:26your target? You have to check it. So,
8:29I will show you by checking the ratio
8:30of AP and PB. You tell me, what is the
8:33X gap between A and P? Look again, kid.
8:36Exactly the same work we did in the
8:38last class. Look at the X gap of A and
8:40P, and similarly, the X gap of P and B.
8:43Look closely at what the gap is between
8:46them. Look, now if I note from here,
8:48what is the gap between A and P? This
8:51is two. This is six. This is 12. This
8:53is two. This is six. This is 12. So
8:55this gap is four. This gap is six. So
8:57that means this gap is four. This gap
8:59is six. So what is the x difference
9:01between them? 4: 6, so what ratio was
9:03found? 2: 3, so what did I do? I used
9:07the same concept. Difference of x
9:09coordinate. I explained in the last
9:11class that if the points are collinear,
9:12you can look at the difference of the x
9:14coordinates. That will also confirm
9:16what your ratio is. See, this is two.
9:18This is six. This was two, x; this was
9:21six. This was 12. Or you could do it
9:22this way, this was two. This was six.
9:24This was 12. So the gap between them is
9:26four. And the gap between these is six.
9:28So what is the ratio? 2: 3. Similarly,
9:31if you remember, we can also do it this
9:33way. Let's look at the difference of
9:35y-coordinates. Difference of
9:37y-coordinates. Now tell me, we studied
9:39in the last class that this is AP and
9:41this is PB. We need the ratio of its
9:43y-coordinate. Look, this y is 3. This y
9:46is 9. This y is 18. Look, how much is
9:48the gap between three and nine? It is
9:50six. And how much is the gap between
9:52nine and 18? It is nine. Look again.
9:54This y is three. This y is six. This y
9:56is 18. This is three. This is six. This
9:58is 18. The vertical gap between these
10:00two is three. And the vertical gap
10:02between these two, look at the vertical
10:04gap between these two again. So this 3,
10:079, 18, this gap of 3 and 9 is six, and
10:11this gap of 9 and 18 is nine. So if I
10:15show this as a ratio, I will say 6: 9,
10:17so again what is the ratio? Again the
10:20ratio is 2: 3, so you can also check my
10:22work. So this is a very basic concept
10:25of ratio. Like we used to study in
10:28O-Level Math that some amount has to be
10:29divided between two children. Suppose
10:31₹ 100 has to be divided between two
10:33children. Dividing ₹ 100 between two
10:36students. The ratio of 2: 3, so in
10:38O-Level Math we used to say, how will
10:40A's share come? The total ratio is five
10:42. So A's share will be 2/5th of the
10:45whole amount. So we will say ₹ 40. If
10:49I ask, how will B's share come? Then in
10:51O-Level Math we used to say 3/5th of
10:54the whole amount. So from here the
10:55answer will come to ₹ 60. So it means
10:58that what will be the total amount? ₹
11:00100, so this is basically the exact
11:02same concept that this ratio was 2: 3.
11:04So what did the total ratio come to?
11:06The total ratio came to five. So what
11:08is this ratio? This is 2/5th of the
11:11whole line. And what is this? 3/5th of
11:13the whole line. 2/5th of the whole line
11:16. 3/5th of the whole line. So the same
11:19result was derived. Clear, beta?
11:21Finally, what result was derived for
11:22you? The same result was derived. Clear
11:25, beta? We studied a simple concept.
11:28This concept is called finding unknown
11:30point using ratio. The reverse approach
11:32is that the points are given to you and
11:35the ratio is given. The ratio is 2: 3
11:37and what was your target? Your target
11:40was to find out the unknown point. Is
11:43that clear, son? To find out the
11:44unknown point. This was our basic
11:46target. Okay? So, I explained one more
11:49small point to you today. Let me tell
11:52you something new. There could be
11:54another method to solve this same thing
11:57. Okay? I will explain the same thing
11:59to you again. Look here. Come on, I
12:01copied the same thing again. Okay?
12:05Copied the same thing again. What was
12:07your target? Your target was the ratio.
12:10Rather, your target was point P. Please
12:12look again. What did we study
12:14previously? What were the coordinates
12:16of point A given? The coordinates of
12:17point A were given, 2: 3, and what hint
12:20were you given about the ratio of AP
12:22and PB? The ratio of AP and PB is 2: 3;
12:25the ratio of AP and PB is 2: 3, that
12:27was your target and the examiner was
12:29asking you to find point P. This was
12:32your target. Meaning you had to find
12:35the coordinates of point P. Okay? You
12:38had to find the coordinates of point P.
12:41This was your basic target. Alright,
12:44now one more option is also possible. I
12:46will have you do this with an even
12:48simpler method. Okay? Come, look again
12:50at what the gap is between their
12:52x-coordinates. Look, what is this ratio
12:54? It's a 2: 3 ratio, right? The total
12:56ratio is five parts. Come, let me
12:58repeat one thing again. We know this
13:00fact, what is the x-gap between A and
13:03PB? We already know this fact that if
13:05this was two. This was your 12, so what
13:07is this gap between two and 12? 10. So,
13:10we already knew the x-difference was 10
13:12. Similarly, this is three, this is 18.
13:15So, the vertical gap between these two
13:17was 15. I showed you this in the
13:20previous question too, that the
13:21y-difference between these two was 15.
13:24Clear, son? Now come, let's try
13:26something new. Our target is this. Our
13:29target is that we have to find point P,
13:31what point P is. Come, now I use the
13:33ratio method. Like the method we used
13:35to study in O-Level math. Unitary
13:37method or ratio method? Unitary or
13:39ratio method. It is O-Level math. We
13:41must have studied the ratio method or
13:43unitary method very frequently. Answer
13:45me one thing. Let me first show you how
13:48to find the x-coordinate. Meaning,
13:50first I will show you how to find the
13:52x-coordinate of P. Let me mark here
13:54that I am first finding the
13:56x-coordinate of point P. So here I am
14:00writing the ratio and here I am writing
14:02the difference. Ratio and difference.
14:05Tell me, what is this entire ratio?
