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AS Level Mathematics 9709 | Coordinate Geometry 3 | Gradient & Slope | Raja Shafqat — Transcript

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0:08Assalamu alaikum, son. In the last
0:10class, we studied ratios and midpoints.
0:13Today, we will continue from there.
0:15Today we will study the fourth point of
0:17coordinate geometry: gradient or slope.
0:19Okay? We have already studied this in
0:20O-Level Math. Obviously, it's the exact
0:22same point. In O-Level Math, in
0:25coordinate geometry, inequalities,
0:26kinematics, graphs, and functions. Look
0:29, first of all, what formula for
0:32gradients did we study before? Let me
0:34start a bit from O-Level Math. Suppose
0:37you have these two different points on
0:39the line. Meaning, this is a line
0:41segment, by the way. It is a line
0:43segment. Point A has coordinates, which
0:46is point A. Let's read the coordinates
0:48of point A as X1 Y1. Okay, son? Let's
0:51read the coordinates of point A as X1
0:53Y1. Here you go. I have labeled point
0:58A's coordinates as X1 and Y1, and point
1:01B's coordinates as X2 and Y2. These two
1:05points are on the line. Okay? This is a
1:09line segment, a part of a line. Our
1:11goal is to find its gradient. Is that
1:13clear, son? So, we also studied in
1:15O-Level Math that if you need the
1:17gradient, what is the gradient itself?
1:18We call the gradient "rise over run".
1:21We must have studied this in O-Level
1:22Math and Physics. What do we call "rise
1:25over run"? It is the gradient. So now
1:28tell me, what is rise and what is run?
1:31Number one, what is run? What is rise
1:34and what is run? Run is change in x and
1:37rise is change in y. When we studied
1:40the distance formula, we saw this as
1:42well. In the distance formula, we find
1:45this hypotenuse. So what did we learn
1:47in O-Level Math about the gradient
1:48formula? Rise over run, meaning change
1:51in y over change in x. So, if I ask you
1:53what the gradient formula is? What
1:55formula for gradients did we learn in
1:57O-Level Math? The gradient formula is
2:01y2-y1 over x2-x1; in O-Level Math, we
2:04used to call this formula "rise over
2:08run". Meaning, indirectly, I can also
2:11use this term for gradients: what is
2:13the gradient formula? The gradient
2:15formula is opposite divided by adjacent
2:17. The gradient formula is opposite
2:21divided by adjacent. Opposite divided
2:24by adjacent. I can say this too because
2:27all these things mean the same thing.
2:28Call it rise over run, or call it
2:30opposite over adjacent. It's the same
2:31thing. So, this is the gradient formula
2:33we studied in O-Level Math. Is that
2:35clear, kid? So, gradient. Alright,
2:37first let's start with a couple of
2:39small points. We learned in O-Level
2:42Math that lines are of three different
2:43types, which we'll study later. Either
2:46the line is vertical, horizontal, or
2:49diagonal, inclined, or sloped. Right,
2:52we studied this in O-Level Math. I'm
2:54telling you without working that
2:56whenever a line's gradient is positive,
2:58its direction is like this. We studied
3:01this in O-Level Math. You must have
3:02studied in kinematics that whenever the
3:04line's direction is of this type.
3:06Whenever a line's gradient is positive,
3:08its direction will always be like this.
3:10Meaning, what is this? Positive
3:12gradient. Increasing slope or positive
3:15gradient. Similarly, if I tell you that
3:17the line's direction is like this. This
3:19line is of this type. The line's
3:22direction is like this. So, in O-Level
3:24Math, this was called a negative slope
3:27or negative gradient. Negative slope or
3:30negative gradient. So, what was this?
3:32This is a positive gradient. And what
3:34is this? Negative gradient. We know
3:37this without needing to work it out.
3:38Still, if any student has a doubt,
3:40let's check it. Suppose I have this
3:43point. These are two points on the line
3:46. I need to find the gradient of this
3:47line. I already know this anyway
3:49without working. The gradient of this
3:51line must be positive. So, this point
3:53is, say, on the y-axis at 0, 2, and
3:56suppose this point is 4, 6; we need to
3:58find the gradient of this line. I know
4:01without working that whenever a line
4:03has this direction, the gradient is
4:05always positive. So, if I calculate the
4:07gradient, this point is x1, y1 and
4:09these point coordinates are x2, y2, so
4:12if I need the gradient, we applied the
4:14gradient formula. What is the gradient
4:16formula? The gradient formula is y2-y1
4:19over x2-x1, so when I calculated the
4:21gradient, the answer for the gradient
4:24came out to be 1, so I knew without
4:26working that whenever my line has this
4:29direction, the gradient will
4:31automatically be what? It will be
4:33positive. I automatically know this.
4:36This point we also studied in O-Level
4:38Math, what is the gradient formula?
4:40Rise over run, y2-y1 over x2-x1, or we
4:43also call it opposite over adjacent.
4:47Later there will be a topic,
4:48differentiation. So you will use this
4:50same formula there, noting that there
4:52is another formula for gradients, tan
4:54theta. Because what is tan theta also?
4:56It is just rise over run. Look here,
4:58what is the formula for tan? Rise over
5:00run. What is the formula for gradients?
5:02Rise over run. So you will study this
5:04concept in differentiation. Clear,
5:06child? So, we have just seen the simple
5:08formula for gradients. Change in y over
5:10change in x, rise over run. Okay? Right
5:13, then in O-Level Maths, we must have
5:16studied what relations exist between
5:19pairs of lines. Look, in O-Level Maths,
5:23we studied pairs of lines. Okay? We
5:26must have studied two things in O-Level
5:28Maths regarding pairs of lines.
5:29Whenever we say that we have two lines.
5:33Okay? I am talking about pairs of lines
5:34. Watch. Pairs of lines means there are
5:37two lines. What are the possible
5:39relationships? Okay? Okay? This too, we
5:42studied in O-Level Maths. Right, look,
5:45we studied in O-Level Maths that there
5:46are two types of relationships. Okay?
5:49In pairs of lines, either the lines are
5:51parallel. Listen closely. We must have
5:54studied in O-Level Maths that when we
5:55talk about two lines, what
5:57relationships can exist between two
5:58different lines? Either the lines are
6:01parallel, or they are non-parallel.
6:05Obviously, we are talking about basic
6:07coordinate geometry. There are parallel
6:10lines or non-parallel lines. Now, if I
6:12define non-parallel in more detail, I
6:15can also say that non-parallel lines
6:18can also be perpendicular. They can be
6:22perpendicular. And non-parallel lines
6:24can also be non-perpendicular. They can
6:28also be non-perpendicular.
6:30Non-perpendicular, non-perpendicular.
6:34So, this means there are two situations
6:37. Either the lines will be parallel, or
6:40the lines will be non-parallel. If we
6:43speak in simpler language, we can also
6:46say that either the lines will be
6:47parallel, or the lines you have, how
6:49will they be? Intersecting lines. So,
6:53what does non-parallel mean in simple
6:56language? I can also say this for
6:57non-parallel. Meaning, what does your
6:59math say? Math says either your lines
7:01will be intersecting or they will not
7:03be intersecting. So, what else can you
7:05call non-parallel in simple language?
7:08You can call non-parallel, intersecting
7:10lines. You can call non-parallel,
7:12intersecting lines. So, this means we
7:14study two types of conditions in math.
