good day good day grade 11s all right so today we'll be looking at ukian geometry right all the nine theorems so if you haven't subscribed please just make sure that you're part of the family right so we're going to take it slowly uh you can play this video at two times the speed if you want to okay so let's just look at all the theorems all right let's start with theorem one so if we look at theorem one the first theorem tells us if we've got a line that is drawn from the center so if we've got a circle and we draw a line from the center and that line is perpendicular to the chord right please note that a chord is a line that moves from circumference to circumference okay so we say that if a line is drawn from the center and it is perpendicular to to the cord then we know that it bcts that chord right so let's suppose that we've got line o well Center o and we've got cord AB all right let's say that it reaches uh or rather the line reaches the uh chord in this case at C right so if that line from the center is perpendicular to the quad remember perpendicular means it forms a 90° then we say that it is uh it bisects the cord so what is our um well what are we given we're given OC perpendicular to a right and our conclusion therefore is that if that line is from the center which is O the center of the circle therefore we know that in this case AC will therefore be equal to CB all right now note that theorem 1 also has got a Converse and I'm going to give you an easier way to state that theorem in fact I might as well do that so when you state the reason in this case we always say line from Center okay so this is line from Center perpendicular to cord okay that's the reason now if we draw again so suppose that we've got cord our Cod and we draw a line that is from the center let's suppose okay I know my Center is not really nicely centered okay so let's suppose in this case that we've got again uh Center o we've got ab and now we are given that AC is equal to CB right so note in this case we can say well a line that is drawn from the center that bisects a chord is perpendicular to that chord okay notice in this case what are we given we're given that AC is equal to CB right so meaning that h a chord AB is bisected okay so therefore we can conclude that it means that OC is therefore perpendicular to a b now note how do we State this reason uh in this case we say that this is the line from Center this time we say bisects the cord all right so that is how we're going to State theorem one right very easy and please note the first theorem the first three theorems rather have to do with the center right now let's look at the second one right so theorem two okay as we swiftly move along now what does theorem 2 State simply States again it has to do with the center that if we've got a line l in this case uh if we've got Center o all right it simply says the angle at the center will be equals to two times the angle at the circumference so supposing that we've got a center which is O right now let's give the name there let's suppose that this is a this is B and this is C and that is our Center which is O right so now what does does that mean it means that angle a o c is equal to 2 * angle a b c okay so now I want to State this very clearly supposing that let's say this angle here is 40° I'm just making an example right so what does that mean it means the angle at the center will be two times that number so which means this angle will be at 80° now please two things that you must uh make sure that o is definitely the center and that b must be at the circumference if it is not at the circumference therefore you cannot use this theorem right I just want to show you other ways in which we can do this right now when you give the reason we always say angle at Center equals to two * the angle at Circle okay yeah all right so that is the reason now note other ways in which they can show you this theorem right is that uh we can actually do this and very deliberately so they can give you the line let's say this is again and this is O all right now they can draw it in this way right now it doesn't look very obvious but note once again you've got the line at center right or angle rather at Center which is this guy over here in red and the angle at the circumference so it reaches the circumference over there so now we know that if we said this is B and C then again uh AB C or rather a o is equal to 2 * angle ABC all right let me write that nicely so 2 * angle ABC right there's another way again so you can actually get this as an obtuse angle uh at times right so there's our Center o so if we draw that as okay firstly you would have a reflex angle over there so supposing this is our Center o right so if I give you that angle over there right and we say uh let's suppose this is O Center o1 and again we've got a we've got B and we've got C right once again it which means that angle a o now note I'm talking about the one that is on the outside right or we can call it 01 so which means o1 angle o1 is equal to 2 * angle okay let's say angle AC CB all right so please note angle at Center is equal to 2 * the angle at circumference so if this was 120 that means angle 01 would therefore be equals to 240 which is twice that all right now let's move on to the next one which is uh theorem number three right so how do we State theorem number three as I said to you the first three theorems have to do with the center right now theorem number three kind of looks like theorem number two so this simply says right if a line or rather if an angle is subtended by a diameter that angle is equal to 90° okay so I know my lines are actually not as accurate as they should be right so if we say this is AB b c right notice if I say to you that AC is a diameter right now remember what is a diameter it is a line that cuts the circle into two equal parts right or you can say that it is a chord that uh divides the Ang the the circle into two equal segments right now I want you to please note if AC is a diameter therefore it means that angle ABC right so angle a b c is equal to 90° so