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AS Level Mathematics 9709 | Coordinate Geometry 4 | Equation of Straight Line/Perpendicular Bisector — Transcript

by Math with Raja Shafqat · 13,132 words · 1,863 segments · language en · Watch on YouTube

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0:08Assalamu Alaikum, son. In the last
0:13class, we used mensuration to find the
0:17area of 2D shapes. I did not explain it
0:21in the last class. Now, we will
0:23continue from there. There was a
0:25mathematician named Heron. Heron
0:27derived a new alternate method to find
0:29the area of polygons like squares,
0:31rhombuses, rectangles, parallelograms,
0:34trapeziums, kites, isosceles,
0:36right-angled, and equilateral triangles
0:38. To find the area of all these, he
0:41derived a new alternate method. Okay?
0:43That alternate method is area using
0:46coordinates. We read this formula in
0:48the language of mathematics. Heron's
0:50formula. Heron's formula. Okay? Now
0:53look, son, what is Heron's formula? I
0:56will guide you slowly. See, the first
0:58thing is that whenever we talk about
1:00Heron's formula, today we will discuss
1:02Heron's formula. Okay? This was a
1:04mathematician whose name was Heron.
1:07Okay? So now, what is Heron's formula?
1:11In Heron's formula, you don't have to
1:14use mensuration. What do you do in
1:16Heron's formula? You find the area in
1:19Heron's formula. Of a polygon. You find
1:23the area of a polygon. But with the
1:25help of what? You find the area with
1:28the help of coordinates. And what is
1:30the name of our topic? The name of our
1:32topic is Coordinate Geometry. So Heron
1:35said that the area of any shape can be
1:37found if you have the coordinates of
1:40that shape. Okay? This was new work
1:42done by Heron. Alright, now I will
1:44explain. By the way, the polygons in
1:49our syllabus are of only two types:
1:51either polygons with three sides or
1:54polygons with four sides. Meaning, in
1:58AS Math, you will find two types of
2:00polygons. Meaning, you will find
2:02triangles of any type—equilateral,
2:04right-angled, isosceles, acute, obtuse
2:06—any type of triangle, okay son? Even
2:09if it's a right-angled one. And
2:11quadrilaterals, meaning squares,
2:13rhombuses, rectangles, trapeziums,
2:15parallelograms, and kites. This is our
2:17syllabus. Now look, what work did Heron
2:21do? Obviously, we know that when there
2:24are three sides, it means triangles,
2:27and four sides means quads.
2:29Quadrilaterals. Okay, son?
2:31Quadrilaterals. So this is our syllabus
2:34. Heron derived a new formula to find
2:36the area. Okay? Now, first of all, you
2:40need to keep two directions in mind.
2:42Then I will start the topic for you. We
2:45must have studied two directions in
2:47O-Level Math. There was a topic,
2:48transformations, in O-Level Math. We
2:50must have heard this word there too.
2:52There are two types of directions in
2:53math. One direction we read as
2:56anti-clockwise direction.
2:58Anti-clockwise direction. We used to
3:01hear the word anti-clockwise direction
3:03in rotation. And we used to hear
3:05another direction, clockwise direction.
3:08So, we must have a sense of these two
3:10directions for Heron's formula. Right?
3:12So, there are two types of directions.
3:14One is anti-clockwise and the other is
3:16clockwise direction. Now look, we must
3:19have studied in O-Level Math what an
3:21anti-clockwise direction is.
3:23Anti-clockwise direction is called the
3:25upward direction. Look here, the one I
3:27am marking with an arrow to show you.
3:29The direction I am marking to show you.
3:31This anti-clockwise direction, we call
3:33it the upward direction. Upward
3:35direction. So, we used to say in
3:37O-Level Math, "Which direction is this?
3:39" This is an anti-clockwise direction.
3:42What does anti-clockwise direction mean
3:45? Upward direction. Upward direction.
3:49Right? Anti-clockwise means upward
3:52direction. When you move in an upward
3:55direction, you say in math language, "
3:58What direction is this?" This is an
3:59anti-clockwise direction. Okay, son?
4:02These are very basic things I am
4:03starting with. Then we used to hear
4:06another word, clockwise direction. Now,
4:08what did clockwise direction mean?
4:10Clockwise direction meant downward
4:12direction. Look, this direction we used
4:15to call clockwise direction in math
4:18language. Right? Then I will start the
4:21topic slowly. We used to say in math
4:23language, "What direction is this?"
4:26This direction is the clockwise
4:27direction. We used to call it simple
4:30downward direction in plain language.
4:32Clear, son? We used to call clockwise
4:35direction simple downward direction in
4:36plain language. Now, use whichever word
4:39you want, clockwise or downward
4:41direction. Let me mark it here. We used
4:44to call clockwise the downward
4:45direction. Right? Upward direction and
4:49downward direction. Upward direction
4:52and downward direction. We used to call
4:55these basic directions. Now look,
4:58whenever Heron's formula is applied,
5:01you will see a sign; in math language,
5:04that sign is called a modulus sign,
5:07though it is also called an absolute
5:09sign. You will use a modulus sign; why
5:13you use it, I will explain. Modulus
5:15signs are these kinds of signs. We call
5:17this sign a modulus sign. Meaning a
5:19pair of parallel vertical lines. These
5:21are shown as a pair of parallel
5:23vertical lines like this. We call this
5:25sign a modulus sign in math. What is
5:27the property of the modulus sign? You
5:30will also study the modulus sign in P3
5:31under the topic of modulus functions.
5:33The property of the modulus is that
5:36whether you put a plus or a minus
5:39inside the modulus. Meaning, whether
5:41the input is positive or negative.
5:43Whatever the input may be. Math states
5:45that whenever you apply a modulus
5:47function, the output is always positive
5:49. It means whatever the input is,
5:52whatever the input is. Okay, whether
5:55the input is plus or minus, but what
5:58the modulus does is, it always shows a
6:00positive output. It always shows a
6:03positive output, so whenever you apply
6:05the (Harris) formula, the modulus sign
6:07will be used in it. The modulus sign is
6:10used because area is always positive.
6:14We already know that area, volume,
6:16perimeter, and length are always
6:18positive, so which sign will be used in
6:20this formula? Always? Modulus. Meaning
6:23the result is always positive. So,
6:26whenever we find the area of any shape,
6:28whether it's a square, rhombus,
6:29rectangle, trapezium, parallelogram, or
6:31kite, the area is always positive. That
6:34is why we use the modulus sign in the
6:37Harris formula. Modulus sign. I will
6:40explain this in more detail now. Is it
6:42clear, son? Modulus sign. Now, the
6:44question is, what is the Harris formula
6:46? Or how do we apply it? For the Harris
6:49formula, you only need to have the
6:51coordinates. The Harris formula has no
6:54interest in the shape. Meaning, if you
6:58are given those four points, you can
6:59find the area with the help of those
7:01four points. Those four points may
7:04belong to a square, a rhombus, a
7:06rectangle, a kite, or a trapezium. So,
7:09the Harris formula is dependent on the
7:11coordinates. It is not dependent on the
7:13shape. Now, let me show you a simple
7:15example. Let's suppose you have this
7:18triangle. Just a random example, I just
7:21plotted a triangle like this. Without
7:23thinking, I don't know what type of
7:25triangle this is. Okay, let's say I
7:27just marked a triangle randomly. I
7:29don't know if it's isosceles or
7:30equilateral. What kind of triangle it
7:32is. I don't know anything. I just
7:34marked a triangle randomly. Suppose
7:36this point here is this point. Its
7:39coordinates are x1, y1, and this is
7:41another point. Its coordinates are x2,
7:44y2, and there is one more point. Its
7:46coordinates are x3, y3, here it is. I
7:48need the area of this triangle. Now,
7:51when mensuration is applied, you first
7:53have to check what type of triangle it
7:56is. It's a right angle, equilateral,
7:58isosceles, or a right angle triangle.
8:00Then it is treated accordingly. Like
8:03right angle is 1/2*base*height,
8:05non-right angle is 1/2 ab sin c, but in
8:07Heron's formula there is no such
8:09complication. In Heron's formula, you
8:11just need the coordinates. The area
8:13will be found out by itself. Now, what
8:15does Heron's formula say? First, look
8:17at Heron's formula. Then, obviously, I
8:19will show you the entire working. First
8:21, just take two minutes to see what new
8:24thing Heron explained. Okay? What is
8:27this Heron's formula? First, just look
8:29at Heron's formula for two minutes.
8:32Then, God willing, I will guide you on
8:33how it is applied. If you want to apply
8:36Heron's formula, then take a look at
8:38what Heron's method is. Heron's method
8:40was this. First, you copy half. Then
8:42you copy the sign, the modulus sign.
8:45Modulus. The name of this sign is the
8:47modulus sign. Now, Heron's formula says
8:49that you can start from any point,
8:51anywhere. But you must follow either
8:53the counter-clockwise direction. Or you
8:56follow the clockwise direction. Heron's
8:58formula says you just have to follow a
9:01direction. How? Suppose I started from
9:04here. I started from A. I am starting
9:06the area from this point A. Is it clear
9:08, son? So Heron's formula is applied
9:10like this. What are the coordinates of
9:12A? First of all, look at the
9:14coordinates of A. The coordinates of A
9:16are X1, Y1. Who is the next point after
9:19A? B and then C. Then you copy the
9:22coordinates of B. x2, y2, then you copy
9:25the coordinates of C, x3, y3, and then
9:28you will end exactly where your area
9:31started (1, 2) 3. That is, Heron's
9:34formula says that where your area
9:36starts, your area must end there. So
9:39that means, where will your area end?
9:41At x1, y1. So if you notice, which
9:44direction did I follow? I followed the
9:46counter-clockwise direction, meaning I
9:48explained the upward direction to you.
