JJ ONLINE MATHS CLASS is live! Circle theorem

What's your name? What? My name is William. William... What? I don't know... What? You came out playing the tamarind, and you had to... I don't know... So I'm trying to trigger this thing. Guys, please bear with me. Okay. Hi. Okay, if you're alive with me, say hi. You're welcome. So I'm trying to get one or two things sorted out. I tried the portraits. The way it is now is better, right? Please confirm for me The way the board is now is better, right? I have 10 people on board. Please, I want you to confirm if the board's position is correct. Yes, it is okay, please, thank you. Okay, Joe, thanks for the confirmation. You're very welcome to my live stream. This is actually my first live stream here on YouTube, okay? You're welcome. And today we are going to discuss circle theory. Circle theory. This is the topic we are going to discuss today, alright? So I would like you to pay close attention to this. Please pay close attention to this so that you will understand me. Okay, I can see that I still have some portion of the board. But let me still adjust this. Timothy, thank you. This is actually the first time I'm going live here, okay? The first time. And I want to say thank you for taking me to 1K subscribers here on YouTube. I know it's not easy, okay? It's not really easy to have a thousand people standing by you, okay? So I want to say thank you to all of you. And yes, Joe, I know. That's why I'm watching with my other phone. So I know the part that is not visible. So I'll try to avoid those parts that are not visible. That's why I drew my line. You can see the line I drew. So it's going to determine where I'm going to write and where I'm not going to write. Please, I've seen through it. I've cut that. But can you hear me clearly and loudly? If you can hear me clearly, please say so. Can you hear me clearly? If you can, please let me know so we can get started, okay? Okay, I believe people can hear me. If you can hear me, just say yes. You can hear me loud and clear. Timothy, can you hear me clearly? Joe, can you hear me clearly? All right. Sorry if I don't pronounce it very well. Welcome to my live stream. Alright, let's begin immediately, right? Should we still wait a little bit, or should we start? I think we should start. Those who are unable to join us on time can watch the replay. Alright, in a special way, I welcome each and every one of you. I welcome you, Joel. I welcome you, Timothy. I welcome you, Miopa. Welcome all the way from Zambia. Timothy, you are most welcome. You're all welcome to my live chat, my live stream, okay? Today we are going to discuss the circle theorem. Trust me, by the grace of God, by the time we have finished with this particular topic today, You will be able to solve any question you encounter that is related to this. based on the theorems that we will be able to cover tonight. All right? Alright, let's begin with the first theorem. Theorem 1. Before we go into the circle theorem, I would like you all to simply allow me to go down to the roots of this circle theorem. Okay? Let me go to the root of this circle theorem. That is because it's when you understand the circle perfectly. that you will be able to understand the circle theorem perfectly. Okay? Alright, now let us discuss the circle and the parts of a circle. First of all, what is a circle composed of? Dots. Just like dots, like this. Okay? Then a line is used to join those dots to form what? A circle. Now, the line used to connect these points is what we call the circumference of a circle. The circumference. Okay? Now, this is the circumference of this circle. Now, if this is the center of a circle, Any line you draw from here to touch one part of the circumference to another across that circle is called a chord. This is a chord. Now, how do you recognize a chord? A chord divides the circle into two parts. It divides the circle into two parts. It's going to divide the circle into major segments. This will be the main segment. That is the larger portion of the circle. And the other part will be the minor segment. Yes. A chord divides a circle into two parts. The major segment and the minor segment. Yes. Alright. Alright, that's a chord. Now, let us discuss another part of a circle. A diameter. If this is the center of a circle and a line is drawn across the circle through the center, It's going to divide the circle into two equal parts. Two equal parts. Okay? Now, what are those two equal parts called? Semicircle A semicircle. Alright? Now, this line that divides the circle into two equal parts is called a diameter. A diameter. Now, a diameter is a chord. But the difference with this particular chord is that it divides the circle into two equal parts. Alright? Alright. Now, let us continue. The writings are not clear. Joel, can you see the writings very clearly? If you can see the writings very clearly, let me know. Joel, you said the writings aren't clear, right? Wow, what's going on there? Okay, let me check something. Let me check something, please. Okay. It's okay now, right? Alright, alright, alright. Thank you, Mojit. Thank you. More brightness. Oh. I can't increase the brightness from there. I can't increase the brightness. Let's just continue. If at any point you cannot see at all, please let me know. Bluish in color. Oh. But my pen is dark in color, so can you see it clearly, please? Alright, please let's handle it. I don't have a good phone. That is the truth. Please handle it like that until I can buy a better phone or a better camera. Okay? Thank you very much. Alright. Now, can you see this theorem complete? Can you see this theorem completed? Can