CIRCLE GEOMETRY

Añadido: 2026-06-02

[00:00:01]circle geometry in this class.

[00:00:07]So we started this topic circle geometry.

[00:00:12]We looked at some important aspect of a circle.

[00:00:22]Look at the definition of a circle and some parts of a circle. Also we look at the theorem one. I state that the angle subained at the center of a circle by a C.

[00:00:44]In this case we choose a C at the center. The angle form at the center here is twice the angle at the circumference.

[00:00:55]So we have angle form at the circumerence if this is C. So angle center is twice if angle at the circumference is X.

[00:01:06]Angle center is 2 X. So that's the first theorem there which is we look at the proof in our last class.

[00:01:21]Don't forget that there's another angle that can be formed at this side.

[00:01:28]This angle is circumference as well. If this angle is Y, the angle at the center is at this reflex side. This angle here will be 2 Y.

[00:01:41]So the understanding of this theorem will help us to prove some other. When we look at this one, two says the angles of an ar in the same segment of a circle are equal. So we have a center of a circle hole.

[00:02:06]We have a cut.

[00:02:12]So this cord here has divide this circle into two segments.

[00:02:18]We have the major segment and the minor segment. All the angle formed by this chord in one segment are equal. So let's form this angle here by this cord or this angle formed by this cord.

[00:02:43]This angle this angle this angle they are equal they are equal.

[00:02:51]So the theorem is proved from the first the construction we join A to the center also B to the center and this angle here is angle at the center.

[00:03:09]So if angle center is 2x don't forget that angle at the center is twice the angle circumference. So definitely from the first term this will be X.

[00:03:23]So this is also at the circumference to this angle at the center. So also this angle is X because this is 2X. This is also at the circumference to this angle at the center. Also this is X which makes this X X here equal. So that make angle in the same segment equal. So whenever we say something like this and there is an angle given in some cases you not see a cut.

[00:04:01]You just see something like this.

[00:04:04]What is subending it is an hack. So instead of drawing a line like this. So no line what we see is just this and label is as a b c d. So if an angle is given here let's say this is 40° and they say find this angle this angle will be 40°. Why they lie in the same segment?

[00:04:33]Likewise this angle B and this angle A they line the same. So let's have the understanding of that theorem. Okay.

[00:04:46]Also we have another theorem this theorem number three and uh this theorem also follows from the first theorem and we have a circle center hole.

[00:05:14]Then a diameter that is a line passing through the center. That's a diameter.

[00:05:20]This a circle center with this diameter.

[00:05:25]Let's diameter is AB.

[00:05:28]So the angle obtained by this diameter at the circumference is a right angle.

[00:05:38]So this angle A C here is a right angle. Why is it a right angle on a straight line?

[00:05:49]And don't forget that angle on a straight line is 180. So and it's twice that of the circumference. So that means we need to divide 180 by 2 to get 90. So that is the third theorem.

[00:06:05]So whenever we come across something like that, we don't need to be told before we know that that angle will be 90°. So if an angle is drawn and they said this line AB is a diameter and form an angle with it center of the circle they form an angle with this and this C.

[00:06:39]So without putting any sign here we should know that this angle will be 90° because this line is a diameter. So angle sub 10 by diameter circum is 90° or we can say angle in a semicircle angle in a semi circle is 90° or right angle.

[00:07:09]Okay. So that is terrain number three here that we are looking at.

[00:07:18]Then let's look at another theorem.

[00:07:22]We call this theorem four.

[00:07:27]Okay. A line drawn from the center of a circle to the midpoint of a C to a C.

[00:07:38]We have a circle center O.

[00:07:46]We have a code AB.

[00:07:53]Okay.

[00:07:55]So a line that is drawn from the center of the circle to this C. This line is perpendicular to this C. That's what the theorem is saying. So now when you check this we can construct by joining A to O and don't forget that this will be a radius line from the center to circumference is a radius also join B.

[00:08:26]So this will also be a radius.

[00:08:30]Now let's leave this middle as N.

[00:08:35]So from here this n is the middle of this chord which implies a n is same thing as b n.