9:50Upward, now this can also happen. There
9:53are multiple options in this. The
9:54interesting thing is the result will be
9:56the same for everyone. It will be 100%
9:58the same. You can start from anywhere.
10:00Just follow any one direction. If I
10:03tell you, "Buddy, I will start from B."
10:05Here you go. So if I start from B, then
10:07what is my first point? My first point
10:09is X2, Y2. Meaning, this time I am
10:12starting from B in a counter-clockwise
10:14direction. Then the next point becomes
10:16C. Here is X3 Y3. Then what is the next
10:18point? X1 Y1 and then where does it
10:20always end? Where you started from. So
10:23remember, it must end at the same point
10:26it started. It must end at the same
10:28point it started. Now, which direction
10:30did I follow in both cases?
10:32Anticlockwise. Anticlockwise. If you
10:35start from C, then A will come, then B,
10:38and back to C. Meaning, Heron's formula
10:40will end exactly where it started. The
10:44rule is that your movement condition
10:46must be cyclic. It should be in a
10:48sequence. Look, now I will change it
10:51intentionally. I will show it like this
10:53. Look, I am changing the direction.
10:56Now I will consider a question in
10:58clockwise. Here, I am considering the
11:00question clockwise. I started from A.
11:02If I start from A, then I will end at
11:05A. So look, what did I do? Copied it.
11:08First, I copied it. My first point is A
11:11, X1 Y1. Then what is my next point? X3
11:15Y3, then what is my next point? X2 Y2,
11:18and then what will be my last point? X1
11:21Y1, so in Heron's formula, the area
11:24will end exactly at the point it
11:27started. I will show the entire proper
11:30working now. I will show it by
11:31expanding it. I will show it by
11:33applying it. Right now, I am just
11:35giving you an idea. An idea. Okay?
11:37First, grasp the concept. Then, God
11:39willing, we will start its practice.
11:40Look, I am changing it again. I am
11:43changing it again. Well, the
11:45interesting thing is, even if a student
11:47doesn't have a sense of direction, the
11:48answer will still be correct. Because
11:51if you consider a triangle, which is a
11:54special case, whether you do ACB, ABC,
11:56or BAC, the result will be the same;
11:59but if you don't have a sense of
12:01sequence in a quadrilateral, the answer
12:04will be 100%wrong. Now, suppose I start
12:07from here. I start from C. Here, I am
12:09moving in a clockwise direction again.
12:12I started from C. What are C's
12:14coordinates? C's coordinates are X3 Y3.
12:17What are C's coordinates? X3 Y3, then I
12:20put B X2 Y2, then I saw X1 Y1, and then
12:23where did I end? Where I started from,
12:27X3 Y3. So, the rule is that it must end
12:30exactly at the point it started. See,
12:33in every question, it will end exactly
12:36where it started. So, in Heron's
12:38formula, you either follow an
12:40anti-clockwise, meaning upward, or a
12:42downward direction. After two minutes,
12:44I will show you an example of this.
12:46Then, God willing, I will also show you
12:47questions from past papers. So, what is
12:49the name of this sign? The name of this
12:51sign is modulus. Whether the sign
12:52inside is plus, or the result inside is
12:54plus or minus. What will the result
12:56always be? Positive. Meaning, whatever
12:59your input inside the bracket is,
13:01whether it's plus or minus. The result
13:03is always positive. Because we know
13:05that area is always positive. Area is
13:07always positive. Now, let me do one
13:09thing. I will show you a quadrilateral.
13:12Whether the quadrilateral is a square,
13:14rhombus, rectangle, trapezium,
13:16parallelogram, or kite. The result will
13:18be the same. Look at this. I'll
13:20randomly sketch a quadrilateral for you
13:22like this. Just random. Okay? I don't
13:26know what shape the quadrilateral is. I
13:27can take any quadrilateral. Obviously,
13:30every quadrilateral will mean there are
13:32four points in every quadrilateral. Now
13:35, whatever the quadrilateral may be. I
13:37am saying repeatedly that Heron's
13:38formula is not dependent on the shape.
13:40Okay, son? Heron's formula does not
13:41depend on the shape at all. I just
13:43plotted a shape randomly like this.
13:45Well, an interesting thing could also
13:47be that you might not even know the
13:48name of that four-sided shape. Clear,
13:51son? Okay? It is possible that you
13:53don't even know the name of the shape.
13:55Okay? Let me repeat what I said.
13:57Suppose I just plotted a shape randomly
13:59like this. This is x1 y1. Okay? This
14:02point is x2 y2. After this, I will now
14:05tell you its proper working. This point
14:08is suppose x3 y3. Suppose this point is
14:11x3 y3 and this point is suppose x4 y4
14:13for you. Okay? This point is suppose x4
14:17y4 for you, here you go. This is x4 y4,
14:20here you go. I just plotted a
14:22quadrilateral randomly like this. Now
14:24that quadrilateral in the exam can be
14:26anything. Square, rhombus, rectangle,
14:28trapezium, parallelogram, kite. Now if
14:31I apply Heron's formula, I have
14:33multiple options. If I follow an
14:35anti-clockwise direction, I can start
14:37from A as well. From B, from C, from D,
14:40from anywhere. If I tell you that, look
14:42, I am following an anti-clockwise
14:44direction for a minute. For one minute.
14:47Look at this. And suppose I am starting
14:49from A. Anti-clockwise direction,
14:51meaning upward direction. Okay? Okay? I
14:54will take a turn like this. First A
14:55will come, then B, then C, then D, and
14:57then back to A. So, how will I show the
14:59points? What is my very first point? My
15:02first point is X1, Y1. What is my next
15:04point? My next point is X2, Y2. What is
15:07my next point? X3, Y3, what is my next
15:10point? X4, Y4 and what is my last point
15:13? X1, Y1. So the interesting thing is
15:16you must end exactly where you started.
15:19If a student starts from B. B, C, D, A,
15:23B; it must end right back there.
15:26Meaning you started from B, then C, D,
15:29A, and ended back at B; another student
15:32starts from C. C, D, A, B, again C.
15:34Clear, kid? So you just have to follow
15:37the direction. You can follow whatever
15:39direction you want. But it cannot be
15:41that first it's A, then C, then B, then
15:43D. You failed the direction. That is
15:46why the sequence matters. Either
15:48anti-clockwise or clockwise. Okay?
15:51Either you take the anti-clockwise
15:52direction or you are taking the
15:54clockwise direction. So that is why the
15:56sequence matters. Clear, kid? So I took
15:58the first example as a random
16:00anti-clockwise direction. In 2 minutes,
16:03I will show you a proper example. Okay,
16:06let's take an example of clockwise.
16:09Let's take another example. Example of
16:12clockwise. Clockwise. Now suppose I
16:14feel like starting from D. This time my
16:16direction is clockwise. Okay? Let's go.
16:19So I started with D first. So what are
16:21the coordinates of D? D's coordinates
16:24are X4, Y4 and then X3, Y3 and then X2,
16:29Y2 and then X1, Y1 and then X4, Y4.
16:33Clear, kid? So that means you just have
16:36to follow a sequence. This time I went
16:38like this. Here is D, after D is C, D,
16:42C, B, A, D; what is this? D, C, B, A,
16:46again D; so the rule is that it must
16:50end exactly where it started. Now first
16:54thing, what is happening here? When the
16:56question comes in the exam, what will
16:58we do? Come on, let me show you an
17:00example. I will apply it properly and
17:03solve it for you. Then obviously, you
17:05will find this in the past papers as
17:06well. Definitely. Suppose I just
17:08plotted a triangle like this. Just like
17:11that, without thinking. I have no
17:13interest. What is the name of the
17:14triangle? Because I am only interested
17:16in the coordinates. The interesting
17:17thing is this. Now look, suppose this
17:19is the point. Alright? The coordinates
17:21of this point are 2, 3; suppose these
17:24are just random points I'm taking. And
17:26it's common sense anyway. If you join
17:28three points, they will definitely form
17:30some kind of triangle. Okay, what if
17:33this point here is C? Suppose the
17:35coordinates of this point are 3, 10,
17:37here you go. Now I need the area of
17:38this triangle. Clear, kid? Now it's my
17:41choice wherever I want to start.
17:44Counter-clockwise direction A, B, C, A,
17:47or I could go B, C, A, B. I could go C,
17:50A, B, C, or I could move this way. A, C
17:54, B, A, or I could move like this. C, B
17:58, A, C, or I could move this way. B, A,
18:01C, B, whatever you want, the answer
18:03will be the same for all. The
18:05interesting thing is, let's just do one
18:07specific thing. Okay, so now I need the
18:10area for this. Let's see, I'll apply
18:12Heron's formula and show you what Heron
18:14did. Here's the half. Suppose I am
18:17following the counter-clockwise
18:19direction; suppose A, then B, then C,
18:21and then A, B, C, and then A. Now tell
18:25me, what are the coordinates of A? The
18:27coordinates of A are 2, 3, look here
18:29kid, 2, 3, then what are the
18:31coordinates of B? 5, 4, then what are
18:33the coordinates of C? 3, 10, and then
18:35it ends at A again, 2 and 3. Here it is
18:38. So if you notice, I followed the
18:40counter-clockwise direction. I followed
18:43the counter-clockwise direction. Okay,
18:45kid? Look here. Mark it here. I
18:47followed the counter-clockwise
18:48direction. Right? Here you go. So to
18:51explain it to you, I'll write it up
18:53here that I wrote A. Here I wrote B.
18:55Then I wrote C, and then back to A.
18:57Meaning, I want to tell you that I
18:59followed the counter-clockwise
19:01direction. Okay, kid? I followed this
19:03direction. Like this, counter-clockwise
19:06, first A, then B, then C, and then A.
