Circle theorem cxc math

Ajouté : 2026-04-29

[00:00:02]All right. So it says the line AC is the diameter of the circle. So just identify that talking this this is the diameter. We know that the diameter is the line drawn from the circumference but it has to go through the center and land exactly on the other on the circumerence.

[00:00:25]All right.

[00:00:28]So you say find the size of the angle ABC.

[00:00:32]So they want us to find this angle here.

[00:00:38]Now this is one of the rules and angle hold before me say this. All right. So angle A B C is equal to 90°. Right? It's basically that's as one of the rule and one way you can look at it is half of the angle which is 180°.

[00:01:06]So angle of the circumference is half the angle at the center. So this angle is a straight angle. So this is 180 and this is 180 as well. So this angle is half or you can see that an angle subended by the diameter is equal to 90°. So even if it's still like this, this angle would be 90°.

[00:01:46]Or if it stay like this, this angle would be 90°.

[00:01:53]Once it's subended from the diameter, it's 90°.

[00:01:58]So this one ABC is 90°.

[00:02:05]So write the rule guys. That's one of the rules also fine.

[00:02:21]But you will catch on as we go along.

[00:02:38]or in some book we see angle at this angle in a semicircle is 90° mean half. So angle in semicircle 90°.

[00:02:59]So make a note of all of those, right?

[00:03:06]And then we can proceed.

[00:03:10]All right. So it says angle O is the center of the circle angle D E F. All right, let me show you something about angles. You see the one in the middle? That's where it's at. So like you say D E F. So this is the angle they referring to. All right. And that angle is equal to 54°.

[00:03:42]And they're asking for find the size of angle D O F. So D O F. So this is the angle that they wanted to find. Now the rule says this is one of the rule. The rule says the angle at the circum friends is twice the angle at the center. Let me write that two way we can write it. So angle at the circumference, everybody must know the circumference. The circumference is literally the distance around the circle, right?

[00:04:21]It's like the line itself. Angle at the circumference is half the angle at the center.

[00:04:36]So that's one way to structure it. CXC like to ask you to basically let me see something. Yeah. CC like to ask you to give the reason. All right. So that's one of the reason you can also write it like that. Say the other way around.

[00:04:56]Angle at the center can tell the word now.

[00:05:05]angle at the center is twice the angle of the circumference. So both of them are saying the same thing center. So yeah. So angle at the center is twice the angle at the circumerence. So that means this angle here is going to be 2 * whatever this angle is. That's why this x and this a 2x. So d o f is equal to 2 * the angle d e f. So that's 2 * 54 = 108.

[00:06:03]One thing when I was going to school, one thing I used to remember it, I always say, well, the angle always going to be wider than this one. So I know that okay, I don't mix it up because sometimes students will actually mix it up. All right, so the one the wider one is going to be the larger one. So the angle at the center is twice the angle at the circumerence or if they give you the one at the center you divide by two to get the other one.

[00:06:38]All right make a note of it please.

[00:06:44]All right so it says O is the center of the circle. The line XYZ is a tangent to the circle. Find the size of angle O Y Z. So of course O Y Z would be this angle right here. All right. This is another rule. Write down this one please.

[00:07:06]angle between a radi which we call radius and a tangent equal 90°. So right here is 90°. Not only that because this one is the tangent and then This is the this one is also 90° and it makes sense because this is a straight angle right and straight angle measure as 180°.

[00:07:49]So O Y Z would equal to 90° and the reason angle between tangent and radi equal 90° guys the you literally will lose mark for for the reasons the reason is off you're going to lose a mark so pay close attention to the reason why is it 90° all right just a All right. So it says I J KL is a cyclic quadrilateral. So the first thing we need to identify what is a cyclic quadrilateral. Now a cyclic quadrilateral is definitely a quadrilateral. However, the four vertices must sit on the circumference. So, you can't have it like this.

[00:08:59]One, two, three.

[00:09:04]This have to dep on this, right? You get the idea. It has to be on the circum.