7:16Either the lines will be parallel, or
7:19the lines will be intersecting. Clear,
7:21son? Now, let me explain this in a bit
7:23more detail. We studied in O-Level Math
7:25that if lines are parallel, what does
7:28parallel lines mean? If lines are
7:31parallel, what did we study in O-Level
7:32Math? We studied in O-Level Math that
7:35whenever two lines are parallel, their
7:37gradients are always the same. Clear,
7:40son? And we will very frequently study
7:42in AS ahead, what does parallel lines
7:44mean? Whenever the gradient of two
7:46lines is the same, we say those two
7:48lines are parallel. Clear? Look, we
7:50must have often studied in O-Level Math
7:53that suppose these two lines are
7:54parallel. If these two lines are
7:56parallel, what will their gradients
7:58always be? They will be the same. Now,
8:00whether they are vertical, if they are
8:01parallel, it could also be this
8:03condition. Both lines are vertical. It
8:06could also be that both of the lines
8:08are horizontal. Right? Both of the
8:10lines are horizontal. Meaning, whenever
8:13two lines are parallel, whenever two
8:14lines are parallel, we have studied
8:16from O-Level Math that their gradient
8:17is always what? The same. So, parallel
8:21lines means the same gradients. We
8:23studied O-Level Math that what does
8:25parallel lines mean? Same gradient.
8:28Right? Same gradient. Meaning the ratio
8:31of rise over run for one line is the
8:33same as the ratio of rise over run for
8:35the second line. Meaning both have the
8:37same gradient. Clear, son? Then we must
8:39have studied one more thing. If lines
8:41are parallel, then okay, their gradient
8:43will be the same. If lines are not
8:45parallel, then there is one more
8:47condition. We call it intersecting
8:49lines. Intersecting lines. Right?
8:52Intersecting lines. Well, there are two
8:56points in intersecting lines too. Right
8:59? Now, if lines are not parallel, then
9:01they will not be parallel, so they will
9:03cut each other. If they are parallel,
9:05then that's fine. Parallel means
9:07non-intersecting. The word parallel
9:10means non-intersecting. Right, son?
9:13Non-intersecting, they won't cut each
9:15other. Now there are two conditions in
9:17intersecting as well. Look, there are
9:19two conditions in intersecting as well.
9:22You see, I showed you two random
9:24diagrams like this. Now if you look at
9:27this condition, then what kind of
9:29intersecting lines are these? What will
9:31I call such intersecting lines? I will
9:34call such intersecting lines
9:35non-perpendicular intersecting lines.
9:38Non-perpendicular lines. What kind of
9:42lines are these? These lines are
9:44non-perpendicular lines.
9:45Non-perpendicular lines. Right? And
9:48similarly, obviously, these lines are
9:50not intersecting. Sorry, these are
9:53obviously not parallel lines, so they
9:55must be intersecting. Now, it's not
9:57necessary that the angle between those
9:59two intersecting lines is always 90 °.
10:02It could also be that the angle between
10:04them is not 90 °. So, what do we call
10:06such lines in the language of
10:07mathematics? Intersecting, or better
10:10yet, non-parallel, or let's say
10:12non-perpendicular intersecting lines.
10:15And this situation right here, what did
10:18we call this in 'O Level Math'? We used
10:20to call this 'perpendicular lines' in '
10:22O Level Math'. We used to call this '
10:25perpendicular lines' in 'O Level Math'.
10:28We also learned in 'O Level Math' that
10:30whenever two lines are perpendicular,
10:32perpendicular means the angle formed
10:34between them, what angle will that be?
10:36That angle will be 90 °. We learned
10:39this in O Level Math. Clear? We learned
10:42in O Level Math that this means for the
10:44intersecting lines you have, there can
10:46be two conditions. Both of these are
10:48intersecting. Either they will be
10:51non-perpendicular or they will be
10:52perpendicular. Now, if we look at
10:55perpendicular lines in a bit more
10:57detail. Then what did we learn in O
10:59Level Math? Look, we learned in O Level
11:01Math that when lines are parallel,
11:02their gradient is the same. What did we
11:05learn in O Level Math? When lines are
11:07perpendicular, what happens? We learned
11:09in O Level Math that whenever two lines
11:11are perpendicular. Whenever two lines
11:13are perpendicular, meaning the
11:15relationship between them is a 90 °
11:17angle. The angle between them will be
11:1990 °. So we learned in O Level Math,
11:21what will be the product of their
11:23gradients? We learned that if this
11:25gradient is m1. And this gradient is m2
11:27. Then we learned in O Level Math that
11:29the product of gradients is always what
11:31? The product of gradients is always -1
11:33. The product of gradients is always -1
11:36. So in simple language, we used to say
11:39that when we multiply the gradients of
11:40these two, the answer will be -1. Or in
11:43simple language, we also used to say
11:45that the gradient of both will be the
11:47negative reciprocal of each other.
11:50Negative reciprocal is common sense. If
11:53one gradient is x, then the other will
11:55be -1/x, so that when they cancel out,
11:57the answer comes to -1. Your answer
11:59will come to -1. So we learned in O
12:02Level Math that whenever two lines are
12:04perpendicular, the product of their
12:05gradients will always be what? It will
12:08be -1. This is also O Level Math. Look,
12:10if I show you a small example. Just a
12:14rough sketch, giving you an idea. By
12:16the way, suppose the angle between
12:17these two lines is 90 °. I just gave
12:20you an idea that, suppose this
12:22condition is clear, if I tell you that
12:25this line has a gradient of 3/2, just a
12:28random example, and I ask you what its
12:30gradient m1 is, then what will m2 be?
12:33So, straight away, what did we use to
12:35do in O-Level Math? If its gradient is
12:383/2, we would say that our gradient
12:40will be what? Its reciprocal. Meaning,
12:43if that gradient is positive, then this
12:45one will be negative. Right? And if
12:48this is 3/2, then it will be 2/3, so
12:50that when we multiply both, the product
12:52of the gradients always comes to what?
12:55It comes to -1. So, we always learned
12:58in O-Level Math that whenever two lines
13:01are what? Whenever two lines are
13:03perpendicular to each other, what will
13:05their product of gradients always be?
13:07The product of their gradients will
13:09always be -1. What will the gradients
13:10always be? The negative reciprocal of
13:13each other. Negative reciprocal. Look
13:16at one more thing, if your math is
13:17somewhat strong. The x and y axes also
13:20have a 90 ° relationship. The x and y
13:22axes also have a 90 ° relationship.
13:24Later, I will also prove to you what
13:26the gradient of a horizontal line
13:28always is. Zero. We must have studied
13:31in O-Level Math that the gradient of a
13:32horizontal line is always zero. So,
13:34what is the gradient of the y-axis?
13:36Always its negative reciprocal, which
13:38is why we say the y-axis gradient is
13:40what? Undefined. Undefined. So,
13:43obviously, if this is zero, then we say
13:46what will its y-gradient be? Its
13:48negative reciprocal. So, when there is
13:50a zero in the denominator, we call it
13:51infinity. Clear, kid? We call it
13:53undefined. Meaning, its gradient is
13:55undefined. So, we learned in O-Level
13:57Math that when two lines are parallel,
13:58what will their gradient always be? It
14:00will be the same. We learned in O-Level
14:03Math that whenever two lines are
14:04perpendicular, in what form should
14:06their gradients be? Reciprocal of each
14:09other. What does the term negative
14:11reciprocal mean? The term negative
14:14reciprocal means reciprocal with the
14:16opposite sign. The term negative
14:18reciprocal means reciprocal with the
14:20opposite sign. It should be a
14:22reciprocal, but the sign should also be
14:24opposite. The sign should also be
14:26opposite. Clear, kid? So, these are the
14:28two conditions we studied in O-Level
14:29Math. Now, I will show you some further
14:32uses of these conditions, regarding
14:34what we need to do with them in A-Level
14:36Math. What else needs to be done? Look
14:39son, now we will require some 2D shapes
14:41for this topic, two-dimensional shapes,
14:44basically there are many quadrilaterals
14:46in my syllabus, there are triangles and
14:49quadrilaterals in my syllabus. There
14:52are triangles and quadrilaterals. Okay?
14:55Now, what will we study in triangles?
14:57Right-angled, isosceles, equilateral.