[00:08:48]So okay and these two triangles are congrent.

[00:08:56]We have all the sides. This O N is common. Side O N is common to them.

[00:09:02]And side A and BN are equal.

[00:09:07]So A and B are also equal because both are radius. So which make these two triangle congress.

[00:09:17]So we can use any of them to find.

[00:09:21]Actually this theorem can be used to find the length of this.

[00:09:25]So when we know the distance from the center to this code don't forget that anytime we know two side of a triangle to get a third side we make use of what is called pythagoras theorem theorem.

[00:09:47]So we say that in this case now a n we pick one of the triangle is right angle. So this is R.

[00:10:10]This is O then this is A.

[00:10:14]So r is hypotenus which is r² = to the sum of the squares of the two other side. So which is a n² + n².

[00:10:28]So whatever is missing there some time will have been given the distance from the center of the circle to the cord and we have been given the radius and the question will say find the length of the cord. So we just put the length from center to record here. Let's say this is four. The radius is five. Using python now we can get a n and a n will be three. So if a n is three that means bn is also three since the distance are the same. So the length of the cord AB will just be 2 * 3 or 3 + 3 you know same thing 2 * 3 that's 6 whatever unit so that is about that theorem four there is the four that's that this is very useful to get the length of a All right.

[00:11:38]So from here we have another theorem and this theorem has to do with a cyclic quad lateral.

[00:11:50]Okay.

[00:11:56]When we have a circle with a collateral inscribed You know has four sides four angles. So each of the angles four angles is touching the circumerence of the circle.

[00:12:12]That's why you can call this a cyclic or lateral cyclic or letter cyclic or the letter.

[00:12:25]So we can't call it a cyclic if the veral is not touching the circumference. So we discover that the four angles are touching the circumference of the circle.

[00:12:44]So now the theorem is saying that the opposite angle of a cateral are supplementary.

[00:12:53]This angle and this angle are opposite here.

[00:12:58]So we label this as y this as x or x y.

[00:13:04]So we say the sum which is x + y = 180°.

[00:13:14]So that's meaning of the theorem. So the opposite angle of add up or sum up to 180° that is their supplementary angle their supplementary angle. So let's take note of that. Let's this be a and this be b.

[00:13:35]So in the same way a and b are opposite.

[00:13:39]So a + b also equals to 180°. to opposite angle of a cyclic as supplementary.

[00:13:48]So we should know that theorem as well before we solve problem.

[00:13:55]And another theorem which follows theorem five we can call this as theorem six is also on collateral.

[00:14:09]And when we have a collateral inscribed in a circle, we call it a cyclic collateral and one of the side being extended that is one side go beyond.

[00:14:25]So we want to look at the angle which is exterior angle here. The says that the angle of equals to the opposite interior angle.

[00:14:37]So this angle here and this angle here are equal. So let's call this angle X.

[00:14:44]This angle will be X.

[00:14:47]So that is what this is talking about.

[00:14:50]So when we solve problem relating to circle we see questions that we don't have to go long way. So when we see that is a collateral and one side is extended like this. So the exterior angle definitely equals to the opposite interior angle. So we have these six theorems.

[00:15:18]Now on circle geometry. So we have some other three theorems which is rel to tangent tent theorem.

[00:15:32]Tent theorem.

[00:15:36]Okay.

[00:15:38]So for the tangent we must know what a tangent to a circle is.

[00:15:51]A tangent is a line that touches a circle at a point.

[00:15:55]So if this is number seven you have a circle center.

[00:16:00]So when we have a line that touches a circle like this at this point. So this is a point of contact and this line we draw is a tangent.

[00:16:12]So the first on tent is that a line drawn from the center of a circle to the point of contact is perpendicular to the tangent. So this line drawn from the center of a circle. Don't forget that a line that is drawn from the center of a circle to the circumference is a radius.

[00:16:31]So this line drawn is definitely a radius. Why is it a radius? Because it's from the center to the circumference here.

[00:16:40]And in this case the tent is touching this radius here.