14:08Five times. Brother, look at these two,
14:10this is two. This is three. So what is
14:12the total ratio? Five times. So tell me
14:14, does the five times ratio show the
14:17x-difference? The five times ratio
14:20represents 10, brother, the difference
14:22between a and b is 10, and what is the
14:25ratio? Five times represents 10. So now
14:28the question is, what will two times
14:30represent? Tell me one thing. Now, if I
14:34tell you that this total ratio is five,
14:36then five times means an x-difference
14:38of 10. So now, what does this two times
14:42ratio mean? Here you go. Now let me see
14:45what the two times ratio means. Two
14:47times means this gap. Two times means
14:51this gap. What is this gap? x-2,
14:53obviously if this is two and this is x,
14:55then what is the gap between them? x-2,
14:58so what will two times show? Two times
15:01will show x-2, is that clear, son? Now
15:03you cross-multiply this. After
15:06cross-multiplying, your question comes
15:08into this form. Now when you simplify
15:12this, you find out what your
15:14x-coordinate is. Your x-coordinate is
15:18directly found to be six. There you go,
15:20your x-coordinate is directly found to
15:22be six. In the previous question, we
15:24got the same answer. Look, what was the
15:26answer previously? Six and nine. The
15:28previous method was the same thing.
15:30This was a bit quicker. What was the
15:32answer from this as well? The same, six
15:34derived. The same answer was derived
15:36from this too. Similarly, if I tell you
15:39my target is the y-coordinate of P, now
15:41I need the y-coordinate of P. I need
15:43this too. First I needed the
15:45x-coordinate, now I need the
15:46y-coordinate. So use the same approach
15:48again. Same approach. What is the same
15:50approach? Ratio and difference. Tell me
15:55one thing. This total ratio is five
15:58times. So what is the vertical
16:00difference? Five times represents 15.
16:02Exactly the same concept. Now the five
16:05times ratio represents the vertical
16:07difference of 15. Brother, five times
16:10represents it. What is the vertical gap
16:11between the two? 15, now what is the
16:14vertical gap between these two? Between
16:16these two, if this is 3 and this is y,
16:17then what is the vertical gap between
16:19them? y-3, just like what was the
16:21vertical gap between these two? 18-3
16:24was 15. So now I need the vertical gap
16:28between them. Look at this. So now I am
16:30checking, what vertical gap will two
16:31times represent? Two times will
16:34represent y-3. Clear, kid? Now if I
16:37cross-multiply, 5 times y-3 equals 2
16:40times 15. Now it's your basic math, 5*3
16:44, so what is the answer for y derived
16:46from here? Nine. So this means you
16:49found out directly what the
16:51y-coordinate of p is. Nine. What is the
16:55y-coordinate of p? The y-coordinate of
16:57p is nine. So, what coordinates did you
16:59find directly? The coordinates were
17:01found directly. 9 6 Clear, kid? Sorry,
17:086 9, so what did we use here? We used a
17:11simple ratio method. Look, what is the
17:15x gap between 2 and 12? It is 10.
17:18Meaning, 5 ratio represents the x gap
17:21of 10. So what x gap will 2 ratio
17:23represent? The x difference between
17:26them is x-2, there is the x. The
17:29vertical gap between 18 and 3 is 15. So
17:32five times represents the vertical gap
17:35of 15. So what vertical gap does two
17:37times represent? This vertical gap is y
17:39-3, so what did we use? We used the
17:43unitary method. Used the unitary method
17:47. Clear, kid? The name of this simple
17:50approach is the unitary method. Now I
17:52am showing you a new option. Option
17:53three for the same question. Now it's
17:56up to you to use any of these three
17:58options in the exam. I am telling you a
18:01third option. In the third option, I
18:04will tell you a small formula. The
18:06third option is called using a formula.
18:09The third option is called using a
18:11formula. Repeating the same concept for
18:13the third time. But this time, I am
18:15using a formula. Look carefully at what
18:17the third concept is. Math says,
18:19suppose you have a line here. This is a
18:23point on the line. Its coordinates are,
18:25suppose x1 y1, take this, this is a
18:27point on the line. Its coordinates are
18:30x1 y1, this point is. Coordinates are
18:32x1 y1 and this is a point, its
18:34coordinates are x2 and y2. x2 and y2,
18:39there it is. And your unknown point,
18:42let's suppose, is somewhere here. The
18:45coordinates of that unknown point are (
18:46x, y). This is your unknown point, and
18:50the ratio you were given, the hint
18:52given to you in the previous part, was
18:54the ratio of AP and PB. I was
18:57considering that ratio as R1: R2, so
19:00let's consider the ratio as R1: R2,
19:02which was 2: 3 in the previous question
19:04. I will express it in the form of R1
19:07and R2. Now I am telling you a small
19:10trick. Now, the trick is this. You will
19:12have to focus a little bit. First, see
19:14the trick that I am showing you. Then,
19:17obviously, I will give you a proper
19:19example of it. If your unknown point is
19:21P, then how will the unknown point come
19:23? Ratio*opposite x-coordinate. Ratio*
19:27opposite x, Ratio*opposite x, similarly
19:32, ratio multiplied by opposite x.
19:35Brother, the ratio's own x is right
19:37here, the ratio's own x is right here.
19:40So what did I say? Ratio*opposite x,
19:42again ratio multiplied by opposite x,
19:47Ratio multiplied by opposite x. Listen
19:50again, the ratio was multiplied by the
19:52opposite x, okay? I will repeat it once
19:55more. Now look again, a bit slowly,
20:00ratio multiplied by opposite x, this
20:04got multiplied by this, plus ratio*
20:07opposite x, ratio*opposite x, r2*x1
20:11divided by sum of the ratio, divided by
20:15sum of the ratio. Similarly, ratio
20:17multiplied by opposite y, Ratio
20:23multiplied by opposite y, again ratio
20:26multiplied by opposite y, again ratio
20:29multiplied by opposite y divided by sum
20:33of the ratio, divided by sum of the
20:36ratio. Now, whoever has a doubt, I will
20:39explain it, okay? Now just watch for
20:41two minutes. Now I am showing you this
20:43same question again, this same question
20:45, okay? We saw the question 2 minutes
20:48ago, I showed you how to solve it twice
20:51, twice. Now I am solving it again for
20:54the third option, I am trying it for
20:56the third time through this trick. Look
20:58, what were the coordinates of your
21:00point A? The coordinates of point A
21:03were 2, 3, your point P was the unknown
21:05point X, Y, and your point B had
21:08coordinates 12, 18. I am trying the
21:10same question again. And what was the
21:13ratio? The ratio was 2: 3. What was the
21:16ratio? The ratio is 2: 3. Now look,
21:18what trick did I tell you? I told you a
21:21trick that if you require coordinates
21:23directly, then what will you do? Please
21:25look at this carefully. Here, what will
21:28you do? Ratio*opposite x, ratio*
21:32opposite x, again ratio*opposite x,
21:35first you need the x coordinate, right?