meaning this angle over here would be equal to 90° right and how do we State the theorem well we shorten it this way we simply say angle at semi Circle right so note that when we say semicircle remember we did say that uh this would be a diameter which cuts the circle into half right so in this case we simply say this is the angle at semi circle right what I want you to also appreciate is the fact that it actually kind of looks like the previous theorem isn't it because think about it what would be the size of that angle there right it would be 180° so so angle at Center twice angle at circumference 90 * 2 gives us 180 right so remember this is angle at semicircle now we're done with the first three circles uh I mean uh theorems right now let's go to theorem four so the next batch of theorems have to do width now note in this case uh let me just remove that line right so the next patch of theorems have to do with the cyclic quad right so theorem 4 I want you to please note how do you state theorem 4 right so if we've got a circle now what is a cyclic quad it is a four-sided figure whose vertices touch the circumference of a circle right so let's say we've got a four-sided figure there that's a b c and d right now the first well theorem number four simply says that the lines or rather the angles that are opposite a cyclic quad are supplementary right so the opposite angles of a cylic quad are supplementary meaning if if I take angle a plus angle C right so a + C would be equal to 180° right note again so it means D plus angle B would also be equal to 180 and how do we State the reason there we simply said say opposite angles of a cyclic quad you don't need to tell us what that means right uh but we know as soon as you say opposite angles of a cyclic quad then we know that they are supplementary which means that they form 180° right now the second one that has to do with a cyclic quad right so that's theorem number five right now what does theorem five say right again it has to do with a cyclic quad okay so which means in this case all the four vertices Right Touch the circumference of the circle now note the moment that they tell you suppose that I gave you this line and I say uh this is AB and I say this is c d e usually when they say something like CD is produced till e what are they trying to tell you that forms a straight line right so what we say there is that the exterior angle of a cyclic quad notice it is the angle outside the uh cyclic quad is equal to the opposite interior angle right so this angle here which is on the outside of the cyclic quad right would be equal to the opposite interior angle right now please I want you to note ladies and gents that once we use uh this theorem right so note that uh how do we State this we just simply say exterior angle of a cyclic quad okay right all right I couldn't fit that all in there right so the exterior angle of a cyclic quad so we know in this case that that is definitely uh equal right all right so which means if we look at angle bde bde would be equal to angle c a c a b right and that is um the angles that have to do with the cyclic quad now I want you to please note as we move on to theorem six right we're almost done so anytime they want you to prove that an angle or rather something is a cyclic quad four-sided figure as a cyclic quad which means you must use any of these theorems right so which means if the opposite angles are supplementary therefore it's a cyclic quad or if the exterior angle is equal to the opposite interior angle then it means that it is a cyclic quad and then theorem six right now theorem six has to do with angle subtended by the same chord right so if we note in this case let's suppose that we've got a b c and d once again right now note we say these are subtended by the same chord or they are subtended by the same AR I'm going to show you which one is the chord here so imagine there's an imaginary line that is there that I've drawn right or another imaginary line there so you can either use that imaginary line or you can say well here is my we call this an ark right so an ark is a part of the circle right so we've got another Arc there at the top okay going to draw it in a blue color so there is another Ark over there now notice we say that the angles that are subtended by the same AR are equal right so meaning angle a would be equal to angle B right uh let's write it nicely so they say in this case case we've got d a c angle da a will be equal to angle d b c so that angle there is equal to that angle or you can just simply say angle a is equal to angle B now note what do you say right the reason that we uh we we give we give is that we always say angle at same segment okay we would definitely understand what you mean when you say these are angle at same segment right now note once again it has to do with the cyclic quad so meaning if I can prove that angles at same segment are supplementary then it means it is a I mean not supplementary rather they are equal then it means it is a cyclic quad right but once again look at this we have another pair of angles at same segment right so this guy here should be equal to that guy over there right so those blue angles over there so which means angle D is equal to angle C okay or you can say ADB right so that's ADB is equal to angle uh B CA so b c a so remember BCA would be that angle over there right once again why because angles at same segment are supplementary now please be very careful ladies and gents that it must be touching the vertices of uh the circle right uh if it doesn't then they might say to you you must prove that it is a cyclic quad and if you can prove right we're going to do lots and lots of questions uh on these uh proofs right so in this case it means that if you find that it is uh the case then you can conclude that it is a cyclic quad right now ladies and gents uh some may not be very popular in fact let me just continue using the white okay so what we can also do on this one is that we can also say that uh angles that are subtend by equal chords are also equal right so if I give you