okay everybody we're going to be starting off today working on part one of theorems with circles again we're we've talked a lot about circles already this year talking about the uh lengths of arcs as well as the area of pieces of circles as well as of course reviewing things like circumference and area um we're now going to be taking it to a new level where we look about look at specifically things that go along with circles as well as some formulas that go that come out of that um it is going to be important that you have a general understanding of these however usually on tests I'll let you guys make sort of a theorem checklist that you can look at um because I don't believe in memorizing everything but I want you to understand the main goal here is to understand what we're doing okay so here's the first page of notes you're going to do um some of the activities that normally we would do um will be done in class so because of that I will be skipping over some things a little quicker however I'll still be explaining the actual theorems themselves but some of the activities we're going to do to understand the theorems those would be things that were already done on your notes but this is the page we're talking about now okay so starting out you already have these things labeled on your notes but we're going to look at one piece at a time what my goal for you guys to do is to Define these in your own words so first we have a diameter okay as you know in from class time that would be defined most likely as a line that cuts through the center of the circle cutting it into two equal halves we have a radius which Cuts halfway across a circle directly to the center that's important we have a chord which connects any two points in the circle a diameter is actually a special kind of cord that also goes through the center we have a secant line which is a line that's basically a chord that continues if you look at it that way but it's a line that crosses two points on the circle and we have a line that crosses only one point on the circle which is called the tangent in addition to that it's not going to show on your notes so you may want to kind of shade it a little bit if you don't already have it but an arc is a piece of a circle so this right here is considered a minor Arc because it's less than 180° a major arc is more than 180° so as you can see it goes just past the diameter so that's called a major arc so if you haven't already done it make sure in your notes that you update all of those things okay the next thing we're going to do this is an activity we did in class so if you're doing the notes for the first time you're going to want to do this on your own and pause the video but um we're going to take any point on the circle and draw the tangent line through that point so you would take a ruler and you would draw any random tangent line I put one right here but the whole point is to try a bunch of different places okay next what we're going to do is we're going to connect this point to the center of the circle and what I want you guys to do is go ahead and see if you notice anything so on your example pause it take a look for a second see if you think you know anything special about that angle what you should have noticed is that actually makes a 90° angle so what the theorem says is that any line is tangent to a circle if and only if the line is perpendicular from the center of the circle to that point this is a book definition but one of the things I wanted to highlight with you guys is the if if the if and only if part kind of give you an idea what that means in geometry and sort of in proof in general if and only if means it works both ways it means if we have a tangent line okay and it connects secondly to the center a line connected to the center then it must be 90° secondly what it's saying is if I have a radius and it makes a 90° angle so first I have the radius and secondly it makes a 90° angle with a line that line must be a tangent line okay now we're going to be moving on to the second page of our notes and again you should have copied down that that theorem so if you didn't already have it in your notes you're going to want to go back and take a look at it we're going to now do a couple practice problems so looking at the first one here the reason that we mainly use it because obviously when you look at something you say oh that's 90 that's you know it's interesting to know that but it doesn't really do much for us in terms of problems what we need to be able to do is we need to be able to go ahead and set that up into an equation and into a problem so usually what 90° tells us is right triangles we know soaa and we also know how to do Pythagorean theorem from that in this case I say hey wait a second because it's a tangent segment because it's a radius therefore it's 90° now I can apply Pythagorean theorem Pythagorean theorem is R2 + 16 = 202 because remember in Pythagorean theorem it's leg squar leg squar hypotenuse squar so since R squ then is equal to 144 remember I got that by taking 200 which or sorry 400 which is 20 * 20 and subtracting uh 16 * 16 which is 256 which then gives me 144 I then take the square root which gives me 12 looking at our second example again you can see we're going to have to notice hey wait a second if that's a tangent line therefore it's a right triangle when it's a right triangle I can go from there that X does not represent the whole part the X represents only the outside so from here we're going to go 62 + 8^2 = x + 62 what that requires you to notice and this is one that people get stuck on is that if that is also the radius right there then that's also six this is a problem the kind of thing that you'll see on a test and that a lot of people get confused on they don't realize that any line any straight line connecting from the center would also have to be six that's a fact that's important for a lot of examples we'll come up with there's only so many pieces of information you know okay now we're just going to go ahead and simplify it 6 s is 36 8 squ is 64 that makes 100 a lot of times people would think oh I have to go ahead and do Pythagorean theorem and or sorry I have to do quadratic formula multiply stuff out do all these things you don't have to do that if we went ahead and multiplied x + 6 * X+ 6 from algebra as you remember you would get x^ s which means you would get two answers however we're not going to get two answers on a problem like this simply because on a problem like this we don't have negative values you know we're not going to end up with a negative value of x okay so from there we're going to take the