21:39So, first you will deal with the x
21:41coordinates. Divided by the sum of the
21:43ratios. Now you need the y coordinate.
21:47Ratio*opposite y. Ratio*opposite y plus
21:52ratio*opposite y, ratio*opposite y
21:55divided by the sum of the ratios, now
21:59simplify this, it becomes 24 + 6 = 30/5
22:02, the answer is six, so what is the x
22:06coordinate of p? It is six. What is the
22:10x coordinate of p? It is six. Again 18*
22:142 = 36, 3*3 = 9/5, 45/5, so what is the
22:19y coordinate of p? Nine, so this is the
22:24direct approach. This is called using
22:27the formula. You used the direct
22:29approach here. Using the formula. So,
22:32today I solved one question for you
22:34using three different methods. The
22:36first two methods—by the way, the
22:38best option was the first method. The
22:40second option is this one. This is
22:42called the unitary method. Unitary
22:44method. The third approach is called
22:46using the formula. So, the small point
22:50we studied today, the name of this
22:52point was finding an unknown point
22:54using a given ratio. Meaning, usually
22:57what happened with the examiner is that
22:59in the last class we studied that
23:01points were given and we had to find
23:03the ratio. Now, what about today? The
23:05ratio was given and one point was
23:07unknown which we had to find. So, I
23:09explained that one small point to you
23:12using three different methods. Clear,
23:14kid? We used three different methods to
23:17check that one point. Done. Now, God
23:21willing, you will revise this once. Now
23:23, apply whichever method you find
23:25easiest. By the way, the easiest method
23:28is the first, then the second, and then
23:30the third. The third is a direct
23:32formula-based method. Meaning if you
23:34need the x coordinate, then ratio*
23:36opposite x, ratio*opposite x divided by
23:38the sum of the ratios. The same process
23:40is for the y coordinate. Ratio*opposite
23:42y plus ratio*opposite y divided by the
23:45sum of the ratios. Clear, kid? So,
23:48these are the two points we studied. In
23:49the same style. I tried the question
23:51three times. The same question, but I
23:53used a different approach all three
23:55times. Come on, now the third point of
23:58this topic. We call this point the
24:00midpoint. Middle point or midpoint.
24:02Okay? We must have studied this in
24:04O-Level Math as well. Come on, let's
24:06get started. This point was also in
24:08O-Level Math. Suppose you have a point
24:11on the line. And another point on the
24:13line segment is given. Here is B. Let's
24:16name this point. A, its coordinates are
24:18X1, Y1. This is a point on the line.
24:20Let's set its coordinates. X2 Y2, X2
24:23and Y2, here it is. Now, what is your
24:26target? You need to find the point that
24:29divides this given line segment into
24:32two equal parts. What does the middle
24:34point mean? The middle point means the
24:38point that divides your line segment,
24:41your given line segment, into...Divide
24:44it into two equal parts. The one that
24:48divides your given line segment into
24:50two equal parts. Meaning, if this is
24:52your A and this is your B, then what do
24:54you have to do? Divide this line
24:56segment into two equal parts. So, what
24:58does a middle point do? Midpoint
25:00technically means the middle point.
25:03Middle point. Meaning, the point that
25:06is exactly in the center of your two
25:08given points. So, what did we study in
25:10O-Level Math? We studied in O-Level
25:12Math that if you need to find the
25:14midpoint, call it middle point or
25:16midpoint. What formula did we use for
25:18the midpoint in O-Level Math? Average
25:21of x-coordinates. Add the x-coordinates
25:24and divide by two. Similarly, what did
25:26we study in O-Level Math? Average of
25:28y-coordinates. Add the y-coordinates
25:30and divide by two. This was the formula
25:33every student studies in O-Level Math.
25:36Meaning, can we also say what a
25:38midpoint is? Average. Average of what?
25:43Average of x-coordinates, add both x's
25:45and divide by 2. Add both y's and
25:48divide by 2. So, this is what we have
25:50studied in O-Level Math. The name of
25:51this concept is midpoint. Midpoint,
25:54middle point, midpoint. Okay, look at a
25:56third-class example. Then I will tell
25:58you something new. See, suppose you
26:01have this line. I am just giving a
26:02random example. Suppose you have a line
26:05and you have a point 2, 3, let's say
26:07this point on the line is already given
26:10. The point doesn't matter, only your
26:12concept is important. The concept
26:14matters. Point A is 2, 3 and another
26:16point is given. Suppose you have one
26:19more point. Let's say the point is 10,
26:2115 and 10, 15, and your target is to
26:24find the middle point. Midpoint. Okay?
26:27The examiner is asking us to find the
26:29coordinates of the middle point.
26:30Midpoint. Your target is to find this
26:33specific point. The middle point
26:35divides your given line into two equal
26:38parts. So, we call this point the
26:40mid-point or middle point. Okay?
26:43Mid-point or middle point. So, what
26:45have we been doing? How have we been
26:46finding the middle point? Mid-point. We
26:48learned in O Level math that you just
26:50apply a simple formula. The same
26:53formula I explained just two minutes
26:54ago. What did we learn in O Level math?
26:57If you need the middle point, what do
26:58you do? You add both the x values.
27:01Divide by 2, and there is your
27:02mid-point. Similarly, you add both the
27:05y values and divide by what? Two, and
27:07there is your middle point. Done.
27:10Meaning, mid-point means average. So,
27:12when I check the coordinates of the
27:14mid-point, what are the mid-point
27:15coordinates? 6, 9, those are your
27:18mid-point coordinates. The mid-point
27:22coordinates are found. Clear?
27:24Technically, mid-point means the ratio
27:27is 1: 1. Obviously, mid means the gap
27:29here is exactly the same as the gap
27:31there. Mid-point means the same gap. So
27:34, this means its mid-point has been
27:36found. The mid-point of this line has
27:39been found as 6, 9, which is your
27:42line's mid-point. All right, son? This
27:44is your line's mid-point. This is the
27:46mid-point of the line segment, by the
27:48way. So that means, your point A was 2,
27:523 and your point B was 10, 15, and what
27:55did you just find? You found the middle
27:59point, meaning you found the mid-point
28:02that divides this line into two equal
28:04parts. There you go, you found the
28:07middle point. What is the mid-point?