15:00In quadrilaterals, we will study square
15:02, rhombus, rectangle, parallelogram,
15:05trapezium, kite. All these properties
15:08are in my course. Now I will guide you
15:10one by one on what basic knowledge I
15:13need for this entire topic. Alright,
15:16look, look, the first thing, the work
15:19we are currently doing is in reference
15:21to the gradient. Let's start by talking
15:25about a square. So look, when we talk
15:28about a square, then rule number one,
15:30what are the corners of a square always
15:32? 90 °. You must have read this in
15:34O-level maths too. Here, the corners of
15:37the square are 90 °, and obviously
15:39these two sides are parallel, which
15:41means their gradient will be the same,
15:43and obviously these two sides are also
15:45parallel. Obviously, these two sides
15:47are also parallel. We already know this
15:49. Here, we already know this. This is
15:52something we studied in O-level maths
15:54that all sides are equal in length. No
15:55doubt. All sides are equal in length.
15:58No doubt. We already know this. And we
16:00kept saying that these two sides are
16:02parallel. Perfect. These two sides are
16:04also parallel. Perfect. The length of
16:06all is the same, the gradient of this
16:08and this is the same, and the gradient
16:10of this and this is the same. So this
16:13means what are its corners? The corners
16:16are always 90 °. Is that clear, son?
16:18If I ask you for a minute, what is its
16:20gradient? I'm just giving an example.
16:23If I tell you, man, its gradient is m.
16:25Suppose, or let's say its gradient is
16:27two. Suppose, I'm just giving an idea.
16:29By the way. Actually, the gradient of a
16:32horizontal line is zero. So I'm just
16:33giving an idea. If this gradient is two
16:35, then obviously what will you say? You
16:38will say that this means the parallel
16:40gradient will also be two. So I'll do
16:42one thing. I'll say the name of this
16:44gradient is m1, then this gradient will
16:46also be m1. If I tell you this gradient
16:49is m2, then you will say this gradient
16:51will also be m2. What is its corner?
16:54Its corner is 90 °, the corner is 90
16:57°, so it means if this is a square,
16:59then the corner of the square is 90 °.
17:02If the corners of the square are 90 °.
17:04Okay? Let's say, what are the corners
17:06of a square? They are always 90 °. So,
17:0990 ° means what will their gradients
17:11be to each other? Negative reciprocals.
17:14So it is confirmed that what will be
17:16the product of the gradients m1 and m2?
17:18The product of the gradients m1 and m2
17:20will be -1. Clear, kid? Both lines are
17:23perpendicular to each other. Now look
17:25at one more special case. We also
17:28learned in O-Level Math that when we
17:30talk about a square, the square's
17:33diagonals also always form 90 °. It's
17:36not just the corners that form 90 °.
17:38We learned in O-Level Math that the
17:41square's diagonals also form 90 °. So
17:43you should also have an idea that it's
17:45not just the square's corners that are
17:4790 °. The square's diagonals are also
17:4990 °. Okay? So if I tell you this is A
17:52, this is B, this is C, this is D. Then
17:54it means that the product of the
17:56gradients of AC and BD should also be
17:58what? It should be -1. So I can say
18:01that the product of the gradient of AC
18:03and the gradient of BD will also be
18:05what? It will be -1. So I will explain
18:09again. These two sides are parallel.
18:11Meaning the same gradient. These two
18:12sides are parallel. Meaning the same
18:15gradient. The relation between these
18:17two sides is 90 °, meaning they are
18:19negative reciprocals of each other, so
18:21if this gradient is three, then this
18:23will be -1/3. Then I mentioned a new
18:25point that in a square, what do the
18:26diagonals also always form? 90 °. If I
18:29tell you for a minute that this
18:31gradient is two. Then you will say what
18:33will this gradient be? -1/2. If this
18:35gradient is two, then what will this be
18:37? -1/2. So what basic knowledge is
18:40needed? Every student should have the
18:42sense of where it is 90 ° in a square.
18:44Meaning what things are parallel and
18:46what things are perpendicular. Just
18:48that, that's it. We need this sense in
18:51shapes. We need this basic sense in
18:54shapes. Clear, kid? Now let's move to
18:57shape number two. This is also O-Level
18:59Math. The name of shape number two is
19:01rhombus. The first shape we looked at,
19:03what was the name of the first shape?
19:04The name of the first shape was square.
19:06Okay? I will mark it here. The first
19:09shape we looked at, the name of the
19:11first shape was square. Okay? Okay?
19:14Here you go. This is also O-Level Math.
19:15The name of the first shape was square.
19:17Okay? We have copied this. This was the
19:19first shape that I explained to you.
19:21The name of the shape is square. Done.
19:23Here you go. I will increase the size a
19:26little bit. Here you go. Let's go. This
19:28first shape you have. The first shape's
19:30name was square. There you go, son. So
19:31I first explained the first shape to
19:33you. Now I will explain the rest to you
19:36one by one. Okay, son? Now I will
19:38explain the rest to you one by one.
19:40This is the first shape you have.
19:41Square. Next, let's look at another
19:44shape. Now the second shape I am doing,
19:47this second shape's name is rhombus.
19:49Okay? Now the second shape I am doing,
19:51this second shape's name is rhombus.
19:52This is also O-Level Math. Okay? The
19:56second shape I am doing, this shape's
19:57name is rhombus. This is also O-Level
19:58Math. Now let's see what the difference
20:00is between the properties of a square
20:02and a rhombus. Let's look at the square
20:04and the second shape I am doing, this
20:06second shape's name is rhombus. So
20:08basically, this is a requirement of our
20:10course. Meaning every student should
20:13know which sides are parallel and which
20:16are not in these properties. Look,
20:19first of all, what do we call a rhombus
20:20? We must have studied in O-Level Math,
20:23and I explained it before while
20:25teaching midpoints, that when we talk
20:27about a rhombus, what is a rhombus? A
20:29rhombus in mathematical language is
20:31called a tilted square. A square that
20:33is tilted. A square that is tilted; so
20:37we learned in O-Level Math that number
20:39one, its four sides are equal like a
20:40square. Okay? We already know this.
20:44Number two, when we talk about its
20:47corners, when we talk about the corners
20:50of a rhombus, to summarize, the corners
20:54of a rhombus are never 90 °. Clear?
20:58But yes, the diagonals of a rhombus are
21:0190 °. Corners are not 90 °. Diagonals
21:04are 90 °. So, if I ask you, what is
21:06the property of a rhombus? What is the
21:08property of a rhombus? So these two
21:10sides became parallel. Done. These two
21:12sides became parallel to each other.
21:14Here you go, these also became parallel
21:15. Now this corner is not 90 °. The
21:20corners of a rhombus are not 90 °.
21:23That is why there is no relationship
21:25between this gradient and that gradient
21:27. Meaning this is that kind of
21:29situation. Look, what would I call
21:31these two lines? I would say these are
21:33non-parallel. These are
21:34non-perpendicular. Finished. So there
21:37is no relationship between their
21:38gradients. No relation. Relations are
21:41either parallel or they are
21:43perpendicular. So, look here, what will
21:45these two gradients be to each other?
21:46Negative reciprocals. What will these
21:49two gradients be to each other?
21:50Negative reciprocals. What is the name
21:52of this shape? The name of this shape
21:54is a rhombus. This means that in both a
21:58rhombus and a square, the diagonals are
22:0090 °. But square corners are also 90
22:04°, and rhombus corners are not 90 °.
22:08That is the difference between a square
22:10and a rhombus. That is the difference
22:12between a square and a rhombus. Is that
22:14clear, son? Let’s look at another
22:16shape. The third shape is called a
22:17rectangle. We also study this very
22:19frequently in O-level math. The third
22:21shape is called a rectangle. Is that
22:23okay, son? The third shape is a
22:24rectangle. We must have studied in
22:26O-level math what the properties of a
22:28rectangle are. Right? What are the
22:30properties of a rectangle? Take a look
22:31at this one too, son. This is a very,
22:34very basic shape and we use it a lot in
22:36math. When we talk about a rectangle,
22:38obviously, what happens in a rectangle?