19:08Now, how is Heron's formula applied?
19:11This is the most important thing, how
19:13is Heron's formula applied? Watch
19:15carefully, kid. This is the interesting
19:17part. In Heron's formula. Actually,
19:19some children use a term. Shoelace
19:21method. Shoelace, like the laces on
19:24shoes. First you expand this way, then
19:26you expand that way. So some kids call
19:28it the shoelace method. Look here, kid,
19:30the first task is to look at my arrows
19:33first. I request everyone to look at my
19:35arrows first. These are upward arrows.
19:37So what is the first thing I will do?
19:39Look at my arrows first. These are my
19:40arrows. Upward arrows. I request
19:42everyone to watch carefully first. Okay
19:44, son? First, focus on my working.
19:46Upward arrows. It goes like this first.
19:492*4, 5*10, 3*3, first these diagonals
19:53are multiplied. So, what will it be
19:55here? 4*2 = 8 plus 5*10 = 50 + 3*3 = 9,
20:00so always multiply diagonally like this
20:04first. Then comes the minus sign. Then
20:07comes the opposite working. Okay? That
20:10was the first part. Then comes the
20:11opposite working. Look here. Now look
20:13at my arrows again. Now my arrows are
20:15downward. That is why some kids also
20:17call this the shoelace method. Shoelace
20:19. Now what will happen? 5*3*4, 2*10,
20:23there you go. Now make it easy. What
20:26will this be? 5*3 = 15, 4*3 = 12, and
20:2910*2 = 20, there you go. Clear, son?
20:32Now let's simplify this a bit. Now make
20:35it even easier. When I make it even
20:38easier, this becomes half. This becomes
20:40modulus. Now what is inside? 59, 60, 67
20:43, there you go. So, this is 67. There
20:47will be some numeric answer, right?
20:48Here it is. Now what is this? 20, 30,
20:5140, 47, there you go, so what did we
20:52get? We got 47, there you go, 67-47 is
20:58what? 20, so it means whether inside it
21:03was +20 or -20. What will the modulus
21:06do? What will the modulus result always
21:08show? The modulus will always show a
21:11positive result. So it means, what area
21:13will be derived? The area derived will
21:16be 10 units squared. This is finally
21:19the area derived. So in Heron's formula
21:21, you just have to follow the direction
21:24. Whether you work with respect to
21:26anti-clockwise or with respect to
21:29clockwise. Okay, son? Whether you work
21:31anti-clockwise or work clockwise.
21:34Ultimately, whether it is + 8-10 or
21:3810-8 inside. Ultimately, what is to be
21:40derived from both? It should be the
21:42same. Ultimately, the answer from both
21:44should be the same. Let me do one thing
21:46. I will try this question again, once
21:49more. But this time, I will take some
21:53other direction. I will start from
21:55somewhere else. Here we go. I want to
21:56show that my result will still be the
21:58same. Look, here you go. When I worked
22:012 minutes ago, I took the
22:02anti-clockwise direction, right, upward
22:04. Now I will do one thing. I will take
22:06the clockwise direction. Here you go.
22:07Let's go. I just randomly started with
22:10C. Let's go. I'll show you the answer
22:12will be the same. First I started, 3,
22:1410 and then 5, 4 and then 2, 3 and then
22:173, 10 there you go. So I started, first
22:20I wrote...what did I write first? I
22:22wrote C, then B, then A, then back to
22:25C. Clear, kid? If you notice, I started
22:28with C, after C I wrote B, then I wrote
22:30A, then I wrote C. Clear, kid? Wherever
22:33it starts, it must end exactly there.
22:35Now see how this area will come about.
22:37How will it be found? Please just watch
22:39my working first. Alright? First it
22:42happens like this: 4*3, 3*5, 10*2, look
22:46, 4*3, 3*5, 2*10, so if I simplify this
22:50, if I simplify this, what will the
22:54answer be? 4*3 = 12 + 5*3 = 15 + 10*2 =
22:5920, there you go, this is the upward
23:03direction. What do we call this sign?
23:06Modulus sign. Then there is the
23:07backward arrow, the reverse. Then it's
23:09like this: 5, 10, 2*4, 3*3, that's what
23:12I told you, right? First it goes like
23:14this, look at these upward arrows, then
23:16the backward arrows, here they are.
23:19Alright? Upward minus backward. So if I
23:22summarize, technically it's happening
23:23like this, first these arrows, then you
23:24have these arrows. Interestingly, even
23:27if you reverse it, the result will
23:28still be the same. Obviously, obviously
23:32. Now what will happen? Now it will be
23:34like this, look: 50 + 8 + 9, see, look
23:37what happened here? 50 + 8 + 9, here it
23:40is. Showed it as 50 + 8 + 9, there you
23:43go, there you go. Now finally simplify
23:46it. When simplified, the order was
23:50reversed. It became 47-67,
23:53interestingly, there it is. So what is
23:5747-67? -20, but I mentioned that the
24:00property of modulus is that the modulus
24:02will show the answer of a negative as
24:04what? Positive. So look, what answer
24:06did you get? 20/2, so ultimately the
24:10area found is also...The area found is
24:1410 units squared. Clear, kid? So you
24:17have many options. You can start from
24:19wherever you want. But the rule is that
24:22wherever it starts, it must end exactly
24:23at that position. This was a
24:25mathematician. His name was Heron. He
24:27did some new work. He did a new piece
24:29of work. It said that you just provide
24:32the coordinates. The area will be found
24:34out. Meaning, if I tell a student, "Hey
24:36, these are four points." Then, if the
24:39student knows Heron's formula, they can
24:41find the area without sketching.
24:44Meaning, they don't need to know if
24:46it's a square, rhombus, rectangle,
24:48trapezium, parallelogram, or kite. This
24:51Heron's formula was a new formula
24:53derived by the mathematician Heron. Is
24:56that clear, son? So, this is not a
24:57mensuration-based formula, because
25:00mensuration-based formulas depend on
25:02length. This formula is dependent on
25:04coordinates. So, the name of this
25:06formula was Heron's formula. Is that
25:08okay, son? In the last session, I
25:10taught you mensuration-based formulas.
25:12Now it's Heron's formula, meaning the
25:14area of a polygon. In our syllabus,
25:16there are only two polygons. Triangle
25:18and quadrilateral. Three sides and four
25:20sides. A shape with three points and a
25:22shape with four coordinates or four
25:24points. Is it clear, son? Now look at
25:27the fifth point of this same chapter.
25:29This point is 100%O-Level Maths. 100%
25:32clear, son? 100%, now let me guide you.
25:36The name of this point is Equation of a
25:37Straight Line. This is 100%O-Level
25:39Maths. By the way, there is nothing
25:41extraordinary in this. Is that okay,
25:42son? This is 100%O-Level Maths.
25:45Equation of a straight line. Look,
25:47let's, first of all, let me start you
25:49off with a few basic things. What did
25:52we study in O-Level Maths? We must have
25:54studied it in O-Level Maths. There was
25:56a topic in O-Level Maths called
25:58Coordinate Geometry. Then in
26:00transformations, inequalities, and
26:02graphs of functions, you must have read
26:03it many times. There are three
26:06different types of lines in Maths. You
26:07often see some lines like these in
26:09coordinate geometry. Horizontal lines,
26:12meaning lines parallel to the x-axis.
26:14Then you also see lines like these in
26:16coordinate geometry: vertical lines.
26:19Vertical lines, meaning vertical, or
26:20parallel to the y-axis. And then you
26:23also see some lines like these in Maths
26:26, which we call tilted lines, inclined
26:28lines, or lines with a slope. We study
26:32all these lines in O-Level Maths. Okay,
26:34perhaps if you recall, we studied such
26:36lines in O-Level Maths. Look, these are
26:39all horizontal lines! Then we studied
26:42lines like these in O-Level Maths, look
26:44at these, the ones I am drawing now; I
26:46haven't explained any equation or
26:47concept yet. In O-Level Maths, we kept
26:50calling these vertical lines, lines
26:51parallel to the y-axis. Then we used to
26:53see such lines as well in O-Level Maths
26:55. Here are the types of lines we used
26:57to see in O-Level Math. We used to call
27:00these tilted lines. Lines with a slope.
27:03Lines with a slope. This is all your
27:05O-Level Math. These are all your lines.
27:07By the way. We used to call all of
27:09these lines. Obviously, these are all
27:11lines. Now I will get you started one
27:14by one. Okay? First, we will start with
27:17slant lines. Our topic name is '
27:19Equation of a Straight Line', which is
27:22100%O-Level Math, 100%okay. So, first
27:25we start with a slant line. Okay? We
27:28must have studied it in O-Level Math:
27:30slant line, slant line. Well, we used
27:33to hear another word for this as well,
27:35you can call it a 'slant line'. Come on
27:37, we also used to hear another word,
27:39call it 'slant' or this too. Actually,
27:42there is a proper word used for this in
27:44Math, and that word is 'oblique'. Right
27:47, okay? Line with a slope. We also call
27:49this a line with a slope. Line with a
27:53slope. Line with a slope. We studied
27:56all of this in O-Level Math. So we are
27:58starting with the first type of line.
28:00Okay? From the first type of line.
28:02Meaning, we are starting with this type
28:04of line. Then I will guide you in more
28:05detail. Here it is. Now my first
28:07interest is in these types of lines
28:08that we used to study in O-Level Math.