28:09Your mid-point came out to be 6, 9,
28:13okay? Mid-point, what does the
28:15mid-point do? It divides your line into
28:17two equal halves or parts, that's what
28:18a mid-point means, right? Does anyone
28:20have any doubts? Let's check. Now, look
28:22, if my mid-point is correct, then the
28:25ratio must be 1: 1. Now tell me one
28:27thing, okay, look, this is two, this is
28:30six. This is 10. 2, 6, 10, how much is
28:33this gap? Four, so then this gap is
28:35also four. So if I look at the
28:36difference of their x-coordinates, what
28:38is the ratio? 1, that's it. Now look,
28:41this is three. This is nine. This is 15
28:43. 3, 9, 15, what is their vertical gap?
28:46Six. So then this vertical gap is also
28:48six. So what is the ratio? 6: 6, 1. So
28:52by the way, we simply found the
28:54midpoint. Done. If you have any doubt,
28:56look, what is the midpoint? Whatever
28:58the gap is from this to this, look, the
29:00gap between two and six is what? Four,
29:01so then this gap is also four. Done. If
29:04the vertical gap from three to nine is
29:06six, then the vertical gap from nine to
29:0915 is also six. That's it. So what does
29:11midpoint mean? Midpoint means it's the
29:13same gap. Clear, kid? So I told you a
29:16small thing. It's nothing extraordinary
29:18. You must have studied all this in
29:20O-Level Math. The name of this point
29:22was midpoint. So if the horizontal gap
29:25between A and M is four, then this gap
29:27is also four. That's it. It's the same.
29:30Similarly, if the vertical gap between
29:32three and nine is six, then this gap is
29:34also six. So it means that the
29:35horizontal gap and vertical gap will be
29:37? The same. Clear? Meaning this run and
29:41this rise, this run and this rise are
29:43both the same. Run, rise, run, rise are
29:46the same, by the way. So, anyway, this
29:48is a very average thing, the midpoint.
29:50Now, look, I'm telling you something
29:51new. That was still an easy approach.
29:53Now, when a reverse question comes in
29:56an exam, a weak student wastes a bit of
29:59time there. Look, I'll give you a small
30:01example. A small one. Okay? I'll give
30:03you a small example. Then I'll tell you
30:05something new. Okay? Suppose a reverse
30:08question comes in the exam. Look at how
30:10the reverse question comes. I will
30:13guide you. Look, suppose a question
30:15came in the exam, suppose you already
30:18have the coordinates of A, 2, 3, here
30:20you go. You are already given the
30:22coordinates of A. And suppose you are
30:24already given the midpoint M. Here you
30:26go. The midpoint is 6, 9 and the
30:28examiner is saying find the coordinates
30:30of point B. This is the statement. The
30:33examiner is saying find the coordinates
30:35of point B. Find the coordinates of
30:37point B. This is the examiner's
30:39statement. Meaning this time you have
30:41point A. The midpoint is already there,
30:43B is not. So, there are two types of
30:45students. The student with a very
30:47average mindset, what will they do?
30:50They will apply the typical formula.
30:52And what will a slightly smarter
30:54student do? I will explain that after
30:56this. See, what will an average student
30:59do? They will use the midpoint formula.
31:02Okay? That student will say, hey, the
31:04same formula we used for the midpoint
31:06earlier, let's use it again. Tell me,
31:08how would this midpoint have come about
31:09? x1 + x2/2, so how would the midpoint
31:12have come about? Add the x-coordinates
31:15of both and divide by 2. Similarly, add
31:18the y-coordinates of both and divide by
31:222. The student will say, brother,
31:24that’s how you get the midpoint. So
31:25what did they do? The student found the
31:28midpoint and equated the midpoint to
31:30the midpoint. This approach is called
31:33equating midpoints. So what did the
31:35student do? Applied the formula and
31:37equated the midpoint to the midpoint.
31:39Okay? In this, what did we do? We
31:41simply used the formula. So what did we
31:44do? We used the concept of equating
31:46midpoints. We equated the midpoints.
31:49Equating midpoints, meaning we equated
31:51the midpoints. Applied the formula
31:52simply. What does this approach mean?
31:55This approach means what did the
31:57student do? The student simply used the
31:59formula. The student simply applied the
32:01formula. Like how we do it normally.
32:03Then what will the student do? They
32:05will equate the midpoint's x to x. The
32:07student will say, hey, we already have
32:09the midpoint's x-coordinate. Similarly,
32:10the student will say, we already have
32:12the midpoint's y-coordinate too. So
32:14what did the student do? Equated x to x
32:17and y to y. Take it, what will the
32:20student do? Will equate x to x. Will
32:22equate y to y. Then what will the
32:24student say? The student will, how will
32:26they solve this? The student will say,
32:28what about the midpoint? Equate x to x.
32:30Here is the midpoint, what? Will equate
32:32x to x. 2*6 = 12, so this becomes 2 + x
32:34= 12, so the student will say, what is
32:36the answer for x found here? The answer
32:39for x is found to be 10. This is your
32:41x-coordinate. The mid-point x, this
32:43gives you the x-coordinate of point B.
32:45The x-coordinate of point B is found.
32:47Then, what will the student do? The
32:49student will equate the y-coordinate to
32:51y. 3 + y/2, 3 + y/2, will equate the
32:55mid-point to the mid-point. Here it is.
32:58So then what happened? 3 + y = what
33:00happened? 9*2 = 18, so from here we
33:02found out what the answer for y is? The
33:05answer for y is 15. That's it. This is
33:07a very average approach. This is a very
33:10average method. A very average approach
33:13. Technically, there is no mistake in
33:16this. The student will say they equated
33:18mid-point to mid-point. Equated
33:19mid-point to mid-point, mid-point to
33:21mid-point. Meaning, equated x to x and
33:24y-coordinate to y. Now, if you have
33:28common sense, this is just a one-step
33:30topic. If you have common sense, this
33:33can be solved in just one step. Okay?
33:35The second option means common sense.
33:38Common sense. Now look, I will tell you
33:41how I can solve the same question with
33:43the second option. I tried the same
33:45question again. Here is the same
33:47question. Look, I am trying this
33:49question again in just one minute. Okay
33:52, so tell me one thing. Mid means an
33:54equal gap. What does it even mean?
33:56Equal. So give me an answer to one
33:58thing. You can solve this in one step.
34:00How? Tell me, what is the horizontal
34:03gap between two and six? You will say
34:06four. And similarly, if I ask you what
34:09the vertical gap is between three and
34:11nine? You will say six. This is nine,
34:13this is three. What is the vertical gap
34:14? Between three and nine, you will say
34:16six. So this means that this is four
34:18and six. Finished. So if this gap is
34:21four and this is six, then this must
34:23also be four and six, right? It must be
34:26four and six. Here it is. So this gap
34:28here will also be what? It will be four
34:30and six. So what do you have to do?