suppose I say to you okay I've got a cord there all right okay so if I tell you that let's say this is going to be a b let's say C let's say this is d E and F all right I want you to note if it is given to me that BD uh rather BC is equal to De it is given so given that BC is equal to De right please I want you to note so I've got equal cords there so therefore I want you to note that the angle that is subtended by those chords would therefore be equal so meaning angle a so therefore it means angle a would be equal to angle F or you can say b a c right so that's b a c is equal to angle DF e so that's DF e like that all right and by the way how do we State this we say these are angles okay that are subtended by equal Cordes right so I want you to please note that Now ladies and gents let's go on to uh the very next one right another way in which they can use this okay uh not very popular as well but um it can be uh used so supposing again that I've got a similar situation right so let's say okay my lines are not behaving all right let's remove that line okay so uh suppose that I say to you there it is there right so I want you to note ladies and gents if I were to give you that scenario now note uh suppose this is given to me as Center o again so there's my Center let's say this is going to to be a b and let's say this is d e right now note if it is given to me that AB is equal to De now please guys I want you to note again that it means that this angle here would be equal to that angle over there only on condition that that is the center right now note those are not vertically opposite angles right so uh we can say right given again given that a b is equal to d e therefore we can say that a o b would be equal to E o d right and why is that once again these are angles that are subtended by equal chords right so uh by the way ladies and gents this is not a very popular one but it does appear now and then okay right and now we go to the very last three right so let's go to theorem uh so we are done with the ones that have to do with the um cyclic quad now we look at the ones that have to do with tangents the next three theorems have to do with tangents right now the first one let's talk about the very first one right and it simply says if I've got a tangent that is drawn right a line L that is drawn from the center will be perpendicular to the tangent at the point of tangency right so meaning I'll have a 90° there right uh in this case if we've got let's say o let's say that is our line a b c right so if I know that OB is definitely the radius and in this case uh AC is the tangent to the circle then I can conclude that therefore OB is perpendicular to AC now we simply call that the tan radius theorem tangent perpendicular to the radius just simply put like that okay right now remember what's a tangent it's is a line that touches the circle only once okay right now let's go on to the next one that has to do with the tangent okay so right so if we get again we can take line that is a tangent so this one simply says that if I have tangents that are drawn from the same point outside the circle those tangents would be equal now please guys I want you to note supposing that I know this is going to be a let's say a let's call this point P let's call this B now from the point of tangency till Point P which is a common point from the point of tangency again which is a till Point P which is a common point then it means that a p is equal to BP these are called tangents from the same point okay right so these are tangents from the same point right so there we go now please I want you to note guys the moment you start having equal um you know uh equal sides if I had to do this please I want you to note the moment I start having equal sides if I were to do this then what type of triangle would a p b b right that would be an isoceles triangle and why do I I know about an isoceles triangle right the base angles of an isoceles triangle are equal but we're going to talk about all of that right once we start you know delving into a uan for now I just want you guys to master these theorems and know them off by heart all right now let's go to the next one right tangents from the same point we know that they are equal and the last one the very very very last one so in this case if I have got a tangent okay right I don't need to have a a center there so let's draw our tangent okay so there's the tangent so we know that a tangent or the angle between the tangent and the Cod will be uh equal to the angle that is OPP opposite that chord right so we call this the tan chord theorem now suppose this is our tangent a b c right and let's say this is d and e right I want you to note guys it means that if I were to take this angle over here right so there's my tangent there is my Cod the angle that is opposite that cord which is that guy over there would be equal to that angle there right so we call this the tan cord theorem so therefore it means that I've got uh angle DBA so angle DB would be equal to angle uh DB and we say that this is the 10 cord theorem and I can tell you this ladies and gents there is absolutely no way that there will never ask the 10 C theorem it's one of the favorites with examiners right so meaning in the same token that this angle over here right there's our tangent that's our cord so which means it would be equal to the angle on the opposite side of that cord so there it is once again so I can say that e no e DB is equal to uh EB C right and that we call the tan cord theorem and please note ladies and gents uh once again if they want us to prove that something is a tangent which means we're going to use the converse of any of these three uh three theorems right uh to prove it okay right so that is how the cookie crumbles ladies and gents um now what we're going to do next is I want to show you how to apply these what's the first thing that you do do when you uh encounter any questions on uclean Geometry all right but we will do that on our next video otherwise for now please see our next video and do enjoy I'll see you next time shop shop