you see the theorem complete? Sorry, guys. I'm experiencing network issues here. A serious network issue. I'm experiencing a network issue. Oh. I'm so sorry. Okay, guys. Let's continue. I'm experiencing some network issues. I'm sorry about that. So let's discuss another part of a circle. This is the most important part that I want people to look at. This is the most important part. Now, if this is the center of the circle, when you draw any line from the center of the circle to the circumference, You are going to have your radius. Okay? You are going to have your radius. Now, if you draw another one from the center to the circumference, you will have another radius. Now, this radius now encloses a portion of a circumference called the arc of the circumference. Guys, this is an arc. If we call this place A and this place B, what is the distance from here to here? Arc of a circle Now, what is all of this? The sector of a circle. Sector of a circle Now, the only thing I want you people to note here is that from here to here is the arc of a circle. and any line drawn from the center to the circumference is a radius. A radius. It is a radius, and they are equal. That is, from here to here is equal to from here to here. What are the two of them? Radii Alright, let us begin the theorem proper. It states that an arc of a circle subtended at the center is equal to twice the angle it subtends at the circumference. What does that mean? The angle it subtends at the center is twice the angle it subtends at the circumference. Now, this is the meaning. If you have a circle, okay, and this is the center of the circle. If this is to be our arc, let's call it AB. This theorem states that when you draw two radii, okay, from A to the center and back to B, What angle are you going to get? The angle subtended by the arc AB at the center. Alright, the angle subtended at the center is θ. Now, if you draw another line from your A and it touches any part of the circumference, any part of the circumference, okay, if it touches it and comes down and stops at B, It means that it is still the same AB arc, AB. by maintaining another angle this time around, not at the center, but at the circumference, okay? Now, let us call that angle alpha. Let's call that angle alpha. Now, this theorem states that 2 times this alpha is equal to theta. That's what it's saying—that two times this alpha equals theta. So, assuming that this alpha is now what? 20 degrees, okay? And you are asked to find theta. The question asks to find theta. It means that our theta is equal to 2 times 20, which is equal to 40 degrees. I hope you're all keeping up with me. Which is equal to 40 degrees, okay? Alright, let's go over some examples. Example. Now, this question states, this is the drawing. Okay, let me draw it here. So, this is the drawing. You have something like this. This is the center of the circle. You have something like this, like this. Something like this. Something like this, okay? Now, this is 35 degrees. This is cubed. This is P. This is S. And this is R. Okay? Now, we are asked to find... Okay, here it is: 40 degrees. Now, the question states, if this zero is the center of the circle, We are asked to find angle POS. POS POS means that we are looking for this specific angle called theta. Okay? Let's call it theta. Angle POS is what we are looking for. Now, I want you to pay close attention to what I'm going to do here. Please pay close attention to what I'm going to do here now. If you look at this drawing now, You notice that from your P, they drew a line or a radius from this P to the center, and they stopped at S. That means PS, from here to here, is what? Our arc. So, what is PS? Arc of a circle And it means that this arc maintains this angle theta, which we are asked to find at the center of this circle. Okay? Now, they drew another line from this P. They drew it and touched the circumference here, and come down to S, meaning that it is still the same PS that is maintaining an angle now at the circumference, Just as I explained earlier. Once that line starts from P and touches any part of the circumference, and stops at S, It means that it is the same PS that sustains what? The angle at the circumference. Now, let's call this place arrow degrees. Now, we are asked to find angle POS, which is this theta. Now, note that this angle at the circumference, which is arrow degrees, That 2 times that arrow's degrees is equal to theta. Is equal to theta. I hope you understand. That means that the main thing we are looking for here, What we are primarily looking for here is what? Arrow Once we get our arrow, We will be able to obtain the theta that we are asked to find. Okay? Alright. Now, let's go ahead and get the arrow. To get the arrow now, I'll draw the triangle out. This is what we have. Like this. Now, what is this? 35 degrees This is Q. This is our arrow. This is P, right? And this is 40 degrees. Now, we are looking for this arrow. Arrow degrees here. I want you people to know that when you see a triangle, There is this theorem about a triangle that The exterior angle of a triangle is equal to the sum of the two opposite interior angles. The sum of the two opposite interior angles. I'm going to show you two methods you can use to find this arrow. Okay? So what I'm trying to say right now is that this arrow here, those arrow degrees Which is the external angle of this triangle? is equal to what? The sum of the two opposite interior angles. Our arrow is equal to 40 plus 35, which is equal to what? 75 degrees Okay? Which is equal to 75 degrees. Alright. If that is the case, Let me show you another way people can get their arrow. Let me show