34:32Move four units ahead from six. And
34:37move six units up from nine. So tell me
34:40, what are the coordinates of B? The
34:42coordinates of B are 6 + 4 and 9 + 6,
34:45so tell me, what are the coordinates of
34:47B found? 10, 15, what are the
34:50coordinates of B found? The coordinates
34:53of B are found. 10 15 exactly the same
34:56answer, try 10 15 10 15 let's make it
35:01even easier, brother. What is the gap
35:03from two to six? Four, four was added.
35:06You add four as well. So 6 + 4 is 10,
35:09done. Three became nine. How much did
35:11it increase? It increased by six. You
35:13increase it by six as well. 9 + 6 is 15
35:15, that’s the method. Look again, the
35:19gap between 2 and 6 is four, so the
35:21next gap should also be four. 6 + 4 is
35:2410, done. Look at the next one. Look at
35:26the y gap. The gap between three and
35:28nine is six. So the next gap must also
35:30be six. Meaning what does this become?
35:3115. That's it. It’s a matter of
35:34seconds. So the midpoint means whatever
35:36is happening, you do the same thing. x
35:38is increasing. x went from two to six.
35:40Four increase. You also increase it by
35:42four. It became 10. What happened to y?
35:44y went from three to nine. It increased
35:46by six. You increase it by six more as
35:47well. It became 15. That's it. Finished
35:50. So instead of using that approach,
35:52the best option is to do exactly what
35:54the x difference is. If it's adding,
35:56then you add. If it's subtracting, then
35:58you subtract. Meaning we used a simple
36:00midpoint approach. Whatever is
36:02happening, you apply the same behavior
36:04or relationship. Right, son? If it is
36:06adding, you add. If it is subtracting,
36:08you subtract. End of story. Let me show
36:10you one more small example. So it was a
36:13small concept. Sometimes things are so
36:17small that you shouldn't waste time
36:19using a long approach. Clear, son? Some
36:22things can be solved in one or two
36:24steps. So you solve it in just one or
36:26two steps. Clear, son? Okay, so that
36:29was one approach. Let me show you one
36:31more example, note it down for two
36:32minutes. Okay? Now look, note it down
36:34for two minutes. Suppose you have this
36:36situation. This point came in the exam.
36:38I am showing you one more random
36:40example. Suppose this is your point,
36:42point A. Its coordinates are 3 10, take
36:44this or let's say 2 10, take any
36:46numerical value, obviously when we do
36:48past papers these questions will be
36:50used there. All concepts will be
36:52applied. Point is already given.
36:54Midpoint is already given. You are
36:57already given the midpoint as well. The
36:59examiner says its midpoint is, suppose,
37:015, 4. Here it is. And the examiner
37:04tells you to find the coordinates of
37:05point B. Here you go. The examiner
37:07tells you to find the coordinates of
37:10point B; that’s your target. Done.
37:12This is your target. Taking a random
37:14example, a smart student would say,
37:16from two to five. X increased by three.
37:18Increase it by another three. Done. The
37:21student will say this is two. This is
37:23five. This is X. This is two. This is
37:25five. This is X. Two became five.
37:26Increased by three. Increase it by
37:28another three. So that means, what are
37:30the coordinates of B? The X coordinate
37:32of B? Eight. Interesting point. 10
37:34became four. It decreased by six, so
37:37decrease it by another six. So if you
37:39subtract six from four, what answer do
37:42you get? The answer derived is -2.
37:44That’s it. This is how you find B.
37:47Meaning, if it is adding, you add, and
37:49if it is subtracting, you subtract.
37:51Look again. If you note for a minute,
37:53what was happening here? If you note
37:55again for a minute. Look, two became
37:58five, increased by three; 10 became
38:00four, decreased by six. So you do the
38:03same. Decrease by six and increase x by
38:06three. Meaning, two became five,
38:08increased by three, so you also add
38:10three. It became eight. 10 became four,
38:13decreased by six, so you also decrease
38:16by six. 4-6 is what? -2. That’s it.
38:19So, whatever is happening, you do the
38:21same thing. Look here. Two became six,
38:23increased by four, so you also increase
38:25by four. It became ten. Three became
38:27nine, increased by six, so you also
38:29increase by six. It became fifteen. Two
38:32became five, increased by three; you
38:33also increase by three, it became eight
38:35. 10 became four, decreased by six; you
38:38also decrease by six, it became -2.
38:40That’s it. So, there is no technical
38:42complexity in the midpoint. By the way,
38:44there is no technical complexity in the
38:46midpoint, clear kids? Now I am telling
38:48you something new, this is the most
38:50important thing. Then I will show you
38:52what kind of questions appear in the
38:54exam. Okay kids, what kind of questions
38:57appear in the exam. So this is just a
38:59basic concept of midpoints that we use
39:01in O-Level Math as well. Look at this
39:03new thing. Now, this concept is called
39:06Properties of 2D Shapes. Our syllabus
39:09requires properties of certain 2D
39:12shapes. You have just studied the point
39:15called the midpoint. So, I will also
39:17talk about the midpoint. Two minutes.
39:18We have studied the midpoint, so we
39:20will only talk about the midpoint. Okay
39:22, now look, I am listing some shapes.
39:24Math says there are four such shapes in
39:27math. Four shapes that share a common
39:31property. Common means it exists in all
39:33of them. There is one common property
39:36in four shapes. Right? So, first I will
39:39tell you which those four shapes are.
39:41There is a common property. First of
39:44all, who are those four shapes? Listen,
39:46then I will guide you. The first shape
39:50among those four is a square. The
39:52second shape is a rhombus. Rhombus. The
39:57third shape is a rectangle. Right? The
40:00third shape is a rectangle. The name of
40:03the third shape is a rectangle. And the
40:06fourth shape is a parallelogram.
40:09Parallelogram. Now we know these four
40:12are quadrilaterals. By the way. If we
40:14have the sense, who are these four? All
40:16four of these are quadrilaterals.
40:18Obviously, these four are
40:19quadrilaterals. So, there is a common
40:21property in these four quadrilaterals.
40:23Obviously, these four are
40:25quadrilaterals. We must have at least
40:26that much sense. So, these
40:28quadrilaterals—or quad, meaning
40:30four-sided figures—have a common
40:32property. What is that common property?