22:41Rectangle corners are 90 °. We all
22:44know that whenever we talk about a
22:46rectangle, the corners are always 90 °
22:49. Rectangle corners will always be 90
22:52°. Is that okay, son? And in
22:54rectangles, what happens is that the
22:56diagonals always have the same midpoint
23:00, but the diagonals do not form 90 °.
23:02Let me summarize it a little bit more.
23:05This is A, this is B, this is C, this
23:07is D. So, I will write down in words
23:09below what its corners are. They are
23:12always 90 °. And its diagonals do not
23:15form 90 °. That is, the diagonals are
23:18non-perpendicular. The corners are
23:20perpendicular. So, if I tell you its
23:22gradient is two. Then you would say its
23:24gradient is also two. Both are parallel
23:27. If I say its gradient is two, what
23:29will its gradient be? -1/2 because the
23:31corner is 90, so if this gradient is
23:33two, then this is also two. What will
23:36these two be to each other? Negative
23:38reciprocals. So then, what will this
23:39gradient also be? It will be -1/2, -1/2
23:43. Clear? So, in this, the corners are
23:4690 °. The diagonals are not 90 °.
23:48What is the name of the shape? The
23:50shape is a rectangle. The shape is a
23:53rectangle. Is that clear, son? So, we
23:55need basic knowledge of which angle
23:57lies where in which shape. What is
24:01parallel and what is perpendicular.
24:04Right? We have no interest in
24:06non-perpendicular ones. By the way.
24:08Here comes the third shape for you, the
24:10rectangle. Fourth shape, fourth. Look
24:13at another shape. We must have studied
24:15this in O-Level Maths too. The fourth
24:17shape is called a parallelogram. Okay?
24:19Okay? The fourth shape is called a
24:21parallelogram. We must have studied
24:22this in O-Level Maths as well.
24:24Parallelogram. All these shapes are in
24:26our syllabus. Okay? Parallelogram, all
24:28these shapes are in our syllabus.
24:32Alright, now let's see what the
24:34properties of a parallelogram are. What
24:38are the properties of a parallelogram?
24:40Look, son. What did we call a
24:43parallelogram? In O-Level Maths. Just
24:45like we studied before, what is a
24:47rhombus? In the language of math, a
24:49rhombus is called a tilted square. Okay
24:52? In math, a tilted square is called a
24:55rhombus. Tilted, a square that is
24:57tilted. What was a parallelogram? In
24:59the language of math, we called a
25:02parallelogram a tilted rectangle. In
25:05the language of math, we called a
25:07parallelogram a tilted rectangle. A
25:09rectangle that is tilted. This is a
25:11parallelogram. Come, let's check it.
25:14Now, suppose you have this condition.
25:16Look, I gave you an idea that suppose
25:19your parallelogram is in this form.
25:21Okay? Suppose you have this diagram.
25:24Here you go. Now you note what
25:26condition exists in a parallelogram. In
25:29a parallelogram, look, what are these
25:31two sides? Obviously parallel. So it
25:33means what should be in a parallelogram
25:36? These two sides of yours are parallel
25:37. Look closely. These two sides became
25:40parallel to each other. Its gradient,
25:42if this gradient is one, then this one
25:44will also be one. And these two sides
25:46are parallel. In a parallelogram, these
25:49sides are parallel. So if I summarize,
25:51if I summarize this whole question.
25:54Number one, the corners of a
25:56parallelogram are not 90 °. Because
25:59it's a tilted shape, right? And number
26:01two, the diagonals of a parallelogram
26:03also don't form 90 °. So if I
26:06summarize, nothing in a parallelogram
26:09is at 90 °. Okay? This means these two
26:12sides are parallel to each other. And
26:14what about these two sides of yours?
26:15These two sides are also parallel. We
26:17already know this. And the third, the
26:19second, and the last thing is that its
26:21corners do not form 90 °. Its
26:23diagonals do not form 90 °. So look,
26:25its diagonal also doesn't form 90 °.
26:28The diagonals of a rectangle also do
26:30not form 90 °. So for which shapes do
26:32the diagonals form 90 °? Number one, a
26:34square and number two, a rhombus. The
26:37diagonals of a square and a rhombus
26:39form 90 °. Is it clear, son? This is
26:42shape number four. The name of shape
26:44number four is parallelogram. Meaning,
26:46what is our interest in right now? Our
26:49interest is basically in those shapes
26:51which are two-dimensional, number one.
26:54And our only basic task is to see what
26:57in the shape is 90 degrees, what is
26:59parallel, and what is perpendicular.
27:01This is what we are interested in. Now
27:04look, I am showing you another shape.
27:06Look, there is one more thing we used
27:08to study frequently, the shape's name
27:11is trapezium. Right, the name of the
27:13shape is trapezium. We must have
27:14studied this very frequently in O-Level
27:16Math, trapezium is used a lot,
27:18especially in kinematics. Right? Now I
27:20need the properties of a trapezium,
27:22what are the properties of a trapezium?
27:24Now come, let’s look at the trapezium
27:26. Number one, obviously these are all
27:28quadrilaterals. What is the meaning of
27:30quadrilateral? All of these shapes are
27:32four-sided shapes. Quad, right? It is
27:34four-sided. Now, what did we call a
27:36trapezium? In O-Level Math, we used to
27:38say a trapezium is a quadrilateral with
27:40two unequal parallel sides. What does
27:43the word trapezium mean? We used to
27:45call a quadrilateral a trapezium. And
27:48such a quadrilateral whose two parallel
27:51sides are unequal. Look, this is your
27:55trapezium. This is your trapezium.
27:57Right? We obviously studied this in
27:59O-Level Math. Likewise, there can be
28:01multiple shapes of a trapezium. You
28:03must have studied this in O-Level Math
28:04as well. There was a topic called
28:05kinematics. Trapeziums like this were
28:07used a lot in that. This is the
28:09trapezium you have. So what does a
28:12trapezium mean? A trapezium is such a
28:14quadrilateral where two parallel sides
28:17are unequal in size. Come, let’s look
28:20at this in a bit more detail. If I talk
28:22about a trapezium, these two sides of
28:24the trapezium are parallel. These two
28:26sides of the trapezium are parallel.
28:27They are parallel but unequal. And the
28:30line that formed a 90-degree angle with
28:31both of these, we used to call the
28:33height of the trapezium. We used to
28:35study the area, right? 1/2*sum of
28:37parallel sides multiplied by the height
28:39. So this is the height. Height was the
28:40line that formed 90 degrees with both
28:42these parallel sides. If these two
28:44lines are parallel. Look at this, look
28:46at the trapezium. These two sides of
28:48yours are parallel. If these two sides
28:49are parallel, then what is their
28:51gradient? It is the same. What is their
28:53gradient? Obviously, it is the same. So
28:54this line that I am dropping, this line
28:56that I am showing you, what is this
28:58line? We have been calling this line
29:00the height of the trapezium. We have
29:01been calling this the height of the
29:02trapezium. Height. So, this is the
29:04height of the trapezium. This is the
29:06height of the trapezium. This is the
29:07height of the trapezium. So, if I ask
29:08you what the gradient of these two
29:09lines would be? You would say the same.
29:11Meaning, if this gradient is two, then
29:13definitely this gradient will also be
29:15two. If this gradient is two, then what
29:17will this gradient be? -1/2, meaning
29:20where the 90 comes, if this is two,
29:22then this will be -1/2. Meaning, if
29:25this gradient is two, then this will be
29:27-1/2. So, where is the 90 in a
29:29trapezium? The line that connects the
29:32two parallel sides forms a 90 °. Clear
29:35, kid? So, what is 90 ° in a trapezium
29:38? We use 90 ° for the line of the
29:40height. Alright, kid? So, what is this
29:42shape? By the way, this shape is a
29:44trapezium. So, what is a trapezium? A
29:46quadrilateral with two unequal parallel
29:48sides. Two unequal parallel sides. Two
29:54unequal parallel sides. Meaning, they
29:56are parallel, but the size is different
29:58. The size is unequal. Clear? There is
30:01a difference in size. So, our interest
30:03in all these properties is what things
30:06are parallel in which shape and what
30:08things are perpendicular in which 2D
30:11shape. Basically, we are interested in
30:13this. So, we have looked at squares,
30:15rhombuses, rectangles, parallelograms,
30:18and trapeziums. There are other shapes
30:20too. Look. I am telling you about a
30:22couple more shapes. Shape number six.