you another way you can obtain your arrow. This is another way. You people know that QPR is a triangle, right? And the sum of the angles in a triangle is 180 degrees. The sum of the angles in a triangle is 180 degrees. Now, if I were to call this place, Let's call this place X. Okay? It means that this X plus 40 plus 35 is supposed to give us 180 degrees. Right? Okay. So, it means that our X plus 40 plus 35 will give us 75. So, X plus 75 equals 180 degrees. Okay? So, how do we get our X then? It means that our X is equal to 180 minus 75 degrees. What is 180 degrees minus 75 degrees? It's going to be 105 degrees. Okay? So, it means that our X is 105 degrees. That means this place is now 105 degrees. Let me redraw it here. So, this is what we have now. This place is 105 degrees. Here is 35 and here is 40. But you people know that we are not looking for this place. What are we looking for? Our R degrees. Right? We are looking for R. But you know that an angle on a straight line is 180 degrees. The sum of angles on a straight line is 180 degrees. Now, this is what we have here. We have a straight line. We have another line here. This is 105 degrees. And this is R. 105 plus R equals 180. 105 plus R equals 180 degrees. Okay? So, how do we get our R then? So, if 105 plus R equals 180 degrees. That means our R is what? 75 That will give us 75 degrees. That's exactly what we have here. Okay? We are still going to get 75. It means the shortest way is simply to say 40 plus 35. And you're going to get your R as what? 75 degrees Alright. Now, let us go ahead and solve this problem. Now, this is what we have. What is our R? 75 degrees What is that? 35 plus 40. So, what are we going to replace this with? 75 degrees Now, if our R is 75 degrees. Now, we are looking at this: our theta is the angle subtended at the center by this arc. That's nonsense. And this 75 is the angle subtended at the circumference by this arc. Now, using the circle theorem. It means that 2 times 75 will give us theta. Right? So, this implies that 2 times 75 equals 150 degrees. Is equal to theta. What angle is that? POS Can you see how I solved it? I believe you saw how I solved it. Okay? Alright. If you have any questions, please let me know. Please let me know if you have any questions. Alright, James, you say you don't understand how I get my 2. It is from the circle theorem. It's as if you joined us late. The angle at the center is twice the angle at the circumference. So, if this is the angle which this arc subtends at the circumference, which is 75. It means that 2 times this 75 will give us this theta. Do you understand now? James, do you understand now? So, the 2 comes from the theorem. The angle at the center is twice the angle at the circumference. Alright, let's solve another example. Let us solve another example. This first question is from what? This is from a Jump question. So, they repeated the question but twisted it and asked a different question there. So this time around, they drew something like this. Something similar, but different nonetheless. So, they drew something like this. Something like this. Like this. They are repeating questions. That is why, if you want to tackle any question, any exam, I would advise you to do what? Please go and read your previous questions. They usually repeat questions. Okay? Alright, so this is what they have. straps This is our R. This is our S. They drew something like this. Again, this is 0, the center. This is P. This is 20 degrees. This is 35 degrees. This is Q. Okay? So, the answer we should find is angle P, S, 0. Now, we Serving the angle P, S, 0. Find angles P, S, O. P, S, O. That means we're looking for this one this time around. You will believe me that this is a very simple listing. Very simple question. Now, how do we address it? We're not going to waste time on this one. Because it's similar to what we just saw. Except that they asked a different question. Okay? Except that they asked a different question. Alright. I'm just sorry about that. Okay? Alright, let's go ahead and find angle PSO, which is this one. First of all, let us call this our small r, r degrees. Okay? If we are to refer to this as our r degrees, we will use the fastest method to obtain our r this time. We are going to use the fastest means to get our r. Chibike, I can't tell you how many there are. But we will keep going step by step, bit by bit. Okay? We will continue to take it bit by bit, please. So for tonight, we are going to do this. Then we fix another one tomorrow again. We continue from where we left off until we complete all the second theorems. Alright? So this is a question from 2004. You have to find angles P, S, and O. That is this particular angle. Now, how do we get it? Let us get our r. Because if we can reach here, we can achieve this. Just follow me and see how we're going to do it. Okay? Now, this is our R. The exterior angle is equal to the sum of the two opposite interior angles. Let's use that one and get it. So it means that our r is equal to 20 plus 35, which is 55. 