40:35Let's see. Math says that the common
40:37property is that the midpoints of the
40:39diagonals are equal. Look, let me
40:41explain this in more detail. What is
40:43the common property? The common
40:45property is this. Let me show you in
40:47detail. Midpoint of diagonal AC.
40:53Midpoint of diagonal AC is equal to the
41:00midpoint of diagonal BD. Now I will
41:07explain this in detail. Suppose a
41:09student has a doubt, I will guide them
41:11on what it means. That is, math says if
41:13one diagonal is AC and the other is BD,
41:15then what will their midpoints be?
41:16Always the same. By the way, if we have
41:18the sense, what does a diagonal even
41:20mean? It is the line that does what to
41:21opposite corners? Joins them. Now look,
41:24I will guide you one by one. Right?
41:26Come, look first so that if any student
41:28has a doubt, they can see it well once
41:31more. Number one, first we are talking
41:34about the square. Right? First, we are
41:36talking about the square. Look, you
41:37have already studied all of this in
41:39O-Level Maths. You have studied all of
41:42this in O-Level Maths. Look, what is
41:43the property of a square? Look, we all
41:45know what a square is, right? There is
41:48nothing very complex about this.
41:50Technically, suppose this shape is a
41:52square. So, in O-Level Maths, we taught
41:55you that the midpoints of its diagonals
41:57are always the same. In a square, the
42:02diagonals—meaning diagonal AC and
42:05diagonal BD—will always share the
42:08same midpoint. It’s a simple thing,
42:11by the way. So what? The midpoint of AC
42:14is the same as the midpoint of BD. Done
42:17. So, that is the first property. It
42:19has a common midpoint. Look, what did I
42:20write? Midpoint of diagonal AC =
42:22Midpoint of diagonal BD. The name of
42:25this shape is a square. Similarly,
42:27another shape also has this same
42:29property. The name of this shape is a
42:31rhombus. The name of this shape is a
42:33rhombus. By the way, you must have
42:34studied this in O-Level Maths too. The
42:36topic was Coordinate Geometry. Rhombus.
42:39Anyway, if there’s a studious kid
42:41here, what does the word rhombus mean?
42:43A rhombus means a tilted square. I mean
42:45, if I explain it in very simple
42:47language, what is a rhombus? You would
42:49say, what is a rhombus? A tilted square
42:51. A square that is tilted. What is
42:54called a rhombus? A rhombus is called a
42:57tilted square. A square that is tilted.
42:59Look, a square that is tilted—meaning
43:01a tilted square—is one where all four
43:03sides are equal. There you go. Now see,
43:05what does that mean? Well, that happens
43:06in a square too, that all sides are
43:08equal. You all know that. Similarly,
43:11the same happens here. It happens in a
43:13square and a rhombus too, but the
43:15difference is that a rhombus is tilted.
43:17I mean, if we say it in simple language
43:19, a rhombus's corners are not 90
43:21degrees. And what are the corners of a
43:24square? 90 degrees. Anyway, right now
43:26we are not interested in the corners.
43:29Nor in any parallel lines. We are only
43:31interested in seeing that its midpoint
43:34is the same. So, it has a common
43:36midpoint, so the midpoint of a square
43:39and the midpoint of a rhombus are
43:41always...The same. There you go. So
43:43right now, I am just telling you the
43:45properties. What was the name of this
43:46shape? The name of this shape was a
43:48rhombus. Here is your rhombus ABCD.
43:51Clear? So, a common property in squares
43:54and rhombuses. A common property in
43:56squares and rhombuses. Look at another
43:57shape. We must have studied this in
43:59O-Level Math too. The third shape is
44:01named a rectangle. The third shape is
44:04named a rectangle. There is also a
44:06common property in a rectangle. There
44:09is also a common property in a
44:10rectangle. There is also a common
44:12property in a rectangle. What is the
44:14common property in a rectangle? That
44:16the midpoint of AC is the same as that
44:17of BD. And look, the same common
44:19property in all of them. Isn't that
44:20right? So, it is the same property that
44:22I am repeating. We have studied this in
44:24O-Level Math as well. By the way. So
44:26the midpoint of AC will be the same as
44:29the midpoint of BD. Meaning, both will
44:31have a midpoint; obviously, the
44:33coordinates of A and C will be
44:34different. Obviously, the coordinates
44:36of B and D will be different. But what
44:37about the midpoint of both? It will be
44:40the same. So in a square, rhombus,
44:42rectangle, and one more shape from
44:44O-Level Math. The name of this shape
44:47was parallelogram. It has the same
44:49common property as well. The name of
44:51the shape was parallelogram. Right? We
44:54must have studied this same property in
44:55O-Level Math too. Look at how. The name
44:58of the shape is parallelogram. Now, for
45:02those who remember, what do we call a
45:04parallelogram in O-Level Math? A
45:06parallelogram is called a tilted
45:08rectangle. A tilted rectangle. A
45:12rectangle that is tilted. Let's
45:14simplify 'tilted' as one whose corners
45:16aren't 90 °. So what is the difference
45:18between a parallelogram and a rectangle
45:20? That a parallelogram is called a
45:21tilted rectangle. It is called a tilted
45:23rectangle. A rectangle that is tilted.
45:28Look at this. We must have studied this
45:30in O-Level Math too. There is nothing
45:32new in this. By the way. I haven't
45:34explained a single new thing. All of
45:36this was in O-Level Math as well. We
45:37also studied in O-Level Math that these
45:39are two parallel sides. These two sides
45:41are equal. These two sides are equal.
45:43We studied this in O-Level Math. But
45:45anyway, right now we are only talking
45:46about the midpoint. So we will only
45:48talk about the midpoint. So the
45:49midpoint of this will be the same as
45:51the midpoint of that. Then we will see
45:53how all this is applied. How it will be
45:56used in AS-Level past papers. So its
45:58midpoint is also always what? It is the
46:00same. Its midpoint is also the same. So
46:03there are four shapes that have this
46:06common property ABCD. So what are the
46:08names of these four shapes? Square,
46:11rectangle, rhombus, and parallelogram.
46:15There is a common property in all of
46:17these. Clear, son? A common property.
46:19What are the midpoints of all these
46:21always? They are the same. Right.
46:22Actually, I am not talking about area
46:24right now. Perhaps some well-read
46:25student knows its area is length*
46:26breadth. L*L, and its area was base*
46:30height. Its area was also length*
46:34breadth, base*height. Its area was also
46:36base*height. This is height, height,
46:38this is base. Base multiplied by
46:41perpendicular height. When we study the
46:43topic of area, then I will talk about
46:44this. By the way, we just studied one
46:46property that the midpoint of AC and
46:48the midpoint of BD are always what?