30:24We must have studied a shape called a
30:26kite in O-Level Math. Right? Now, let's
30:28see what the properties of a kite were
30:30in O-Level Math. Right? Now, let's look
30:32at the kite. If we talk about a kite.
30:36Right? We must have studied the kite in
30:38mensuration in O-Level Math as well.
30:40Right? In O-Level Math, we used to
30:42study kites in mensuration. Mensuration
30:45. Alright, kid? Now, look, let's also
30:47look at the kite in detail to see what
30:49properties a kite has. Alright, kid.
30:52What are the properties of a kite?
30:53Let's go. What is obviously in a kite?
30:55A kite is also a quadrilateral. We
30:57studied this in O-Level Math as well. A
30:59kite is also a quadrilateral. And we
31:00are using all these properties in AS
31:03Math now. Okay? So, what used to be in
31:06a kite? If any student remembers, what
31:08was in a kite? There was a pair of
31:10isosceles triangles. Meaning, there
31:12were two isosceles triangles. If we
31:14remember. We used to say it has two
31:17isosceles triangles. Clear, kid? The
31:19kite used to have a shape like this.
31:21Right? Look again. Kites used to have
31:24this type of shape. Let me explain that
31:26once again. What was in a kite? A kite
31:29used to have a pair of isosceles
31:31triangles. A pair of pairs of isosceles
31:34triangles. Let's see how a kite is made
31:37. I drew an isosceles triangle. Here,
31:40take this. I drew this isosceles
31:43triangle. You have an isosceles
31:49triangle drawn here. Okay? Okay? If I
31:52join these two, it becomes a proper
31:54isosceles triangle. And if I join these
31:56two, then what does it become? An
31:58isosceles triangle. So what did we
31:59study in O-Levels? We studied in
32:01O-Levels that a kite has two isosceles
32:03triangles. One at the top and one down
32:05here. And we also know that a kite has
32:08the property that these two sides are
32:10equal. What is the property of a kite?
32:13Look, the property we studied in
32:15O-Levels is that these two sides of the
32:17kite are equal. Okay? Wait a minute,
32:21son. What property did we study in a
32:24kite? That these two sides are equal.
32:26The top two, that's why it's an
32:28isosceles triangle. Okay? Look, this
32:31means that these two sides are equal.
32:34Here and these two sides are equal.
32:37Meaning the top is isosceles and the
32:39bottom is also isosceles. Here. It is
32:41obviously isosceles for you. So the
32:44first thing is, when we talk about its
32:46diagonals, the kite's diagonals also
32:49always form 90 °. Later we will hear
32:51another word too. That word will be
32:53perpendicular bisector. I haven't used
32:55that word yet because we will use it
32:57when the time comes. So what is this
33:00shape? This shape is a kite. ABCD, this
33:04is your kite. So if I summarize, the
33:07corners of a kite are not 90 °. The
33:10corners of a kite are not 90 °. And if
33:12we look at it in more detail, the
33:14diagonals of a kite are 90 °. So now
33:17look, you have studied 90 ° diagonals
33:19in three shapes. Number one, diagonals
33:21in a square are 90 °. Number two,
33:23diagonals in a rhombus are 90 °. And
33:25number three, diagonals in a kite are
33:2790 °. So as soon as we hear the word
33:2990. It immediately clicks as
33:30perpendicular. If I tell you for a
33:32minute that its gradient is two. Then
33:34you will immediately say what is the
33:35gradient of this vertical? -1/2, so the
33:37corners of a kite are not 90 °. The
33:41diagonals of a kite are 90 °. Okay? So
33:43I am showing you all these shapes once
33:45so that you remember that, hey, all
33:46these shapes are in our syllabus. And
33:48we have to use all of this. Is that
33:50clear, kid? Now I am showing you one or
33:52two more things. All these are also in
33:54the course. Look at another small shape
33:56of this type. We used to study this in
33:58O-Level Math. Isosceles triangle. We
34:01kept studying it in O-Level Math.
34:02Isosceles triangle. In fact, we also
34:04studied it in AS in circular
34:05measurement. All right? We also studied
34:08it in AS in circular measurement. So,
34:10the name of the triangle was isosceles
34:12triangle. Isosceles triangle. All right
34:16? This is also in my course because in
34:18this topic of coordinate geometry, this
34:20isosceles triangle will also be used.
34:22Is that clear, kid? Now, what used to
34:24be the property of an isosceles
34:25triangle? In fact, if you notice, this
34:28same isosceles triangle is also in the
34:30kite above. This same isosceles
34:33triangle is also in the kite. This is
34:35also isosceles, and obviously, this is
34:36also isosceles. So, what property of an
34:39isosceles triangle did we study in
34:41O-Level Math? Look, we studied in
34:43O-Level Math that if a triangle is
34:45isosceles, then these two sides of it
34:47will be equal. If these two sides are
34:49equal, then what will these two base
34:51angles be? These two base angles will
34:53also be equal. Take these as alpha and
34:56alpha; when we drop a line from here,
34:58what was the property of an isosceles
35:01triangle? That when you drop a line in
35:04an isosceles, it always used to bisect
35:07the base. Later, we will hear a term:
35:10perpendicular bisector. And it also
35:12used to equally do what to these angles
35:14? It used to divide them. This was the
35:16property of an isosceles. So, if I
35:18summarize, number one: in an isosceles
35:21triangle, neither the corners are 90,
35:23and number two: in an isosceles
35:26triangle, neither the diagonals are 90.
35:29Meaning, in an isosceles triangle,
35:31neither the corners are 90 nor are the
35:32diagonals 90. So, where is there a 90
35:35in an isosceles? Whenever you drop a
35:37line between the isosceles, that is,
35:38between the two equal sides, that line
35:40does what to the base? It bisects it
35:42and does what equally to the angle? It
35:45divides it. Look here, if I say this is
35:47isosceles. Then when I drop a line, it
35:49will do what to the base? It will
35:51bisect it and will also do what equally
35:53to the angle? It will divide it. Is
35:56that clear, kid? So, what is the name
35:57of this triangle? Isosceles triangle.
36:00So, the shapes we have studied so far
36:02—square, rhombus, rectangle,
36:03parallelogram, kite, trapezium,
36:05isosceles triangle—are all part of
36:07our course. Is that clear, kid? And we
36:09obviously have to use all of these. We
36:12have to use all of them. Look at
36:13another shape. These are also all from
36:15O-Level Maths. Look at one more shape.
36:16Shape number eight. All these shapes
36:18are in our course. Alright, son? How
36:20many shapes have we seen so far? We
36:22have seen seven shapes. Look at shape
36:24number eight. Here it is. One more
36:26small condition. In fact, this is a
36:28very easy case. Well, there are
36:30questions on this in our past papers
36:31too. We must have studied this in
36:33O-Level Maths as well. Right-angled
36:35triangle. That is why I am saying this
36:37is the easiest triangle. We call this a
36:39right-angled triangle. There are two or
36:42three questions on this in our past
36:43papers. Okay? Right-angled triangle.
36:47Right? When we talk about a
36:48right-angled triangle, it's obvious
36:50from its name. When a triangle is
36:52right-angled, it means its largest
36:54angle is 90 °. So when we talk about
36:56any right-angled triangle. Like I
36:58marked this triangle. This is a
37:00right-angled triangle. If I say to you,
37:02"Hey, this triangle is a right-angled
37:03triangle," or "This triangle is a
37:04right-angled triangle." Then what
37:05clicks immediately from "right angle"?
37:07What will its corner be? 90 °. Right,
37:09son? What is all our interest in?