28:10And obviously, we will study it now in
28:12A-Level Math. Our interest is in this
28:14type of line. So we have studied before
28:16that whenever a line is of this type,
28:18its gradient is always positive. And we
28:19have also studied before that whenever
28:21a line is of this type, its gradient is
28:23always negative. We studied two things
28:25in O-Level Math. Now please watch very
28:28carefully. We studied two things in
28:30O-Level Math. Some students use these
28:32equations in O-Level Math as well. y =
28:33mx + c Okay? This is called the
28:38slope-intercept form. By the way, maybe
28:41some student remembers the name, so in
28:42the language of Math, it is called the
28:44slope-intercept form. Slope-intercept
28:47form. Even if we don't know the name,
28:50no issue. This is what we studied in
28:52O-Level Math. This is called the
28:54equation of a tilted line, diagonal
28:56line, slant line, line with non-zero
28:58slope, use any word you like. Okay?
29:01Well, this is what we studied in
29:03O-Level Math. mx + c, if we remember
29:04O-Level maths, what was 'm' in this? We
29:07called M the gradient of the line, rise
29:10over run, and then what was 'c' in this
29:13? In the language of math, we called 'c
29:16' the y-intercept of the line. I am
29:19talking about O-Level maths; I will
29:21explain O-Level maths obviously, and in
29:23this condition, what were 'x' and 'y'?
29:26'x' and 'y' were called any point on
29:28the line. What were 'x' and 'y'? A
29:30point on the line. Meaning every point
29:32that is on the line was called 'x' and
29:34'y' in math. You studied this in
29:36O-Level maths. Right? I am telling you
29:38something very basic. You could say a
29:39given point on the line, any point on
29:41the line, every point on the line. You
29:44must have studied in O-Level maths that
29:46whenever, let me just show you some
29:47rough sketches. Just giving you an idea
29:49. Look, all these lines that cut above
29:53the x-axis. Look, all these lines that
29:57cut above the x-axis. These lines that
30:00are above the x-axis, we used to say
30:02all of them have positive y-intercepts.
30:04We know that in O-Levels, we used to
30:06say any point where a line touches the
30:07y-axis is called the y-intercept. Now,
30:10all these y-intercepts that are below
30:13the x-axis, these are all negative
30:15y-intercepts. Look, all these
30:17y-intercepts are negative. Obviously,
30:19we would say, man, their y-intercepts
30:22are negative. And this line passing
30:25through the origin, we used to say in
30:27O-Levels that this line's y-intercept
30:29is zero. Remember? Meaning if a line
30:32cuts above the x-axis, we say its
30:34y-intercept is positive. If a line cuts
30:37below the x-axis, we used to say its
30:39y-intercept is negative. When a line
30:41passed through the origin, we used to
30:43say its y-intercept is zero. This is
30:45about O-Level maths, by the way. Right?
30:48And obviously, what used to be the
30:49gradient of this type of line? The
30:51gradient used to be positive. In two
30:52minutes, we will see examples of all
30:54these. Then there was this second model
30:56in O-Level maths. Right? This was the
30:58second model. I have drawn exactly the
31:00same thing again. Right? This is all
31:02O-Level maths. Obviously, 100%. But
31:06obviously, we will cover the complete
31:07course. So, I will revise these basic
31:09things as well. Look at these lines
31:12cutting above the x-axis. What is the
31:15y-intercept of all these? The
31:17y-intercept of all these is positive.
31:20These lines cutting below the x-axis,
31:22these lines cutting below the x-axis,
31:25what did we say their y-intercept was?
31:27Their y-intercept is negative. This is
31:30also O-Level math. And the line that
31:33passed through the origin, we used to
31:36say its y-intercept is zero. Okay,
31:39child? The line that passes through the
31:41origin, we used to say its y-intercept
31:42is zero. Look, this line I am
31:44highlighting on purpose. We used to say
31:47that this line has a y-intercept of
31:50zero. This is your O-Level math. Right?
31:52We will look at this in a bit more
31:54detail now. Right now, I am just doing
31:55a quick revision. Clear, child? We will
31:58obviously look at it in detail after a
31:59while. The name of this form was
32:01slope-intercept form. We must have
32:02studied this very frequently in O-Level
32:04math. Right, then we also used to study
32:06another form. We used to show this form
32:08like this. There was another point too.
32:10This was called the point-gradient form
32:12. Okay? Come on, look at this. We used
32:14to study one more form of O-Level math.
32:16y-y 1 equals m times x-x 1, here it is.
32:19There was one more form we used to
32:21study as well. Here is that form. Okay?
32:23Students study this form in O-Level
32:25math and Add-Maths as well. In this
32:27form, y-y 1 equals m times x-x 1, look,
32:31what was m? What did we obviously used
32:34to call m? Gradient or slope. Gradient
32:37or slope in O-Level math. And this x1,
32:41y1, we used to call it in math language
32:47, "given point on the line". This is
32:51your O-Level math, by the way. And what
32:54were x and y? "Any point on the line".
32:57What are x and y? We used to call x and
33:02y, "any point on the line". Is it clear
33:07, child? So, we studied both of these
33:09things in O-Level math. Sometimes it
33:12also happened that to confuse us, the
33:14examiner would show the equation of a
33:17line like this too. Look, he used to
33:20say, show it like this. This will
33:22happen in A-Levels too. It will say
33:24find the equation of the line passing
33:25through A and B in this form. Sometimes
33:27it will ask to show it like this. The
33:29examiner will provide it like this as
33:31well. The required form could be like
33:32this too. The examiner will ask you to
33:34express your line in this form. Alright
33:37? We will do it. Sometimes the examiner
33:39says to express your line like this.
33:42You will find all these styles in the
33:43paper. It is possible the examiner asks
33:45you to show it like this. To express
33:47the equation of a straight line in this
33:49form. ax + by = c, show it like this.
33:53So, this thing will be required in
33:54A-Level Maths in Coordinate Geometry.
33:57Sometimes they will say, well,
33:58sometimes they won't mention anything.
33:59Sometimes it will say find the equation
34:01, express the equation of the line in
34:02this form. Sometimes it will say
34:03express the equation of the line in
34:04this form. Is it clear, child? Now
34:06let's deal with them one by one, what
34:08we used to study in O-Level Maths. This
34:10is 100%O-Level Maths. mx + c and y-y 1
34:13= m (x-x 1), this topic is 100%O-Level
34:18Maths. Okay, now look, we used to see
34:20three situations in O-Level Maths,
34:22three. Alright? I will guide you one by
34:25one on what the situations were and
34:28what we used to do in O-Level Maths.
34:30Okay, child? So, let me revise O-Level
34:33Maths for you once, regarding what you
34:35have studied in O-Level Maths. Okay,
34:38child? There used to be three models.
34:39Model number one. Alright? So that this
34:42whole thing gets covered. Model number
34:44one was that, suppose you have this
34:45point. Then I will guide you. One point
34:48is given on the y-axis. Suppose this
34:51point is 0, 2, there you go. So, I am
34:53now giving three different examples.
34:55And this point on the line is given: 4,
34:586, the examiner used to say, "Find the
35:00equation of the line, find the equation
35:02of the line passing through A and B,"
35:04meaning the examiner is saying in this
35:06form that you tell the equation of this
35:08particular line. Okay, let's go. Now
35:10that student, okay, there is no
35:12restriction. Number one option, number
35:15one option, number one. Okay, now look,
35:19child, option number one. Okay, what do
35:21I need? The equation of the line. Now
35:24the student who uses this equation mx +
35:26c, mx + c, that student will straight
35:28away say, "Man, the y-intercept is
35:30right here, the y-intercept is two." So
35:33they will immediately say, "Buddy, c is
35:34two, here you go." They will
35:35immediately say, "Buddy, c is already
35:37given." The student will immediately
35:39say, "c is already given, done." So, in
35:40this condition, what will the student
35:42do? Find out the gradient. If the
35:44student is smart, they know the
35:45gradient without working. Right, they
35:48know the gradient formula without
35:50working? The gradient formula is y2-y1.
35:54What formula for gradient did we study?
35:56y2-y1 over x2-x1, there you go. Okay?
36:00So, by chance, what is the gradient?
36:01One. What did this gradient turn out to
36:04be by chance? +1, there you go, the
36:06gradient is one. It had to be something
36:08, obviously. So then, what would the
36:10student do? You need the equation of a
36:11straight line. So what will the student
36:13do? Put one in place of m. Put two in
36:15place of c. This is your equation of a
36:17straight line found. This is your
36:19O-Level Math. Finished. This is
36:22situation number one. This is the first
36:24situation. There you go, kid, this is
36:26the first situation. Whatever the
36:28coordinates or points are, I’m just
36:30giving you an idea. The idea is that in
36:32this condition the intercept was given,
36:35and one point was also already given.
36:38Okay? Now, the second option used to be
36:40this. The second option. Okay? This is
36:43obviously such a basic thing that every
36:44student must know it. I think. Here you
36:47go. Look at one more example. Now
36:49suppose there is a student who uses
36:51this equation. Now suppose there is a
36:53student who uses this equation. y-y 1 =
36:55m times x-x 1, a student uses this
36:58equation. This is also a perfectly
37:00valid equation. So what will this
37:02student do first? Find out the gradient
37:04. Obviously, y2-y1 over y2-y1 over x2-
37:07x1, here it is. So first they will find
37:10the gradient. Its gradient came out to
37:12be one. Now which point is x1 y1?
37:15Whether you use this as x1 y1 or you
37:17use that as x1 y1, the result from both
37:19will be the same. Clear, kid? So now
37:22use whichever point you want. Now
37:24suppose a student says I will use this
37:27point as x1 y1. So y-y 1 = m times x-x
37:301, there you go. Now when the student
37:33simplifies this further, y-x = x-4,
37:35there you go. When the student
37:37simplifies this further, what result is
37:39derived? y = x + 2, meaning both
37:42results were derived as the same. Same
37:45derived. Now, if any student has a
37:46doubt, they can definitely use the
37:48other point. Here you go. Now if a
37:50student says, "No sir, I will use this
37:52point x1 y1," then you can use this. So
37:54, what will the answer be? y-y 1 equals
37:56m times x-x 1, here it is. So when you
37:59simplify this, what will the answer be?