46:51Both are always the same. Okay? This is
46:54a common property, a common property.
46:56Okay, now let me show you an example.
46:59Let me show you a small example so that
47:00we can apply what we are supposed to do
47:02with this. How will it be used? I will
47:05copy a small example. Then, God willing
47:07, I will have you solve it using two or
47:09three methods. Clear? I took a small
47:11example. Okay? Suppose this is a shape.
47:14Now you can take any of these shapes.
47:16You can take a square, a rhombus, a
47:18rectangle, or a parallelogram. Take
47:21whichever shape you want. Okay? I will
47:23show you a small example so that your
47:25concept gets cleared first. Then I will
47:28show it to you two or three times using
47:30different methods, and obviously, you
47:32will see all this in past papers too,
47:34obviously. Okay? It is not that we will
47:37only see this in concepts. Obviously,
47:40we will practice this in great detail
47:42in past papers. Come on, suppose you
47:44have this shape. Okay? Suppose you have
47:48this shape. Wait, let's do one thing.
47:56Let's copy this same shape. So we don't
47:58have to put in too much effort. Come on
48:00, I am giving a random example. Then
48:02obviously we should do this work
48:03properly and neatly. Okay? Suppose you
48:06have a shape like this. Come on, take
48:08this. It is a shape just like this.
48:10Come on, it is a parallelogram. Shape.
48:12Come on, the shape is a parallelogram.
48:14Come on, let's name it a parallelogram.
48:15So just look at the concept because
48:17this same concept will be used
48:18throughout. The name of the shape is
48:20parallelogram. Okay? Now I will solve
48:23this same question for you two more
48:25times. This used to be in O-Level Math
48:27too. There was a topic in O-Level Math
48:29called vectors. In vectors, you used to
48:30study an approach too. I will explain
48:32that to you after two minutes. I'll
48:34explain after two minutes. Okay, let's
48:36assume that A has coordinates. 2 3, 2 3
48:39, these are the coordinates for A. All
48:41right? I will fill in some coordinates.
48:44I will make one set of coordinates what
48:45? Unknown. Then I will show you how
48:48this concept applies in the exam. All
48:50right? First, let me show you one or
48:52two examples. Suppose in the exam, B's
48:53coordinates are also given as 4 6, here
48:55you go. B's coordinates are also given
48:57as 4 6, there's another point. I am
48:59also given the coordinates of C. C's
49:01coordinates are, suppose, 5 13. Let's
49:04just do it randomly; our main goal is
49:06to understand the concept of what to do
49:08. 5 13, and suppose we have a point
49:11that is missing. Okay? There is some
49:14point missing. Make some point, C or D,
49:16unknown. For any one of them, come on,
49:18make any one of them unknown. Here you
49:20go. Suppose this is it, I will make one
49:22point unknown. Here you go. Come on,
49:24suppose the examiner is saying that C's
49:25coordinates are unknown. And is asking
49:27us to find C. Is asking us to find C.
49:30Come on, I will show you this same
49:32example tried twice. Twice. I will copy
49:35this same example ahead. And I will
49:37show you with a new approach as well.
49:39Come on, let's begin. The examiner is
49:41saying find the coordinates of point C.
49:43Now this approach is valid. In squares,
49:45in rhombuses, in rectangles, in
49:47parallelograms too. In all four shapes,
49:49whenever any of these shapes appear,
49:51this approach is valid. Okay, look,
49:53it's asking us to find the coordinates
49:55of C. So, the student who remembers the
49:57condition will say that the midpoints
49:59of both are the same. So, first, what
50:01will they do? They will find the
50:03midpoint of BD. Midpoint of BD, the
50:06student will say, let's find the
50:08midpoint of BD. All right? Come on,
50:11look at the approach first, brother,
50:12how do you get the midpoint? x1 + x2/2,
50:15so what will the student do? The
50:17student will add both x values and
50:18divide by 2. Similarly, the student
50:20will add both y coordinates and divide
50:23by two. Simple, here it is, the
50:25midpoint is obtained. Then, the student
50:28will say this is the midpoint of BD.
50:30Then the student will say, similarly,
50:32what will the midpoint of AC be? Use
50:34the same approach. Midpoint of AC.
50:38Similarly, look at the midpoint of AC.
50:40Midpoint of AC, what will be the
50:42midpoint of AC? Similarly, there is
50:44only one formula for the midpoint. X1 +
50:47X2/2, Y1 + Y2/2, done. Then what will
50:50the student do? The student will say
50:51both midpoints are the same. So then
50:53what will the student do? The student
50:54will equate the midpoints. 3.5 and this
50:56was eight. What will the student do?
50:58The student will equate the midpoint
50:59with the midpoint. Here you go. At the
51:01very beginning, I showed you this exact
51:03example. Here it is. What will the
51:05student do? The student will equate the
51:06midpoint with the midpoint. Here you go
51:07. 3 + y/2, this is the approach we saw
51:10at the start today. Then what will the
51:13student do? The student will equate the
51:14x-coordinate to the x-coordinate. Right
51:16? This is simple math, by the way. 3.5*
51:192 = 7, 7-2 = 5, so the student will say
51:21what is the x-coordinate? It came out
51:23to be five. There you go, your
51:25x-coordinate is found. Similarly, then
51:27what will the student do? The student
51:29will also equate the y-coordinates to y
51:30. 3 + y/2, the student will say = 8, 8*
51:332 = 16, 16-3 = 13, so the student will
51:36say what are the coordinates? 13, this
51:39is your y, so finally the student will
51:42say what are the coordinates of C? The
51:46coordinates of C are 5, 13, 5, 13, that
51:49is the approach. What did the student
51:52do? Equated the midpoint with the
51:54midpoint because we know that whether
51:56it is a square, rhombus, rectangle, or
51:58parallelogram. What are the midpoints?
52:00Always equal for diagonals. They are
52:02always the same. So you found both
52:04midpoints. Equated both of them.
52:06Meaning technically, what did we do?
52:08Equated the midpoints with each other.
52:10So this was found, and the x-coordinate
52:12was found, and the y-coordinate too.