37:12Parallel lines and perpendicular lines.
37:14So if I tell you that this gradient is
37:16m1, then you will definitely say this
37:18gradient will be -1/m, meaning both
37:20these gradients are negative
37:21reciprocals of each other. Meaning if
37:23this gradient is two, then what will
37:24this gradient be? It will be -1/2.
37:27Clear, son? So what kind of triangles
37:29are these two? Right-angled triangles.
37:31So in a right-angled triangle, the
37:33largest angle is obviously 90. So in a
37:37right-angled triangle, the angle
37:39between the base and the height must be
37:4290 °. So in a right angle, there is a
37:44very clear message that in a
37:45right-angled triangle, what corner will
37:47you find? Right angle means you will
37:49get 90 °. What does the word "right
37:51angle" itself mean? The word "right
37:52angle" means perpendicular. What does
37:55the word "right angle" mean? The word "
37:57right angle" means perpendicular.
37:59Perpendicular. So this is also in our
38:01course. We saw shape number eight, the
38:03right-angled triangle. I will explain
38:04one more small point. Then we will move
38:06forward. Point number nine. Shape
38:09number nine. All these nine shapes are
38:11in our course. All nine shapes are in
38:14our course. Right? I am showing one
38:16more small shape. Only these nine
38:18shapes are in my syllabus. And these
38:19will be used very frequently. We also
38:21studied one more question. We also
38:24studied that in circular measurement.
38:26Equilateral triangle. Okay? Equilateral
38:29triangle. We saw questions about it in
38:30circular measurement too. Equilateral
38:32triangle. When we talk about an
38:34equilateral triangle. Okay? When we
38:37talk about equilateral triangles in
38:38math, what is the property of an
38:40equilateral triangle? What basically
38:43happens in an equilateral triangle? In
38:44an equilateral triangle, you basically
38:47have three what? There are three pairs
38:49of isosceles triangles. Let's just
38:51understand that there are three pairs
38:53of isosceles triangles in an
38:55equilateral one. There are three pairs
38:57of isosceles triangles in an
38:59equilateral. Look here, if I sketch an
39:01equilateral for you. Suppose you have
39:04this triangle. In O-Level Math, we used
39:06to call an equilateral a regular
39:07polygon too. In O-Level Math, we also
39:10called it a regular polygon. Suppose
39:12you have this equilateral triangle. So,
39:14what did we learn in O-Level Math? What
39:16does equilateral mean? That all sides
39:18are equal. Here you go. This is what we
39:20studied in O-Level Math. Second, we
39:21studied that when we divide its angle
39:23180 by 3, then all these angles will be
39:26what? They will be the same. So this
39:28means that these two sides are equal,
39:30so this pair is also an isosceles one
39:32for you. Meaning, if these two sides
39:34are equal, then these two will also be
39:36the same. And if these two sides are
39:37equal, then these two will also be the
39:39same. And if these two sides are equal,
39:41then these two will also be the same.
39:43So this means that all three sides and
39:45all three angles are the same. So, we
39:48learned one more thing in O-Level Math:
39:50when we drop a line in an isosceles.
39:52Look, when we drop a line in an
39:55equilateral one as well, what happens?
39:58What basically happens in an
39:59equilateral? You have three pairs of
40:02isosceles triangles. When you drop a
40:04line from here too, what will happen?
40:05This will also form 90 °. When you
40:08drop a line from here too, what will
40:09happen? This will also form 90 °.
40:12Clear, beta? So first, keep this in
40:14mind that no doubt we know that all
40:15three sides are equal. No doubt. Okay?
40:18We already know this fact. So this
40:20means that when you drop a line, these
40:22two will be equal. When you dropped a
40:24line from here, these two will also be
40:26the same. Meaning if this is alpha and
40:28alpha, then this is also alpha and
40:29alpha. This is also alpha and alpha.
40:31Meaning we know that when we talk about
40:33an equilateral, all sides are equal and
40:35all angles are equal. So, where will
40:37there be a 90 ° angle in an
40:39equilateral triangle? And when you drop
40:41a vertical line, what will it do to the
40:43base? It will bisect it. We read this
40:46as equilateral triangle. Equilateral
40:48triangle. Is it clear, son? Equilateral
40:51triangle. Well, so far we have tried
40:53nine shapes. Is it clear, son? These
40:55nine shapes we tried. What were all
40:57these shapes? We call these
40:59two-dimensional shapes. All these
41:01shapes are in my syllabus. Okay? And
41:03later on, you will study one more shape
41:05. Its name is circle. That is, of
41:08course, a separate complete topic of
41:10coordinate geometry named Coordinate
41:11Geometry of Circles. The very name of
41:14the last point you will study in
41:16coordinate geometry, what is it?
41:17Coordinate Geometry of Circles. So, by
41:20the way, we don't need circles right
41:22now. But later you will study a whole
41:24concept of coordinate geometry named
41:27Coordinate Geometry of Circles. Okay?
41:30Coordinate Geometry of Circles. This is
41:32a complete concept you will study later
41:34. Look, when we talk about a circle,
41:37there is only one condition for 90 °.
41:40Maybe you remember. We will use this
41:43later on. We will use this later on.
41:45Let's say this is a property. We don't
41:47need it right now. By the way, in this
41:49topic. We studied in circles where a 90
41:51° angle could be possible. So, we read
41:54about it in two or three properties.
41:56Especially the most used property was
41:58what we called the semi-circle property
42:01. We call it the angle subtended by a
42:03diameter is always 90 °, right? We
42:06said that when we join both endpoints
42:08of the diameter on the circumference,
42:10what does it form? It forms 90 °. Is
42:14it clear, son? We had also studied this
42:15in a property of circles. And then,
42:17maybe you remember. We had also studied
42:18another property of circles. We had
42:20studied another property too. I am just
42:22telling you whether there is a 90 °
42:23angle in a circle or not. I just want
42:25to tell you this. Okay? Where is there
42:27a 90 ° and where is there not one in a
42:28circle? Look, let me show you another
42:30example. We had studied another
42:32property as well. If you remember the
42:34circle property, we studied that a line
42:36through the center bisects the chord.
42:39Okay, son? We had also studied another
42:41circle property that a line through the
42:43center bisects the chord. If I tell you
42:45, look, I have a chord here. Here you
42:47go. I have a chord here. So we studied
42:50another property that if we drop a line
42:52from the center, what will it always do
42:54to the chord? It will bisect it. So
42:56that means in the properties of a
42:58circle too, we saw a 90 ° angle. But
43:00we will study all of this later in the
43:02Coordinate Geometry of Circles. In the
43:04Coordinate Geometry of Circles. Then we
43:06also studied 90 ° in the tangent kite
43:08condition. We also studied the tangent
43:10kite condition. We studied the 90 °
43:11angle in that too. So for now, we will
43:13not talk about circles at this moment.
43:15That is an entire topic in itself where
43:17the whole game is about circles. Is it
43:19clear, son? Let's proceed. So basically
43:22, for now, we need nine shapes. What
43:24happens in these nine shapes? We only
43:26need to see what is parallel and what
43:28is perpendicular. Which are those nine
43:30shapes? Square, rhombus, rectangle,
43:33parallelogram, trapezium, kite,
43:35right-angled, equilateral, isosceles.
43:38These are the nine shapes in my
43:40syllabus. Is it clear, son? So, what
43:42did we study today? Today, so far, we
43:44have only talked about the gradient.