38:01x + 2, here it is. This means no matter
38:03which x1 y1 point you take, what result
38:05will finally be derived? The same
38:07result will be derived. This was your O
38:09Level Math. Clear, son? So it means
38:12whether you use y = mx + c or use this.
38:14But the interesting thing is that in A
38:17Level Math, this is the examiner's
38:19favorite equation. Clear, son? Okay,
38:21shall I show you another model if I may
38:24? I am showing you all the
38:25possibilities or situations. Look, let
38:27me show you another example. Suppose
38:29you have this condition. Here you go.
38:31Now suppose you have this one point.
38:33Take any coordinates for it. Suppose
38:35this point is 2 4 and this is another
38:37point. Its coordinates are 6 10, here
38:39you go. The examiner says, "Find the
38:41equation of the line passing through
38:43points A and B," okay? Now, the student
38:45who uses this mx + c equation in O
38:48Level Math, what will that student do?
38:51Look, if a student uses mx + c, what
38:53will they do? Look, here is the
38:57equation y = mx + c, there it is. Now
39:01the issue is that you neither have c
39:03nor do you have the gradient. Okay? So
39:06here, you will have to make both
39:07efforts yourself. So what did the
39:09student use to do first? In O Level
39:10Math, they used to find the gradient.
39:12What is the formula for the gradient?
39:14The gradient formula is y2-y1 over x2-
39:16x1, here you go, y2-y1 over x2-x1. So
39:19the student will say their gradient is
39:216/4 or 3/2. If you want to simplify it,
39:24go ahead, 3/2, there you go, that is
39:26the gradient. So then what did we use
39:28to say in O Level Math? We used to say,
39:30"Hey, the equation is mx + c," so the
39:31equation became something like this.
39:33Here you go. Then we used to find c. We
39:36know what x and y are. Any point on the
39:39line. Any point on the line. Now you
39:41can take whichever point you want.
39:42Suppose a student says to me, "Sir, I
39:44will use this point." No issue. Now you
39:46use this point. Okay? What did you do?
39:49You substituted the point. Here you go.
39:51You substituted the point. Here you go.
39:53Let's simplify it even further. So from
39:55here, the two and the two got cancelled
39:56. From this, we found out what the
39:57answer for C is? One. What did the
39:59answer for C become? One. There you go.
40:01The answer for C is found to be one.
40:03When the answer for C is found to be
40:04one, a student will say there is a
40:05Y-intercept, and there is a gradient
40:07too. What was your target? Your target
40:09was the equation of the line. So, what
40:11is the equation of the line? mx + c. So
40:15now, what will be its final answer?
40:17Finally, the answer will be 3/2x and
40:20finally + 1; this will be your final
40:23answer. Clear, kid? This is your answer
40:25. Alright. Now, the student who uses
40:27the second option. Suppose a student
40:30uses the second option. Option number
40:32two. Okay? So their answer will
40:34obviously be the same too. What was the
40:36second option? y-y 1 = m times x-x 1,
40:40finished. Here it is. So now, this
40:43student obviously has to find what
40:44first? Gradient. y2-y1 over x2-x1,
40:48meaning gradient is a must, obviously.
40:516/4 = 3/2, so the gradient for this
40:53student will also be what? It will be
40:55the same. Here it is. Now, which point
40:57is x1 y1? Any point. x1 y1 is called a
41:00given point on the line. Whether you
41:03take this one or you take that one. Is
41:05it clear, kid? So now suppose a student
41:07says, "Sir, I will use this." Okay?
41:09Let's say a student says, "Sir, I will
41:10use this point." Some student tells me,
41:13"I will use this one." Now what will
41:15you do? What is the equation? y-y 1;
41:17what is the gradient? 3/2 x-x 1, here
41:20you go. What will x-x 1 be? Here you go
41:23. Okay? You considered a point as x1 y1
41:26. 2, 4, here you go. y2-y1, y-y 1 = m
41:31times x-x 1. Meaning, what point did
41:34this student take as X1, Y1? 2, 4; if
41:37you don't want to take this, you can
41:39ignore it; take 6, 10. The result from
41:41both will be what? It will be the same,
41:42obviously. Okay, now make it a bit
41:44easier. When you make it easier, this
41:47is Y-4. This is 3/2x. This multiplied
41:51to become three. Now if this four goes
41:53there and gets added, it will become 3/
41:552x + 1, that's it, done. The result
41:58will be the same, you check it. That is
42:01why I am forcefully showing both
42:02options. Look, in front of you, I have
42:05shown both options, and I know that
42:06finally, the answer from both methods
42:08will be what? Same. What will the
42:11answer be from both methods? Same. In
42:13O-Level Math, such a third-class
42:14question used to come too. Such a
42:16third-rate thing. If you note this
42:19question. Suppose you have a question.
42:21Call it C, call it A. Let's say this
42:23point is 0 3 and the gradient is
42:24already given, here it is two.
42:26Sometimes it is written, "Find the
42:28equation of the straight line whose
42:29y-intercept is 0, 3, and gradient is
42:31two". So now in O-Level Math, if you
42:33look, this is the easiest example. By
42:35the way, if I tell you that I require
42:38its equation, then this is the easiest
42:40question. What is your target? Your
42:43target is the equation of the line mx +
42:45c, here it is, mx + c. So what will the
42:47student say? The student will say the
42:49gradient is two and the y-intercept is
42:513, so simply, what has been found here?
42:53The equation of the straight line. Done
42:55, this is the equation of the straight
42:58line. If a student uses the second
43:00option, obviously their result will
43:01also be what? It will be the same. If a
43:04student uses the second option. Look,
43:06what is the second option? y-y 1, m
43:09times x-x 1, here it is. Well, now from
43:13this too, the result will obviously be
43:15the same. All the examples are in front
43:17of you. The answer is the same for all.
43:18So look, the student will say the
43:19gradient is already given. Here it is,
43:21the gradient is two. So x1 y1 is what
43:23we call the given point on the line. So
43:25that is the only point, 0 3, and there
43:27is no other option. So what will the
43:30student do? y-y 1, m times x-x 1, here
43:34it is. So when you simplify this, what
43:37answer will be derived? The answer
43:39derived will be 2x + 3, so finally,
43:42note why I did this question repeatedly
43:44—so that I could show you that
43:46ultimately, the answer from all will be
43:49what? It will be the same. Is that
43:51clear, kid? So this is what we used to
43:53study in O-Level Math. We used to
43:55classify this type of line. A line with
43:57a slope, a slanted line, a diagonal
44:00line, a tilted line. Or for this line,
44:03another word is also used in Math:
44:05oblique. Oblique. Right? So I just
44:08showed you the first example. We
44:10started with a slanted line. Here it is
44:13. We started with a slanted line. We
44:15started with a very basic thing, a
44:17slanted line. Like we used to study in
44:18O-Level Math. Right? Slanted line, line
44:21with slope, oblique, clear. Okay, then
44:23we used to study one more thing in
44:25O-Level Math. This was a simple matter.
44:28Right? Then in O-Level Math, we used to
44:30study another type of line as well.
44:32Look at that one too, child. Okay? The
44:34second type of line, we must have
44:35studied this in O-Level Math as well.
44:37Okay? We used to call this a horizontal
44:39line. Okay? We must have studied this
44:42in O-Level Math too. Horizontal line.
44:44Okay? This was in O-Level Math as well.
44:46Interestingly, this is also going to be
44:48used in our syllabus. Obviously, that
44:50is why we are studying it. We aren't
44:51doing anything extra. Clear? Whatever
44:54the syllabus requirements are, we will
44:56check those conditions one by one. Then
44:59finally, we will start the practice.
45:01Alright, son? Now your target is the
45:02second one, the horizontal line. This
45:05was also a type of line. The previous
45:07type you studied, what was the name of
45:09the previous type you studied? It was
45:11called a line with a slope. Tilted line
45:14, diagonal, inclined, slope, oblique,
45:16there you go. We call it a horizontal
45:19line in O-Level Math. See, what is the
45:21meaning of a horizontal line? We say in
45:23O-Level Math that a horizontal line
45:26means a line parallel to the x-axis, a
45:29line parallel to the x-axis. Clear?
45:33Also, let me show you an example. We
45:37have studied in O-Level Math, all these
45:39lines, all the lines I am drawing, here
45:41it is, all these lines, we used to call
45:43all these lines, what are they? These
45:46are all lines parallel to the x-axis.
45:48All these lines, the gradient of all
45:51these lines will obviously be zero. We
45:53studied this in O-Level as well. I will
45:56repeat. What is the reason for that?
45:58The reason is that we studied in
46:00O-Level that the gradient of the x-axis
46:03is always zero. For a gradient, a rise
46:06is a must. So, in a horizontal line,
46:09there is no rise at all. In a
46:10horizontal line, there is no rise at
46:12all. Tell me one thing, if I give you a
46:15small example. Just a very basic
46:17example, that I considered two points
46:19on the x-axis. Well, first of all,
46:22there is no need to calculate this. We
46:24have known since childhood that the
46:26gradient of the x-axis is always zero.