52:14Now let me show you a new approach. So
52:16this approach we used to use in
52:18higher-level math in vectors. So I
52:20showed you one option first. I will
52:22write here Option Number One. Come on,
52:24take this. Right? I will show you an
52:27option first. I wrote Option Number One
52:29here. Here, I solved it once. I will
52:32try it again for the second time. What
52:35is this approach called in the language
52:36of math? This approach is called the
52:38midpoint method. Meaning we did no new
52:40work here. The approach we used in this
52:42, what is that approach? Simply used
52:44the midpoint method. Used the midpoint
52:46method. Just like we do. That's the end
52:48of it. If I were to use a term for this
52:51approach, I would say I used the
52:52midpoint method. The question is solved
52:55. Now, look at something new. Here, I
52:57have copied the same question twice.
53:00Copied once, then copied again. There
53:05you go. Now let me show you a new
53:07approach. Here it is. I copied the same
53:10question twice again. Come on. Same
53:13question. The same. What is the name of
53:15the shape? The name of the shape is
53:17parallelogram. Right, kid? The name of
53:19the shape is parallelogram. Your target
53:21is C. Now I am solving this twice again
53:25. Option number two. Alright, kid, come
53:28on. Option number two. You have also
53:31studied this in O-Level Math. Perhaps
53:33you might remember it. This is option
53:35number two. Option number two has been
53:38called the vector method in O-Level
53:40Math. Vector method. Vector method.
53:45What did we call option number three as
53:47well? Vector method. Meaning options
53:49two and three are technically the same.
53:51By the way. You will get the idea now.
53:54We used to do such questions in O-Level
53:56Math Paper 1. We used to call them
53:58grid-based questions. Grid-based
54:01questions. Let's check. Now, we just
54:04need a little bit of common sense. We
54:06know that this side and this side are
54:08both the same. We obviously know this.
54:12So, tell me one thing. What is the gap
54:14between its x and y coordinates? Listen
54:16carefully and watch closely. What is
54:18the gap in its x and y? Okay? We first
54:21saw what? Its x and y gap—whose? AB's
54:25. This is AB, brother. Between 2 and 4,
54:28what is the x-gap? You would say the
54:30x-gap is two. Done. What is its x-gap?
54:33You would say two. And tell me, what is
54:36the vertical gap between three and six?
54:38You would say three. So, what is the
54:40vertical gap between this? Done. It is
54:42right in front of you. So if this gap
54:43is two and three, then this gap will
54:45also be two and three. So we used to
54:48copy this same thing in AS/O-Level Math
54:50that if its difference is two and three
54:52, then its difference will also be two
54:53and three. We studied this in O-Level
54:56Math. So then this gap will also be two
54:57and three. Here it is. So now tell me
55:00one thing. You already have D. You
55:03already have D. What were the
55:05coordinates of D? 3 10, here it is.
55:08What was your target? Your target was
55:10the coordinates of C, here it is. What
55:11were the coordinates of C? What are the
55:14x and y coordinates? You know what the
55:17gap between x and y is, right? You
55:19already found that out. What is the gap
55:22between x and y here? You already found
55:24that out. What was this gap? Two. What
55:26was this gap? Three. So that means how
55:29will you move from D to C? Two to the
55:32right and three up. So that means you
55:35will go two to the right from three and
55:37three up from ten. So what exact answer
55:40was derived? The exact same answer was
55:42derived. What are the coordinates? 5,
55:4513. 5, 13. Done. Meaning if this gap is
55:49two and this gap is three, then this
55:51gap will also be two and this gap will
55:53also be three. 2 + 2 is 4. And add
55:57three to three, it is six. So 3 + 2 is
56:015. In 10, add three to 10. Meaning if
56:04you add three to 10, so if you go
56:05vertically, what do you get? It became
56:0613. That's it. The same answer was
56:08derived. Similarly, these two sides are
56:11also parallel. These two sides are also
56:13equal. So we kept saying look at the
56:15gap between these two. This is also the
56:17same vector method. Obviously, look at
56:18the gap between these two. What is this
56:20gap? And what is this gap? Look,
56:22between these two, this is two. This is
56:24three. So this is two. This is three.
56:26How much is the x gap? One. Done. How
56:28much is the x gap between them? One.
56:30How much is the x gap between them? One
56:32. And how much is the vertical gap
56:33between them? This is three. This is 10
56:35. This is three. This is 10. The
56:36vertical gap is seven. There you go, 1
56:39and 7, so that means from here too it
56:41will be the same. From here too you
56:43will move the same amount. If there is
56:45a gap of one and seven between them,
56:46then how much will the gap be in this?
56:48The gap in this will also be one and
56:51seven. The gap in this will also be one
56:54and seven. So if you notice one thing,
56:57you already have B. What do you need?
57:01You need C. You already have B. Sorry,
57:03yes, you already have B. B is 4, 6, and
57:06your target is C. Here it is. So you
57:09just found out two minutes ago what the
57:11gap between their x and y is. We now
57:13know that the gap between its x and y
57:14will be the same as the gap between its
57:16x and y. Brother, the distance between
57:18this and this will be equal to this and
57:20this. So, how much is this gap? This
57:21gap was one. This gap was seven. Look
57:24here, this gap was one. This gap was
57:26seven. So that means you will go one to
57:29the right from four and go up from six.
57:32So tell me, what are the coordinates of
57:34B? The coordinates of B are 5, 13, so
57:37in A-Level Maths, we call this approach
57:39the vector method. We keep calling it
57:42the vector method in maths. So, what
57:44was your target? Your target was C. So
57:46tell me, what are the coordinates of C?
57:50What coordinates did you get for C? The
57:52coordinates of C are 5. 13, here you go
57:56, 5, 13, these are the coordinates of C
57:59for you. By the way, we moved forward
58:02from D here, didn't we? Right? So, when
58:06we moved forward from D, whose
58:07coordinates did we get? C's. Here, I
58:09wrote the number letter wrong by
58:12mistake. The concept behind the letter
58:14was correct. There you go. So if this
58:17gap is of two and three, then this gap
58:19is also of two and three. Meaning, from
58:21here go two forward to the right and
58:23three up. If this gap is of one and
58:25seven, then from here go one to the
58:27right and seven up. Add one to four.
58:30Add seven to six. The answer is 5, 13.
58:34Right means addition to x. Up means
58:37addition to y. Simple. So which
58:40approach did we learn today? Today we
58:41learned the midpoint approach. Midpoint
58:43approach. There are still four or five
58:45more points in coordinate geometry. So
58:47first our basics will be completed.
58:49Then, God willing, we will finally
58:51start its past papers. But first, we
58:54will study all the concepts one by one.
58:57When 100%of everything is covered, then
58:59we will start the past papers. Okay,
59:01son?

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