[00:09:10]So, you check that when you get it. You have to check that. Ensure that all four vertices is on the circumference. So that's the first thing. The next thing now the opposite sides are supplementary. What do what I mean by supplementary?

[00:09:28]Both of them add up to 180°. So this call this I J call this J and this one K.

[00:09:42]So L + J is going to give you 180°.

[00:09:49]And of course I + J.

[00:09:57]Yeah. No, this J. This K.

[00:10:02]So I + K is also 180°.

[00:10:06]So that's two thing. You check if the vertices are on the circumference. All four of them can't be three, can't be two, can't be one. All four of them must be on the circumference of the circle.

[00:10:19]The next thing, the opposite sides are supplementary. So they add up to 180.

[00:10:25]And it makes sense because the all four angles add up to 360. So if two add up to 180 and the next one add up to 180, you get back 360.

[00:10:36]All right. So let's go to this question directly. Now it says find the size of angle I J K. So this is this angle here. And of course if J + 124 = 180° all we do we just transpose. So it will be 180° - 124 make 30 and then 50 well six. So 56°.

[00:11:07]All right. Find the size of angle J KL.

[00:11:11]You guys What's up with that one? What will be the size? You guys get no chance to do that one, guys. And what's up?

[00:11:20]I no J K L.

[00:11:24]All right. So, it says find the size of angle J K L. So, J.

[00:11:34]All right. I will find first J K L which is this angle here.

[00:11:43]So J KL which is this angle. So we know that 91 plus the K is going to give us 180. The two are supplementary.

[00:11:54]So we just substitute well not substitute transpose.

[00:12:00]It's a positive. So minus 91 from both sides. So that is give us 89°.

[00:12:07]So when all of them add up give you 360° and the two angles must give you 180 as well. The opposite sides please remember it's opposite you know opposite ones must give you 180. So the opposite angles are supplementary.

[00:12:33]Safe.

[00:12:39]>> All right. A little bit different. So probably let me see. Find the size of angle M. Oh, so far. Sorry.

[00:12:52]All right. So it says A M B and C D M D are triangles. Angle A MB is 39° and angle DC cm= 42°.

[00:13:05]Find the size of M D C I'll do this one.

[00:13:17]I don't think we have done one like this. All right.

[00:13:22]So, AM B and CBM. So angle AB M. So we have A B M which is this angle here is 39° and angle D cm. So we have D C M this one is 42° and they want us to find M DC. So remember we said that is the third angle the mega one rather. So it's angle D. So we want to find this angle. So this angle here MDC is equal to A BC.

[00:14:06]Angle in the same segment are equal. So angle in the same segment it says another rule are equal.

[00:14:21]So that mean is equals to 39°. I can't ask this one.

[00:14:28]See for that.

[00:14:32]Yeah, you guys do that one. B A M. What will be the size of B A M? What?

[00:14:41]All right. So find the size of angle B A M.

[00:14:46]So that's B A M which is this angle. Now this angle is equal to so B A M is equal to DC cm.

[00:14:59]So both of them equal 42.

[00:15:02]Same reason angle in the same segment.

[00:15:11]Remember to write the reason guys just 42.

[00:15:15]So these two equal and these two equal also just to add to that this angle here and this angle here equal and they are said to be vertically opposite.

[00:15:34]Makes sense, right? Because all three angles in the triangle must add up to 180. So if this and this equal and this and this two have equal to as well.

[00:15:49]All right. Um A M is equal to C M D.

[00:16:02]They are vertically opposite. Make a note of that please.

[00:16:09]All right. So it says so is the center of the circle. Two tangent to the circle touch the circumference at point m and point n. T is the point Jesus.

[00:16:34]All right. So T is a point where the two tangents meet. Angle OTM is equal to 51°.

[00:16:45]Find the size of angle N to O.

[00:16:49]I want you guys to try that one before I give the solution. All right.

[00:17:01]Who is the center of the circle? The two tangents of the circle touch the circumference at M and N. T is the point where the two tangents meet. And it says angle OTM which is this angle here is equal to 51°.