46:28Check it, when you move horizontally on
46:31the x-axis, moving horizontally means
46:33there is no rise. There is only run and
46:35nothing else. Run, meaning your
46:37horizontal change is occurring. So, if
46:39you force the gradient formula, you
46:41will say y2-y1 over x2-x1, and you will
46:44say, well, the gradient will always
46:46come out to be zero. So we know that if
46:49the gradient of the x-axis is zero,
46:51then what should the gradient of every
46:53line parallel to the x-axis be? It
46:56should be zero. Therefore, if we talk a
46:58bit sensibly, we studied mx + c today,
47:01right? So that means m is not even
47:03possible, right? m became zero straight
47:05away. So when m is 0. When m is zero,
47:09that's why we learned in O-Level math
47:12that a horizontal line means y =
47:15constant. y = constant, and common
47:18sense also says that when you draw a
47:21horizontal line, say y = 3, there is no
47:23change in y on it. What is changing on
47:26it? When you mark any point, they will
47:28all have one thing in common.
47:30Everyone's y-coordinate will be fixed.
47:32So what will change in all of them? The
47:35change in all of them will be in x. So,
47:37can I also say what a horizontal line
47:39means? A horizontal line means y =
47:43constant. y = constant. And if your
47:47math is a bit stronger, you must have
47:49learned in O-Level math that any line
47:52above the x-axis means y is greater
47:54than zero. Because when we move upwards
47:57above the x-axis, y always increases.
48:00On the x-axis, y is always zero. And
48:03whenever we move under the x-axis, y
48:05always decreases. We say y is always
48:07negative. y is always negative. So what
48:10does this mean? y = constant? Meaning,
48:14on a horizontal line, there is only a
48:17change in x. Let me show you a simple
48:20example again. Suppose, I have written
48:23a line...a line y = 3, now look, this
48:26is one, this is two, this is three,
48:28this is four, this is five, this is six
48:30; these are your values on the x-axis.
48:33Now notice one thing, all these values
48:35on the x-axis! Now notice one thing, as
48:38you extend all these points, look, here
48:40it is, here it is. If I explain this to
48:44you in detail, then for any point on
48:46this line, its x will keep changing,
48:48but y is fixed, right? If I ask you
48:51what this point is? Then you will say
48:53sir, its coordinates are 4, 3, if I ask
48:55any child what this point is? So the
48:57child will immediately say its
48:58coordinates are 6 3, so what does that
49:00mean? On a horizontal line, y doesn't
49:02change at all. y is always constant.
49:05What changes? x is always what changes.
49:08That is why we don't really consider
49:09the horizontal line that much. By the
49:11way. So we always say a horizontal line
49:13means no change in, no change in y. No
49:15change in y. y is always constant. y is
49:19always constant. Okay, then we used to
49:21study one more thing in O-Level Math.
49:23Okay? We used to study one more thing
49:25in O-Level Math. Maybe we remember. Its
49:27name was vertical line. Okay? We used
49:30to call these vertical lines in O-Level
49:33Math. We must have studied this in
49:34O-Level Math too. Especially there was
49:36a topic, reflection. Transformation. In
49:39transformation, a child studies this
49:41very frequently. Vertical lines. Well,
49:43everything is important. When will our
49:46math be strong? When we make sense of
49:48everything. Okay, what did we study in
49:51O-Level Math? In O-Level Math, it was
49:54used a lot in inequalities,
49:56transformations, especially reflection.
50:00Vertical line. We used to say, man,
50:02what is the meaning of a vertical line?
50:04Vertical lines mean a line parallel to
50:06the y-axis. A vertical line means a
50:08line parallel to the y-axis. Look at
50:10this. If you remember a little bit of
50:13O-Level Math. If you remember a little
50:15bit of O-Level Math, these used to be
50:17vertical lines. Look, I have
50:19deliberately drawn some lines in yellow
50:21color. I just drew some lines randomly.
50:24Now you note one thing. All these lines
50:27are parallel to the y-axis. All these
50:30lines are parallel to the y-axis. Look,
50:32all these lines are parallel to the
50:35y-axis. Okay? We already know these are
50:37parallel to the y-axis. And maybe some
50:39child remembers coordinate geometry.
50:40Even today we studied previously that
50:43the gradient of the x-axis is always
50:45zero. So the gradient of the y-axis is
50:48always -1/0, meaning we call it
50:51undefined. Meaning for a vertical line,
50:53we don't even use the word gradient.
50:55Mostly we don't use it at all. In fact,
50:58just as we say that in a horizontal
51:00line there is no change in y, y is
51:03constant. So similarly, what did we
51:05study in O-Level Math? Let me show you
51:07a small example we studied in O-Level
51:09Math. First, suppose you have this one,
51:11this two, this three. Here you go.
51:13Suppose you have this three. Now, if I
51:16draw a vertical line from this. See, I
51:18have drawn a vertical line. Here it is.
51:20This vertical line passes through three
51:22. Now, notice one thing. Okay, I have
51:24drawn the vertical line. Here it is.
51:26This vertical line is passing through
51:27three. So, in O-Level Math, we used to
51:30call vertical lines x = constant. Just
51:33like horizontal lines are called y =
51:35constant. So, this was called x =
51:37constant because x is fixed on it.
51:40Meaning, look at what is changing here?
51:42If you check, the x-coordinate is the
51:44same. The y-values are continuously
51:46changing. Notice. If I ask a student, "
51:49Hey, what is this point?" Let's say if
51:52I tell a student, "This is 1, 2, 3, 4,
51:565, -1, -2, -3, okay?" I gave an example
51:59to the student. If I ask a student what
52:01this point would be. Then what will the
52:03student say immediately? The student
52:05will say, "Sir, 3, 1; 3, 1 is right
52:07here." If I tell a student, "Hey, what
52:09would this point be?" Then the student
52:11will say, "1, 2, 3, 4, 5," so the
52:13student says, "3, 5," which means what
52:16is fixed on this? What is fixed on this
52:20? The x-coordinates are fixed on this,
52:23the x-coordinates are fixed, while y is
52:26changing. So, when you talk about a
52:29horizontal line, x is changing on a
52:31horizontal line, and common sense says
52:33y is changing on a vertical line.
52:35Common sense. If it's a vertical line,
52:37y will change. x will be constant. If
52:40you talk about a horizontal line, x
52:42will change. y will be constant. Clear,
52:45kid? So, we used to call this a
52:47vertical line. We don't even talk about
52:49the gradient of vertical lines in math.
52:52So, if I summarize a bit more, we used
52:55to say that all these vertical lines,
52:57what did we call all these vertical
52:59lines? We used to call all these
53:02vertical lines x > 0. Does any student
53:04remember what we called the y-axis? x
53:07is 0, and these vertical lines I am
53:09drawing on the left side, we used to
53:11call them all x < 0; we studied this in
53:13O-Level Math, and common sense says
53:15that on the left side, x-values are
53:17negative, and on the y-axis, x is
53:19always zero. On the y-axis, x is always
53:23zero. You learned this in O-Level Math.
53:26Learned in O-Level Math. Similarly, we
53:28kept calling this y > 0, here we kept
53:30saying y < 0, and here we kept saying y
53:33= 0, so we learned in O-Level Math that
53:35what is the x-axis called? We call the
53:38x-axis y = 0, y = 0, so there is
53:42nothing extraordinary in this. So today
53:45we just looked at the equation of a
53:47straight line. We looked at oblique,
53:49slant, inclined, slope, and tilted
53:51lines. Then we looked at a horizontal
53:54line, a line parallel to the x-axis.
53:57Then we looked at a vertical line. A
53:59vertical line means a line parallel to
54:01the y-axis. A line parallel to the
54:04y-axis. Finally, one last point of
54:06coordinate geometry. Then, InshaAllah,
54:09we will first practice all this
54:10coordinate geometry. Clear, beta? Here
54:13is one more last point. The name of
54:15this point is. By the way, this point
54:17is also carried forward from O-Level
54:19Math. This point is also carried
54:20forward from O-Level Math. The name of
54:22this point is the equation of the
54:23perpendicular bisector. Perpendicular
54:25means 90 degrees. Bi means two. Sector
54:28means to divide. So now I will explain
54:31what a perpendicular bisector is. This
54:34is, by the way, carried forward from
54:35O-Level Math. Okay? Perpendicular
54:37bisector. Okay? We must have learned
54:39this in O-Level Math. Look closely.
54:41What was O-Level Math? Suppose I tell
54:43you, look, I have a line here. Okay? I
54:47am just giving a random example. Random
54:48, this point we also studied in O-Level
54:50Math. Look at this. So suppose this
54:52point is your A. This point is your B.
54:55If the examiner asks me to find the
54:57equation of the perpendicular bisector
55:00of line AB. The examiner asks me to
55:05find the equation of the perpendicular
55:08bisector of the line through A and B.
55:12So, what did we use to do in O-Level
55:16Math? Look at this. What we used to do
55:19in O-Level Math, look closely, beta.
55:21What does a perpendicular bisector do?
55:23Number one. In O-Level Math, we kept
55:24saying this line is the perpendicular
55:26bisector. In O-Level Math, we kept
55:28saying that this line I drew is the
55:30perpendicular bisector. Why? How? What
55:34did we learn in O-Level Math? Number
55:35one. A perpendicular bisector, what is
55:38the job of a perpendicular bisector?
55:41The job of a perpendicular bisector is
55:43to bisect the given line. Meaning, to
55:46divide it into two equal parts. The
55:48word perpendicular means 90 degrees.
55:51Look, the word perpendicular means 90
55:53degrees. So, if anyone asks us what a
55:56perpendicular bisector does? So, what
55:58is required for a perpendicular
56:00bisector? Look, what are the qualities
56:02of a perpendicular bisector? What are
56:05its properties? So, whenever we talk
56:08about a perpendicular bisector. Okay?
56:10Perpendicular bisector of a line
56:12through A and B, let's say, or call it
56:14line AB. Line through A and B is
56:17actually the proper term. So, the first
56:18thing, where does the perpendicular
56:20bisector of any line pass through? Rule
56:23number one: a perpendicular bisector
56:26always passes through the midpoint of
56:29the given line. We learned in O-Level
56:32Math that to perpendicularly bisect,
56:34see, if a line has to be bisected,
56:36where will it always be bisected from?