43:46Alright, son? We talked about the
43:47gradient or slope. We talked about rise
43:49over run. And in different shapes, we
43:52saw where there is a 90 ° angle and
43:54where there isn't. Okay, now the next
43:57point in coordinate geometry is called
43:59Heron's formula. Alright? Heron's
44:02formula. This formula is a coordinate
44:05geometry formula. Obviously, our
44:07topic's name is also coordinate
44:08geometry. Look, when we talk about
44:10mensuration in math. Alright? Look
44:12closely for 2 minutes. When we talk
44:15about mensuration in math. We must have
44:17studied mensuration many times in math
44:19before this. Look, whenever we talk
44:21about mensuration. Alright? So let me
44:23remind you of a small thing. Come on,
44:27take a minute to look, son. Wait,
44:31whenever we hear the word mensuration
44:33in math, when you find the area in
44:35mensuration—the area in all of
44:36mensuration that you must have studied
44:38in O-Level math. In mensuration, your
44:42area is always dependent on length. In
44:46mensuration, your area is always
44:48dependent. Dependent on length. In
44:54mensuration, there is no work for
44:56coordinates. In mensuration, the whole
44:59game of area is dependent on length.
45:02Look, if I give you different examples.
45:04Let's do a quick recap, let me revise
45:06it a bit for you. Let me revise it a
45:09bit. For different shapes, if we talk
45:12about a square, here you go. If I talk
45:14to you about a square, what must we
45:16have studied in O-Level math? When we
45:18studied O-Level Math, we must have seen
45:20the area of a square, because all these
45:22formulas will be used by me now. These
45:24shapes, I have to use all nine of these
45:27shapes. So, what are the corners of a
45:29square? They are 90 degrees. So, what
45:31did we study in O-Level Math? That the
45:33area of a square is something everyone
45:35knows. So, what is the area of a square
45:36? Its sides are both the same, so we
45:38say that both its sides are the same.
45:41Meaning, its base is the same and its
45:42height is also the same. So, in O-Level
45:44Math, we used to say base*height, or we
45:46would ask, what is its area? Everyone
45:49knows it was also called l². Or I can
45:51also call its area base*height. Base*
45:53height is the same thing. Now, if any
45:56student says that the formula is base*
45:58height, that is also fine. Base*
46:00perpendicular height. These two lines
46:02are parallel, so who are all these
46:04heights? These are all perpendicular
46:06heights. Let me explain some basics.
46:08Then I will introduce the new thing,
46:10what is Heron's formula? What is the
46:12new work we will do today? Right? What
46:15is the new work I will do today? Now,
46:18look at the second point. We must have
46:20studied one more shape. The name of
46:22that shape was a rhombus. The name of
46:23that shape was a rhombus. Here it is.
46:25So, let me tell you about some shapes
46:27once again. Then I will tell you what
46:29today's new work is. We studied another
46:32shape, a rhombus. A rhombus is called a
46:34tilted square in the language of math.
46:36A square that is tilted. So, we studied
46:38in O-Level Math that all sides of a
46:40rhombus are equal. We just studied this
46:42a little while ago. So, when we talk
46:43about the area of a rhombus, what is
46:45the area of a rhombus? Look here: base
46:47multiplied by perpendicular height,
46:49perpendicular height, perpendicular
46:51height, perpendicular height. So, all
46:53these lines that I dropped, I wrote 'h'
46:55next to them; here it is. And what did
46:58I write down here? Base. So, if anyone
47:01remembers the formula for the area of a
47:04rhombus, it is base multiplied by
47:06height, or if someone's mensuration is
47:09a bit stronger, there was another
47:12formula for the area of a rhombus. If
47:15someone's mensuration is strong and
47:16they remember, we used to study another
47:18formula for the area of a rhombus too.
47:20I will guide you. Look, there was
47:22another formula for the area of a
47:23rhombus as well. In O-Level Math, we
47:26also learned that there is another
47:27formula for the area of a rhombus: 1/2*
47:29product of two diagonals. All right? In
47:32O-Level Math, we also learned that
47:34there is another formula for the area
47:36of a rhombus, and that formula is 1/2*
47:38product of two diagonals. All right? So
47:41, I will mark it like this. There was
47:43another formula for the area of a
47:45rhombus. So, here it is, we used to
47:47write it like this in O-Level Math. The
47:49formula for the area of a rhombus was
47:511/2*product of two diagonals. Let me
47:53label it like this. A B C D, so we used
47:56to call it 1/2 D1*D2. If a student's
47:59mensuration is strong, 1/2*product of
48:02two diagonals, or wait, I can also
48:05write it as 1/2*AC*BD. So, we used to
48:08use this formula for a rhombus as well.
48:11I am giving this quick recap for a
48:13reason. Then I will tell you about a
48:14new option regarding what we are
48:15studying today. So, what do we study as
48:17the area of a square? Base*Height. What
48:19is the area of a rhombus? Base*Height.
48:21So, for a square, the base and height
48:23are obviously at 90 degrees. That is
48:25why we have no issues. So, what do we
48:27call height? Height is called the
48:28perpendicular distance. So, obviously,
48:30this is the perpendicular distance. So,
48:33in this condition, this is the
48:34perpendicular distance. Obviously, this
48:36is not the perpendicular distance. So,
48:38its area was not side by side for a
48:41rhombus. It is base*height, right? Base
48:43*height. Okay, look at one more thing.
48:47So, I also mentioned another formula
48:49for a rhombus. 1/2*product of two
48:51diagonals. Okay, look at one more shape
48:53. Let's look at a third shape, one more
48:55shape. We must have also studied this
48:56in O-Level Math. The name of the third
48:58shape is rectangle. I am telling you
49:00some basics so that I can then tell you
49:02a new formula called Heron's formula.
49:04Then let's see what Heron's formula is.
49:06When we talk about a rectangle, a
49:08rectangle is also a very basic shape.
49:10When we talk about a rectangle, math
49:12says that the corners of a rectangle
49:14are always what? 90 degrees. We already
49:16know this. So, we learned in O-Level
49:18Math that the area of a rectangle is
49:20also what? The area of a rectangle is
49:22also base*height. And normally a
49:24student says length*breadth. Because
49:26obviously, this is the base. If a
49:28student calls it length*breadth, that's
49:30fine. If a student says base*height,
49:33that is also fine. Base*height will
49:36work. Because the height is obvious in
49:38it, right? That's why we don't have an
49:39issue. Call it base*height or length*
49:42breadth. So, the area of a rectangle is
49:44obviously known to every child. There
49:47is hardly any child who doesn't know
49:49the area of a rectangle. Then we used
49:51to study another shape in O-Level Math.
49:53The name of the shape was a
49:55parallelogram. The name of the shape
49:58was a parallelogram. This is also a
50:00shape we must have studied. Very
50:01frequently in O-Level Math mensuration.
50:03What did we use to call a parallelogram
50:05? We used to call a parallelogram a
50:07tilted rectangle. A rectangle that is
50:09tilted, look at these two parallel
50:11sides. So, what was the area of a
50:13parallelogram as well? Base*height.
50:14Look, I am drawing a line with a yellow
50:17color. All these lines, what is all
50:19this? The perpendicular distance
50:22between two parallel lines. So, what I
50:25call height, what is height? Height is
50:27the vertical distance between two
50:29parallel lines. You see, what is its
50:31height? Look, these two lines are
50:33parallel. This is the vertical distance
50:34between them. This is its height. So,
50:36if any student remembers what the area
50:38of a parallelogram was? Then we used to
50:41say the area of a parallelogram is base
50:43multiplied by perpendicular height.
50:45Base multiplied by perpendicular height
50:47. This shape's name was a parallelogram
50:50. So, why am I telling you this basic
50:52mensuration once, so that I can tell
50:54you a new point today, that there was a
50:57mathematician named Heron. He derived a
51:00new method, which is based on
51:02coordinate geometry. Okay? Okay? Now,
51:05let's look at another shape, shape
51:06number five. Suppose you have a shape
51:08named trapezium. We must have studied
51:10this a lot in O-Level Math too. I am
51:12giving you a small recap of what we
51:14studied in O-Level Math and what we
51:16have to study today. Okay? Suppose you
51:19have a shape here. The shape is like
51:21this, a trapezium. We must have studied
51:23this a lot in O-Level Math too. You
51:25must have studied this shape many times
51:26. The name of the shape is a trapezium.