[00:16:45]Now so the angle between this radius and this tent is 90°.

[00:16:51]So we should know that is similar to this why this now is becoming a target. So all right that's what we have there.

[00:17:06]The angle between the radius and the tangent is 90°. So if this is 90° this other side 90° they both add up to 180°. So that is that very very important theorem. Now let's consider the next theorem on taggent that's that's to do with a tangent and excuse me we have like three cuts cuts in a circle and a tent.

[00:17:46]So let this be the tangent to this circle center.

[00:17:53]So the center is not really needed. So let this be the point of contact.

[00:18:00]And if there's a cord on this point of contact in this circle, the angle between this cord and this tent is equals to the angle between other two cuts. So we have another cord like this.

[00:18:20]This is a C. Don't forget we also have another C like this where they form an angle here.

[00:18:28]Okay. So these two C attach to the force C and forming an angle here.

[00:18:36]So this angle between the force let's call this C number one cord two C three.

[00:18:43]So the angle between chord one and tangent is equals to the angle between two and three.

[00:18:53]So if this is x, this angle is x.

[00:18:58]So it want to look like this particular number but it's not. So this has to do with tangent and from the point of contact you have a cord not just one chord you have chord one chord two chord three. So that the angle between chord one and the tangent is equals to angle between cord two and chord three. So this theorem on tangent is very very important as well.

[00:19:36]So we should note that and when we start applying this you'll be able to see clearly. Now when we look at this other side this angle also between this chord and this tent is equals to this angle here between this cord and this other cord.

[00:20:04]Okay. So that is thating which is number eight making number two on tenting.

[00:20:12]Then also we have another is the last. Yeah.

[00:20:23]Calling number nine.

[00:20:25]So we have a circle.

[00:20:29]Okay. Let's make it.

[00:20:34]Let's put it here.

[00:20:38]The number nine. You have a circle.

[00:20:44]Okay. And there is a tent two tents coming from a point P outside here meeting at this point P external point P.

[00:20:59]Okay. So we have this at the center of the circle. Don't forget from this first term if we draw a radius now touching this tent this angle here will be 90° also touching this tent here this angle is also 90° the theorem says that if come from external point the circle these two tangents are equal in length so they are equal in length and if A line is drawn from the center to the point P. This line will bisect the angle between this point. The word bisect in mathematics means to divide.

[00:21:48]So this line that is drawn from the center, we divide this angle here into two= half. Also we divide the angle here into 2= half. So we need that theorem and we should be able to apply it as well for us to be able to solve questions.

[00:22:17]So we have n theory. Once we able to master this n theorem in this class, we can lay our hands on questions and solve and solve question. Okay.

[00:22:32]So we write the question down now and try it before you meet in our next class. So if you are just joining this class, you are just coming across this channel.

[00:22:46]This is master t class where we solve mathematics.

[00:22:53]So try so that you can access our video easily from here. The question I want to give let's try it against the next class.

[00:23:12]Okay. So we have this circle.

[00:23:18]Okay.

[00:23:23]This is center of the circle.

[00:23:29]This angle here is touching.

[00:23:32]So we are given an angle at the center here 130° and we have 26° here. This angle is A B then this as C.

[00:23:53]So there is a line drawn like this like a construction line just like a distraction something which is also useful when you look at the proof of theorem one yesterday last class rather so you discover that we did a construction like this and the something there's a line that also went like this this a standard exam question like a y so this line I think these two lines is not affecting our solution and we can use it to get our solution easily as well. So this one 30° so and then this 26°. Now these question says we should find this angles. Number one angle. Don't forget this is O. This is center of the circle.

[00:24:52]Say find angle B C.

[00:24:57]No, find angle O B C.

[00:25:04]When you say angle BC, you know what it means. So you go from O to B then to C. B is the main thing here.

[00:25:14]So O BC you find this angle and two find angle B O C you go from B to A to C which means this angle. So this is an interesting question which when we understand it theorems it is easier to solve. Thank you for watching. I will stop here today.

[00:25:49]So see you next class and bye for now.