[00:17:17]And it wants us to find the size of N.

[00:17:21]So N would have been this angle here.

[00:17:24]and NTO is equal to M to O basically bisect the angle. So this is 51° as well. Um I thought they ask for one here.

[00:17:43]So if they had asked for here what guys A and B what be the size of these two angles separately separately sometimes that's what they ask for in the exam you have enough information to answer it all right so to find NOT or MOT both of them are equal basically Because remember this is a radi.

[00:18:22]So once you figure out this 90° then it becomes a k have been 90 plus the 51 um 5 and 9 or 14 and then you subtract this from the 180 to give you 39°.

[00:18:43]So both of them equal 39.

[00:18:48]Make sense? All three angles add up to 180. All four angles add up to 360 cuz this is a quadrilateral as well foursided.

[00:19:02]So and not equal 39°.

[00:19:08]Let's see if that one do already. No, we have not done this one. All right. So our new rule the line DCE is a tangent.

[00:19:19]So this is a tangent angle B C E just put in that is equal to 61. So B C E which is this angle here is equal to 61°.

[00:19:34]Angle B C A D C A is equal to 68°.

[00:19:46]And they want us to find the angle C A B which is this angle right here.

[00:19:55]C. All right. That's kind of easy because we can see that this one is a angles on a straight line. So all of that add up to 180.

[00:20:05]>> So we find C.

[00:20:09]>> Oh, sorry.

[00:20:17]Oh.

[00:20:28]All right. So, C A B is equal to the co interior opposite. So, this angle here and this angle right here would be equal.

[00:20:39]So, this one is 61.

[00:20:43]And also this angle here and this angle here equal.

[00:20:54]Sorry about that guys. Literally a ACB.

[00:21:00]If you wanted to find ACB, two way angles on a straight line or angles in a triangle.

[00:21:06]So either way you could find it that way.

[00:21:14]And that um it's actually ACV is equal to 180 - 61 + 68.

[00:21:26]So 9 129. So you want one more for 30.

[00:21:30]So that be 51, right? So this will be 51. Angles on a straight line add up to 180. Also angles in a triangle add up to 180.

[00:22:06]So please remember this guys. This one equal to this one and this one equal to this one. I feel uh right to carry this What's going on?

[00:23:31]So we call it the alter segment theorem >> right as you can see it's angling a semicircle right see there diameter that means say FGH would have been 90° and this is 90° and FG which is this length here equal to this length it mean that it's an isos right and if the two of them equal is 180 - 90 = 90 / 2 + 45° I try number 9 them and JK M are triangles. Angle M JK= 33°.

[00:24:26]Angle M L I= 47. Find the size of angle L I M and find the size of angle M KG.

[00:24:37]I think that's might be the last one for tonight. I wanted us to kind of go a little further.

[00:24:46]So we got it. All right. So it says I L M and J K M are triangles. As you can see angle M J K. So you have M J K that's 33 and angle M I M L I M L I which is this angle here= 47°.

[00:25:14]Find the size of L. So L would be this.

[00:25:18]So this angle equal to this one. So that's 33. So L= 33. Angling the same segment equal and of course find the size of angle M K. So M K which is this angle here. And this one is equal to this one which is 47° and it's the same reason. So please make a note of that.

[00:26:34]This is all >> saying if class end. Yeah. Yeah.

[00:26:49]Hey guys, see you guys on Sunday.

[00:26:55]>> Good night.

[00:26:56]>> Good night. Good night. Take care.

[00:27:00]>> Next question.

[00:27:04]Well, Wednesday know what time >> again 6 to 8.

[00:27:19]>> Okay. All right, sir.

[00:27:21]>> All right.

[00:27:23]All right. Take care, sir.

[00:27:24]>> Yeah, man. All right. Take care.

[00:27:26]>> Finally.

[00:27:29]>> All right. Take care, guys.

[00:27:31]Keep going. All right.

[00:27:33]Yeah.