51:28Okay, child? The name of the shape is a
51:30trapezium. So, what happens in a
51:32trapezium? We studied a trapezium a
51:33little while ago as well. A trapezium
51:35is called a quadrilateral with two
51:36unequal parallel sides. So, we also
51:39call this a trapezium, a quadrilateral
51:41with two unequal parallel sides. So, if
51:44I make it a bit easier. Okay? Here you
51:47go. So, what is its corner? Obviously
51:4990 degrees. So we know that these two
51:51lines are parallel. What is the
51:53relationship between them? Of 90 °.
51:55Okay, son? There is a 90 °
51:57relationship in that. Look, let's
51:59assume these two lines for a minute.
52:02Let's say these two lines are parallel.
52:05So, what formula did we study in
52:06O-Level Math? We used to say that these
52:09two lines are parallel. The formula was
52:121/2*sum of parallel sides. And what did
52:14we call the height? We used to call it
52:17the height. We called the height the
52:20line that forms 90 ° with both
52:22parallels. We used to say, "Hey, this
52:25is the trapezium's height." This is the
52:27height of the trapezium. We studied
52:29this earlier in O-Levels. This is the
52:31trapezium's height. Suppose this is
52:32your A. Suppose this is your B. So, if
52:34I ask you what the area of a trapezium
52:36is? What did we study in O-Level Math?
52:381/2*sum of parallel sides multiplied by
52:41the height. We studied this formula in
52:44O-Level Math. What was this formula
52:46called? This formula was for the area
52:48of a trapezium. We have studied all
52:50these basic formulas in O-Level Math.
52:52Shape number six. Then we must have
52:54studied another shape in O-Level Math,
52:56a kite. So, all the formulas I am
52:58showing you are based on mensuration.
53:01These are mensuration formulas that we
53:03have studied since childhood. A kite,
53:05we even studied kites today. When we
53:08talk about a kite. Okay? When we talk
53:11about a kite. Like this shape, here is
53:12a kite. So, what did we study in
53:14O-Level Math? Well, very few children
53:17mostly know the area of a kite. What is
53:19a third-rate answer? Children's
53:21third-rate answer. We will add the
53:22areas of the triangles. The kite's area
53:24will come out. If the examiner ever—I
53:27often ask kids in class what the area
53:30of a kite is. Then a useless answer
53:32also comes up. "Sir, we will add all
53:33the sides." If the child is sensible,
53:35this diagonal is 8. This diagonal is 12
53:37. Show me how will you find the
53:39triangles? Finding the kite's area is
53:40impossible. No matter how smart you are
53:42. Now this largest diagonal is 12. The
53:44smaller one is 8. Now how will you find
53:46the area? How do you know what the
53:48height of this triangle is? What is its
53:49height? So, it means this is a
53:51senseless point. So, if we are asked
53:53what the area of a kite is? Then we
53:55learned in O-Level Math that the kite's
53:57area is 1/2*product of the two
53:59diagonals. So if I say this is A, this
54:02is B, this is C, this is D. Then I can
54:04also say what are the diagonals of the
54:06kite? The diagonals of the kite are AC
54:08and BD. The diagonals of the kite are
54:11AC and BD. So what is the area of a
54:13kite? 1/2*product of two diagonals.
54:17Like I told you about the rhombus here
54:19as well. 1/2*product of two diagonals.
54:21This is 1/2*product of two diagonals
54:23for the rhombus too. This is for the
54:25rhombus. Okay? So this is the area of
54:27the kite. These are all the formulas
54:29that every child learns in O-Level Math
54:31. Look at another example. Come on, if
54:33I tell you about a right-angled
54:35triangle, everyone knows this too. So
54:37all these basic examples I gave you are
54:39the ones every child comes from
54:41studying in O-Level Math. When we talk
54:44about any right angle, what is the area
54:46of a right angle? Okay? If I talk to
54:49you, man, that I have a right-angled
54:50triangle here. Here you go, this is a
54:52right-angled triangle I have. If I tell
54:54you, man, how will its area be
54:55calculated? Then obviously you already
54:57know, you say, what is the area of a
54:59right-angled triangle? 1/2*base*height.
55:01Here you go. So if I tell you, what is
55:03the area? You will say 1/2*base*height.
55:081/2*base*height. Finished. This is also
55:10all O-Level Math. Similarly, we often
55:12see another triangle, isosceles. Here
55:15it is. Okay? If I tell you, man, I need
55:17the area of an isosceles triangle. We
55:19must have studied this in circular
55:21measurement too. In circular
55:22measurement. So if I tell you, man, I
55:24have a triangle here. I am just giving
55:26a random example. Okay? Okay? I am
55:28giving a random example like this, that
55:30I have an isosceles triangle here.
55:31Isosceles. Okay? We must have studied
55:34this before as well, studied very
55:36frequently in circular measurement.
55:37When we talk about an isosceles
55:39triangle, isosceles. So in an isosceles
55:41, if I tell you, man, these two sides
55:43are equal. These two sides are equal.
55:45Then the child will immediately say,
55:47sir, these two angles will be the same.
55:50So if you have an isosceles, you can do
55:53this for isosceles: 1/2*a b sin C. You
55:56can use this approach that we used to
55:59study for non-right angles, for
56:02non-right angles. Otherwise, if you
56:06have a bit more strength, then if these
56:08two sides are equal, what will this
56:09line that I dropped be? What will this
56:13line do? This line will bisect your
56:15base. We have already studied this,
56:17right? When we talk about triangles, we
56:20have already studied that it is
56:23isosceles. Right? Whenever we talk
56:25about isosceles, when we drop a line in
56:27an isosceles, what does that line do to
56:29the base? That line bisects the base.
56:33We already know this, right? What does
56:34that line do to the base? That line
56:36bisects the base. Clear, kid? That line
56:39bisects the base. So now you can also
56:42say that this is the height and this is
56:43the base. Do 1/2*height*base. Will the
56:46result be the same from both? It will
56:48be the same. So, we will apply the
56:50formula according to the situation.
56:53Okay, kid? So if this angle is alpha,
56:54then obviously I will say that this
56:56angle is also what? Alpha. Right? That
56:58covers isosceles. Similarly,
57:01equilateral triangles—these are all
57:03those nine shapes that we discussed a
57:06while ago regarding gradients. These
57:09are all the shapes we discussed
57:10regarding gradients. Similarly, when we
57:12talk about equilateral triangles. We
57:15say that if we need the area of an
57:17equilateral triangle, what will the
57:19area be? So we know what equilateral
57:21means. All sides are equal. We already
57:23know what equilateral means. I can say
57:26that an equilateral has three pairs of
57:30isosceles triangles, or three pairs of
57:33equal sides. When we talk about
57:37equilateral, all sides are equal.
57:39Everyone knows that. There is nothing
57:41extraordinary in this. What do we say
57:43in this? We say all sides are equal.
57:46All sides are equal. All angles are
57:48equal. Everyone knows this is 60. This
57:50is 60. This is 60. Everyone knows all
57:52these sides are the same. So if I tell
57:55a student, "I need the area of an
57:56equilateral triangle." Then the student
57:59can easily say 1/2*product of two sides
58:02*sin theta. 1/2 ab sin c, 1/2 ab sin c,
58:06we used to study this formula in
58:08O-Level Math too. Okay, kid? So, in all
58:12these nine formulas that I wrote, we
58:14didn't apply Heron's formula in any of
58:16them. Now, what is this Heron's formula
58:19? Now we have to start Heron's formula.
58:21There was a mathematician whose name
58:23was Heron. Heron—some people also
58:25call it Hero. Actually, his name is
58:27Heron. It is Heron's formula. This
58:29mathematician did something new. This
58:31mathematician introduced a new method
58:34to find the area of all these shapes
58:37with the help of coordinates. So, the
58:40name of this mathematician was Heron.
58:42In the next class, God willing, what
58:44will we do? We will apply Heron's
58:46formula for mensuration to find the
58:48area with the help of coordinates. Okay
58:50, kids? Then we will see what Heron's
58:52formula is. Okay? Goodbye.

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