56:38From the middle. So, what is the job of
56:40a perpendicular bisector? To break your
56:42given line into two equal parts. Is
56:45that clear, son? Number two: if the
56:47gradient of your given line is m, then
56:48what will be the gradient of the
56:50perpendicular bisector? The negative
56:51reciprocal. What does the word
56:53perpendicular even mean? Negative
56:54reciprocal. So, the second thing I
56:56would tell you is, what is the gradient
56:58of a perpendicular bisector? If someone
57:01asks us what the gradient of a
57:02perpendicular bisector is? Then what is
57:05it always? It is always the negative
57:08reciprocal. The negative reciprocal of
57:12the given line, the negative reciprocal
57:16of the given line. So, it is very clear
57:19that you have this line here. You have
57:21to bisect this line. "Bi" means to
57:23divide into two parts. So, when is
57:25something divided into two parts? When
57:27it is broken from the center. So, where
57:29does a perpendicular bisector always
57:31pass through? Through the middle, and
57:33it always forms 90 degrees with the
57:35line. With the given line. So, the
57:37gradient of the perpendicular bisector
57:38is always the negative reciprocal of
57:40the given line. Is that clear, son?
57:42Negative reciprocal of the given line.
57:43So, always remember these two things.
57:45Whenever the word "perpendicular
57:47bisector" is used, you must first have
57:50an understanding of these two things.
57:52Is that clear, son? If you have a sense
57:54of these two things, then we will start
57:56with perpendicular bisectors slowly.
57:58Perpendicular bisector. By the way, for
58:00a perpendicular bisector. Let's first
58:02use the word "perpendicular bisector"
58:04once more. "Perpendicular" means 90
58:06degrees. "Bi" means two. "Sect" means
58:09to divide. Okay? Okay? "Bi" means two,
58:12"sect" means to divide. Okay? So, what
58:14does a perpendicular bisector do? It
58:16divides your given line into two equal
58:18halves, two equal parts. Actually,
58:20another word is also used for a
58:22perpendicular bisector. So, let me
58:24write a word here first. So, another
58:26word is also used in math for a
58:29perpendicular bisector. And that word
58:32is line bisector. A perpendicular
58:35bisector is also called a line bisector
58:37. Because the perpendicular bisector
58:39itself is a line. All right? And what
58:42is its job? To bisect a given line.
58:44That is, a line that bisects a given
58:47line, divides it into two equal halves
58:51and forms a 90-degree angle with it, is
58:54called a perpendicular bisector.
58:57Another name for it is line bisector.
59:00If you are a bit smart, when the
59:02perpendicular bisector divides the
59:04lines equally, it means this is its
59:05mirror image and this is its mirror
59:07image. So I can also say that the
59:10perpendicular bisector is a line of
59:13reflection. If you have a little bit of
59:16common sense. Where will the
59:17perpendicular bisector pass from?
59:18Through the center. It divided it into
59:20two equal halves. The distance from the
59:23midpoint to this side is the same as
59:24the distance to the other side. So this
59:27is its mirror, and this is its mirror.
59:30So for the perpendicular bisector, I
59:32can also use this word. Line of
59:34symmetry. Let's say the perpendicular
59:37bisector did this: it divided your
59:39given line into two equal parts. So the
59:42point that was on the right has its
59:43mirror here, and the one on the left
59:45has its mirror here. That's what a
59:46perpendicular bisector means, right? So
59:48what is the perpendicular bisector
59:50itself? The perpendicular bisector
59:52itself is a line. And what is its job?
59:55To bisect a given line. Clear, kid? To
59:58bisect a given line. I'll show you a
1:00:00small example. Then we'll start slowly.
1:00:03Suppose I show you a random O-Level
1:00:05math style question. O-Level math style
1:00:08. Suppose you have this point and this
1:00:10point. I am just giving a random
1:00:12example. I am just giving a random
1:00:14example. Suppose you have a point here.
1:00:16The coordinates of that point are 2, 4.
1:00:18There you go. And another point. The
1:00:20coordinates of that point are 6, 10.
1:00:22There you go. 6, just a random example.
1:00:25The examiner says, "Find the equation
1:00:27of the perpendicular bisector of the
1:00:30line through A and B, find the equation
1:00:32of the perpendicular bisector of the
1:00:34line through A and B, of the line
1:00:36through A and B." Now, Insha'Allah, I
1:00:38will have you practice all of this, and
1:00:40show you examples of everything. First
1:00:43thing, okay? Now we need the
1:00:45perpendicular bisector. First thing,
1:00:47what does a perpendicular bisector do?
1:00:49It divides your given line into two
1:00:51equal parts. Straight away, what is the
1:00:54first thing I did? The first thing
1:00:56means, what is it supposed to do? You
1:00:59must always bisect the given line. Form
1:01:02a 90 ° angle. That means you must
1:01:03always divide the given line into equal
1:01:05parts. Is that clear, son? I will show
1:01:07it like this. Here you go. Now, show
1:01:09the 90 ° angle like this. Plot it a
1:01:10bit more properly. Here you go. That
1:01:12will work. So, what is the job of a
1:01:14perpendicular bisector? To divide the
1:01:15given line into two equal parts. Is it
1:01:17clear, son? Now, look, this white line
1:01:21is called the perpendicular bisector of
1:01:24AB. Okay? So, below this white line, I
1:01:27will write that this is my target. Who
1:01:30is this line? This line is the
1:01:32perpendicular bisector. What is another
1:01:34name for it? Line bisector. Meaning the
1:01:36line that divides your given line into
1:01:38two equal parts. What is the first
1:01:41thing you will do? The first thing you
1:01:43will do is find out. The midpoint of AB
1:01:45. The midpoint of AB. So, what is the
1:01:47formula for the midpoint? x1 + x2/2,
1:01:49what is the formula for the midpoint?
1:01:52x1 + x2/2, y1 + y2/2, look, this is the
1:01:57midpoint formula. What is the midpoint
1:02:00of AB? x1 + x2/2, y1 + y2/2, here it is
1:02:07. So, what is the midpoint? The
1:02:09midpoint is 4, 7; this is your midpoint
1:02:12. So, this means that your line, the
1:02:16white perpendicular bisector, will pass
1:02:18through which point? It will pass
1:02:20through the midpoint, that is, through
1:02:224, 7. Is it clear, son? This is a very
1:02:25basic thing. Number two, if I ask you
1:02:28what the gradient of the given line is?
1:02:30What is the gradient of the given line?
1:02:33Then you will use the simple formula
1:02:35for gradients. y2-y1, x2-x1, here you
1:02:38go. So the student will say, "Sir, it's
1:02:40simple, here is the formula." 6/4, 3/2,
1:02:42there you go, this is the gradient of
1:02:44the given line, meaning what is the
1:02:45gradient of the given line? The
1:02:47gradient of the given line is 3/2, so
1:02:49that means what will be the gradient of
1:02:51the perpendicular bisector? The
1:02:53gradient of the perpendicular bisector
1:02:55will be the negative reciprocal, i.e.,
1:02:57-2/3, here you go. Now, what is your
1:03:00target? You need the equation. Now your
1:03:02target is achieved. Your target is
1:03:05achieved. Because you need two things
1:03:07for the equation. A point and a
1:03:10gradient. So, which point will be used?
1:03:12The midpoint will be used as the point.
1:03:14Here you go. The point will be the
1:03:16midpoint. What gradient will you take?
1:03:18The gradient of this will be -2/3.
1:03:21There you go, that's it. What is your
1:03:24target? The equation of a straight line
1:03:26. So, what will you say now? Use the
1:03:28y-y 1 = m (x-x 1) formula, and that's
1:03:33it. Now, simplify this a little bit.
1:03:36Just simplify it. Is that clear, son?
1:03:38Now, simplify it however you like. What
1:03:41does this become now? 3y-21 = -2 x + 8,
1:03:45there it is. So I can show it like this
1:03:48as well. 3y = -2 x + 29, here it is.
1:03:52Right here. Sometimes the examiner asks
1:03:53to show it this way too. Show it as ax
1:03:56+ by + c = 0. That's not an issue. You
1:03:59bring the x here first. Here is x, then
1:04:02you write y. Here is y, then you bring
1:04:04the constant here. Take this. A student
1:04:06would say, this is the equation. That's
1:04:08it. And if you want to write it in the
1:04:11form mx + c, just write -2/3x + 29/3,
1:04:14here it is. So, this is not rocket
1:04:16science. This is exactly O-Level Math.
1:04:19It is exactly O-Level Math. So, for the
1:04:22perpendicular bisector, just keep two
1:04:23things in mind. Where does the
1:04:25perpendicular bisector always pass
1:04:26through? The midpoint of the given line
1:04:28. And what is its gradient? The
1:04:30gradient of the perpendicular bisector
1:04:32is always the negative reciprocal of
1:04:35the given line. Okay, now I'm telling
1:04:37you something interesting. Okay? Now I
1:04:39am telling you something new. Okay?
1:04:41Finally, God willing, we will start
1:04:43practicing this now. Look at a new
1:04:45thing. Now I am showing you some 2D
1:04:47shapes. Okay? I am showing you some 2D
1:04:50shapes. Properties of 2D shapes. First,
1:04:52you look at them for a minute or two.
1:04:56Properties of 2D shapes. We have
1:05:01studied some properties of 2D shapes
1:05:03slowly from time to time. We studied
1:05:06their properties as the right time came
1:05:08. Now I am showing some 2D shapes. The
1:05:12perpendicular bisector is used in some
1:05:14three or four 2D shapes. Let's see how.