55 degrees. Okay? I believe everyone will understand this one. Okay, now we have gotten our r to be what? 55 degrees. Now, it means that our O, let us call it theta. That means our theta is going to be equal to 2 times 55, right? That will give us 110 degrees. Why? Because the angle at the center, which is theta, is equal to 2 times the angle at the circumference. The circumference. This arc, P.S. Maintains this angle, theta, at the center. Okay? And also maintains this angle, r, at the circumference. Alright? So, that means our theta is 110 degrees. But that is not what we are looking for. We are looking for this. Just don't mind the way I drew this. Don't mind the way I drew this. Now, this is what we have here. This, this, and this. This is what we got at 110 degrees. What is this to us? Center Now, this is P and this is S. Pay close attention to what I am about to say now. When you look at this triangle, you notice that your line extends from the center to the circumference right here. From center to circumference. Meaning that this place is what? Radius Is the radius of a circle. Now, you drew another line from this zero to S again. Meaning from here to here. That is, from the center again to the circumference is another radius. This is another radius. And one thing about the radius of a circle is that what? They are equal. Excuse me. Chibike, thank you very much. Alright, the radius of a circle is what? They are equal. Now, why did they give us this? Why is it important that we are analyzing this radius? You know, in an isosceles triangle, the two sides are equal. Now, once two sides of a triangle are equal, it means that the name of the triangle is an isosceles triangle. That means the base angles of that isosceles triangle are equal. What does that mean? When you have this, this place and this place are equal. This is how they mark it. It shows that this place and this place are equal. As when this place is 100. OK. Now, notice that the sum of the angles in a triangle is supposed to be 180 degrees. Right? So, it means that 80 remains for this particular angle to be completed. OK. Now, since this is an isosceles triangle, that 80 will be divided by 2. By 2. So, it means that this place is 40 and this place is also 40. What are the base angles? Equal. What are the base angles? Equal. Yes. So, that is the essence of stating that this is a radius and this is a radius. Since these two sides are equal, this is an isosceles triangle. And remember that this is what we are looking for. This S. What is that? Base angle of this isosceles triangle. Now, let's go ahead and get it. Since this place and this place are equal, we are going to assign the same variable to that location. So, let's call it xx since they are equal. So, we are going to have that x plus x plus 110 equals 180 degrees. That is the sum of the angles in a triangle. Right? Okay. Now, x plus x is what? 2x It's just like saying 1 orange plus 1 orange. It's going to give you two oranges, right? Yes. So, x plus x is 2x. Alright. Plus 110 will give us 180 degrees. Okay? Alright. Now, 2x is going to be equal to 180. This plus 110 will cross over to become minus 110 degrees. Okay? So, this 2x will now be equal to 180 minus 110, which will give us what? 70 degrees Excuse me. But you know, we're looking for x, not 2x. Now, how do we obtain our x? We are going to divide this by 2 and divide this by 2. Why do I choose 2? I chose 2 because I know that it's only when I divide this by 2 that, if I cancel and cancel, I'm going to have only x remaining here. So, our x now, this x here that is remaining, equals 2 divided by itself, we are going to have 1. Divide 7; we are going to have 33 words remaining. 1. Divide 10, and we will have 5. So, we're going to have 35 degrees. That means our x, which is angle PSO, is what? 35 degrees So, angle PSO is equal to 35 degrees. Alright. Finally, I think with this, you all understand what I did, right? So, do you have any questions? Joel, thank you very much. Okay, Joel. You're welcome. I'll do that. I'll have warm water. James, I'll have the bitter cola. Thank you very much. Thank you. Alright. I think I'll just give you the one you're going to solve on your own. Solve it and drop the answer you got in the comment section because I'm going to post This lives as a replay. Please share your own answer in the comment section. Is this time okay for everyone when we come again tomorrow? Is 8:30 PM okay, or should we move it to 9 PM? Let me know in the comment section. Is 8:30 PM okay, or should we move it to 9 PM? Look, I'm going to hand in the assignment right now. Just let me know if you like the time or if we should increase or decrease it. Time, please. So, let's go over the assignments. Okay. Assignments They also received the same question. It's not time-consuming, James. Seriously, it's very easy because I'm analyzing it. That is why it seems time-consuming. It's not time-consuming. And they are repeating it over and over again. I'm going to give it to you again, but they've twisted the question once more. This particular one is from a different year. A different year. Now, this is what they have here. The same thing. The same thing, but what? Different question. I have something like this. I have this. So, this time around, this is S. This is P. This is R. This is 0, the center of the circle. This is Q. So, they said the temperature here is 35 degrees. It's 30 degrees here. This is 35. 35 degrees. This is 30 degrees. So, we are asked to find angle OQP. Find angle OQP. This one here. OQP That means they are searching for this angle. Find angle OQP. That means they are searching for this angle. This should be very easy. They asked the same question again. They only changed the numbers. They changed the numbers. That is precisely it. Alright, everybody. Thank you so very much for joining my life. I should add options. Okay, sorry. These are your options. Options Your A is 35 degrees. Your B is 30 degrees. Your C is 130 degrees. Your D is 25 degrees. Okay, these are your options. Please solve this and drop the answer for me in the comment section. When we come tomorrow. Once again this time. Before we start tomorrow's live stream. Lesson for tomorrow. Thank you so very much for joining me. 8:30 PM is okay. Have a lovely day. Bye.