1:05:17Number one. Number one. First of all,
1:05:19if we talk about a square. Okay? So
1:05:22look carefully, son. Because we have
1:05:25studied the shapes repeatedly. Square,
1:05:27rhombus, rectangle, parallelogram,
1:05:30trapezium, kite, right-angled,
1:05:33isosceles, equilateral. Look, I am
1:05:36talking about the square again now.
1:05:38Clear, son? Now I will explain to you
1:05:40what a perpendicular bisector is. Well,
1:05:42we have already studied the properties
1:05:44of a square before. The corners are 90
1:05:46degrees. Now, if I explain this in more
1:05:49detail, both diagonals of a square
1:05:51bisect each other. Is that clear, kid?
1:05:55Keep this in mind: both diagonals of a
1:05:58square bisect each other. So, I can say
1:06:02AC and BD are perpendicular bisectors
1:06:06of each other. This AC and BD are
1:06:10perpendicular bisectors of each other.
1:06:15Perpendicular bisectors of each other.
1:06:18What do these two do to each other?
1:06:20They bisect each other. So that means,
1:06:22where is AC? What must AC be doing? If
1:06:24it bisects BD, where must AC pass
1:06:26through? Through the midpoint of BD.
1:06:28What must the gradient of AC be? The
1:06:30negative reciprocal of BD. What must BD
1:06:33be doing? BD must be bisecting AC. What
1:06:37will the gradient of BD be? The
1:06:39negative reciprocal of AC. So, remember
1:06:42that in a square, both diagonals do
1:06:44what to each other? They bisect each
1:06:47other. Clear? So, these properties
1:06:49should be known as well. Look at one
1:06:51more thing. One more shape. We had
1:06:53studied one more shape. Let's look at
1:06:55that again too. Rhombus. A rhombus also
1:06:58has perpendicular bisectors. Let's
1:07:00check. A rhombus also has perpendicular
1:07:03bisectors. We have studied before, what
1:07:05is a rhombus? A rhombus is a tilted
1:07:06square. A square that is tilted. So,
1:07:09what happens inside a rhombus? Both
1:07:11diagonals of a rhombus also always
1:07:14bisect each other. What does bisect
1:07:17mean? To divide into two equal parts.
1:07:20To divide into two equal parts. Clear,
1:07:23kid? This is regarding the rhombus.
1:07:25Here is A, B, C, D; so if I talk about
1:07:29a rhombus, it has the same property. In
1:07:32a rhombus too. Okay? So this property
1:07:35applies to the rhombus as well. This
1:07:37property is for the rhombus and this
1:07:39property is for the square too. Here
1:07:41you go. These are the common properties
1:07:44between these two. Clear, kid? What was
1:07:46the name of the shape? The name of the
1:07:48shape was square and the name of the
1:07:50shape was square and rhombus. Okay, kid
1:07:52? Now I am explaining one or two more
1:07:54things. Okay? So, let me give you all
1:07:57the basic knowledge once. Then, God
1:07:59willing, we will start our chapter.
1:08:02Alright, now I am explaining one more
1:08:04thing. We must have often seen another
1:08:05shape in math. Kite, about the shape.
1:08:08Kite. Kite. Okay? Now what happens in a
1:08:12kite? Note this. Kites also have
1:08:14perpendicular bisectors. You must have
1:08:16read this in O-Level Math too. The
1:08:18topic name would be Mensuration. Okay,
1:08:20kid? The topic name will be Mensuration
1:08:23. Suppose you have a shape here. The
1:08:25name of the shape is a kite. So what
1:08:27happens in a kite? Note this down. Now,
1:08:29I'm telling you an interesting thing.
1:08:31Suppose this is A, this is B, this is C
1:08:33, this is D. See, in a kite, this
1:08:35triangle is small and this one is
1:08:37larger. Obviously. This line here, the
1:08:41longer diagonal, bisects the smaller
1:08:44diagonal. It divides it into two equal
1:08:47parts. The smaller one cannot bisect
1:08:49the larger one because the midpoint of
1:08:51the larger one would be somewhere else.
1:08:53So, when we talk about a kite, what
1:08:55happens in a kite? In a kite, we say BD
1:09:00is the perpendicular bisector of AC. In
1:09:07a kite, they don't both bisect each
1:09:10other. This longer diagonal, the larger
1:09:13one, what is its job? To bisect the
1:09:16smaller one. So BD is the perpendicular
1:09:18bisector. Meaning BD bisects AC, BD
1:09:22bisects AC, here it is. So what does BD
1:09:25do? It divides your AC into two equal
1:09:28parts. Done. It divides it into two
1:09:31equal parts. So, in a kite, there is a
1:09:34perpendicular bisector too. Meaning, in
1:09:37a kite, the longer diagonal is the
1:09:39perpendicular bisector. If you wonder
1:09:41who BD is? Then a student would ask who
1:09:43is BD? BD is the longest diagonal. BD
1:09:46is the longest diagonal. So the longer
1:09:49diagonal or the longest diagonal
1:09:51bisects the smallest one. It bisects
1:09:53the smallest one. Take a look at one
1:09:56more small thing. Look at another small
1:09:58thing. We study another triangle. The
1:10:01name of the triangle is isosceles
1:10:02triangle. We must have studied it many
1:10:04times under different conditions.
1:10:05Isosceles triangle, isosceles triangle.
1:10:09Note a property within an isosceles
1:10:11triangle as well. We studied a topic
1:10:13before this called circular measurement
1:10:15. We studied it there too. If I tell
1:10:17you this triangle is isosceles. Then
1:10:19the student will immediately say that
1:10:21these two base angles will be equal.
1:10:23Every student knows this. What is its
1:10:25property? That when you drop a line,
1:10:28okay? When you drop a line, where will
1:10:31you drop it from? Between the two equal
1:10:33sides, it will always do what to the
1:10:35base? It will bisect it. What will it
1:10:38do to the base? It will bisect it.
1:10:40Change the triangle. Change the
1:10:43triangle. Suppose you have this
1:10:45situation. There you go. Change the
1:10:47triangle. Take another isosceles. There
1:10:49you go. Suppose this and this are both
1:10:50equal. So, math says that these two
1:10:51angles will then be equal. Math says
1:10:53then these two angles will obviously be
1:10:55what? They will be equal. Look, the
1:10:56base angles will be what? They will be
1:10:58equal. There you go, these two base
1:11:00angles will be equal. If this is alpha,
1:11:01then math says this is also alpha. So,
1:11:03when you drop a line from here between
1:11:05two equal sides, what does that line
1:11:07always do to the base? It always
1:11:09bisects the base. What does it always
1:11:12do to the base? It bisects it. So, I
1:11:14will give this a name. This is A, this
1:11:16is B, this is C, this is D. Let's name
1:11:17this one too. A, B, C, D. So, who is AC
1:11:22here? AC is the perpendicular bisector.
1:11:25AC is the perpendicular bisector. Okay?
1:11:27Now, whether the situation is like this
1:11:29, math will say these two angles of
1:11:31yours are equal. So, when I drop a line
1:11:34from here, what will this line do to
1:11:35the base? This line will bisect the
1:11:39base. Clear, son? You can name this A.
1:11:41Name this B, this C, this D. This is
1:11:44all isosceles for you. So, what is AC
1:11:47here? I can say here that AC is the
1:11:52perpendicular bisector of BD, that's it
1:11:58. So, there is a perpendicular bisector
1:12:02in isosceles as well. Clear, son? Look,
1:12:05in this, both of these are bisecting
1:12:08each other. This is bisecting this. AC
1:12:11is bisecting BD. Okay? Similarly, BD is
1:12:15bisecting AC. In this, BD is bisecting
1:12:19AC. In this, AC is bisecting BD. Clear?
1:12:22Who are all these? Perpendicular
1:12:23bisectors. And just one last thing. Now
1:12:26, God willing, our coordinate geometry
1:12:28basics are done. Now, God willing, we
1:12:30will practice this. Now, look at one
1:12:32more thing. This point is named
1:12:34equilateral triangle. You must have
1:12:35studied this in O Level math as well.
1:12:37You also studied it in circular
1:12:38measurement in A Level math.
1:12:40Equilateral triangle. What does
1:12:42equilateral mean? Look, equilateral
1:12:44means all sides are equal. We already
1:12:47know what an equilateral triangle means
1:12:49, one where all sides are equal. We
1:12:52used to call it a regular polygon in O
1:12:53Levels too. So, this means this and
1:12:55this are both equal. Note this. This
1:12:57and this are both equal. So, when we
1:12:59drop a line, it will bisect the base.
1:13:01Obviously, this and this are also equal
1:13:03. So, when we drop a line from here, it
1:13:05will bisect the base. When we drop a
1:13:08line from here as well, what will it do
1:13:09to the base? It will bisect it. Clear,
1:13:12kid? So that means it has three
1:13:14perpendicular bisectors. Alright, kid?
1:13:18It has three perpendicular bisectors.
1:13:21Okay? When we talk about what? An
1:13:24equilateral triangle. Alright, kid?
1:13:27When we talk about an equilateral one,
1:13:29it has three perpendicular bisectors.
1:13:31When you drop a line, it will bisect
1:13:33the base. Alright, kid? And what will
1:13:36it form with the base? 90 °. So I told
1:13:39you this so that at least you know
1:13:41which shapes have perpendicular
1:13:43bisectors. Clear, kid? So today we have
1:13:46finally completed all these seven
1:13:49points that are considered basic
1:13:51coordinate geometry for AS. Clear, kid?
1:13:55In the next class, God willing, we will
1:13:57start practicing basic coordinate
1:13:59geometry from here. Alright, kid? Allah
1:14:01Hafiz.

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