square root of both sides which is going to give me 10 square root of 100 is 10 and when you take the square root of something squared the square symbol just goes away okay from there I'm going to subtract six from both sides and I get X = 4 on both of these problems I know I should have done this earlier I'm being mean but they were also both 34 five right triangles so if you remember back to what we did earlier in the year you would have noticed that 6 is 2 * 3 8 is 2 * 4 so that would mean the hypotenuse would have to be 2 * 5 which is 10 you can see where that comes into play in the problem 10 = x + 6 so then that would have to be four left over and the top one of course was a 345 4 * 3 is 12 4 * 4 is 16 and 4 * 5 is 20 okay the next thing we're going to look at is we're going to pick a point anywhere on the outside of the circle and we're going to draw two tangent lines from there there should only be two tangent lines that occur based on whatever point you pick I want you to pick your own Point don't pick mine and we're going to try to discover another theorem here you're going to need a ruler for this as well after you do that what I want you to do is measure the two segments okay so what you should have noticed hopefully is that the segments are equal to to the same length and that is true of any point you picked so hopefully you picked a point different than mine if you want to test it again just pick another point on the other side of it uh closer than the circle or whatever you're going to notice that that's always true now it's hard to draw a perfect tangent line but that is in fact the rule if you draw the tangent line that hits exactly one point so here we go the theorem States tangent segments from a common Point outside the circle are always congruent so let's kind of make some sense of that we're talking about tangent segments okay so by that we mean the two purple segments that connect at one point each right there and right there from a common point so that common point would be right there Point P outside the circle so the point can't be in the center the point has to be outside the circle obviously if we take a point inside the circle those are not going to necessarily be equal to each other it's kind of obvious but outside the circle and including t tent points it's always the same okay it's a very simple idea but it does lead to some practice problems so let's take a look all right so here's a practice problem for you you may have to fill in the 3x + 10 and 7 x - 1 on your notes if not you're going to go ahead and set it up so what the theorem says is because they are both tangent segments and because they're from a common Point therefore we just set them equal to each other okay so all you're doing is you're solving a equation with variables on both sides we'd subtract 3x from both sides add six and divide by four and we get xal 4 if I then ask you to to figure out exactly what the segment was you would just take that four and plug it into one of them 3 * 4 + 10 and you would figure out that that's going to be each one is going to be 202 okay now we're moving on to the next page of our notes as you can see this is page three so make sure you have that page available okay so we're going to start with the central angle in Define it so definition of a central angle is an angle made from the center and two points on the circumference anytime we have an angle we have three points to create it so what it's saying is one of the points the vertex is always going to be in the center and the other two points will be on the angle on the circle itself and that's called a central angle it's easy way to remember it comes from the center okay an inscribed angle is the next one we're going to talk about an inscribed angle all three points are on the circumference of the circle so if we pick any three points in the circle and we connect them that creates an inscribed angle these definitions will be important as we go forward very important so make sure you know them okay when we talk about the measure of an arc we know about angles angles are measured in degrees well we can also measure an arc in degrees so for example if I take a look at AC right there and I say hey that piece right there that that piece that I just highlighted in red what's the measure of that Arc well in order to do that we need to know the measure of the central angle so if I take a look from the center and I say hey that's 60° that means then that the ark is 60° as well so first of all here's my question when I say AC how do I know that I mean that Arc and not the giant Arc that goes around the other side of the circle so obviously there has to be some way to do it the way we do it is we go ahead and throw a third letter on the circle now if I want to talk about the large Arc which has to be 300 because the full circle is 360 then I would use something such as ABC or I could also use CBA a CB would be different that would be including the red one plus a little bit of extra so you want to make sure that you're always using the right terminology but basically what we do is when we talk about Minor arcs we use two letters when we talk about major arcs we use three letters you may even see three letters for a minor Arc but in general that's the method and that right there in purple is the major arc ABC okay so the next thing we're going to do is we're going to compare inscribed angles and central angles so what we're going to do is create a central angle we're going to say which is going to be bigger if I take that same angle and I move it back to here that's the same angle copied over and then I just simply extend it out until it hits the circle as you can see that inscribed angle makes a bigger Arc the question is how much bigger is it you know twice as big is it three times as big is it half again as big one and a half times so we're going to need to go through a process and kind of take a look at it so what we're going to do is the first thing we're going to do is we're going to go ahead and we're going to copy down onto your paper and inscribed angle so you're going to plot any y r and s on there like I did and connect them onto your notes any three points will be fine okay from that you're then going to connect the center out so it looks like my shape could be a wider triangle a narrower triangle now what I want you to do is I want you to trace the angle I only want you to trace the angle right here okay I want you to notice that angle right there onto your tra onto some tracing paper if you don't have tracing paper at home simple you know computer paper works or even lined paper so you're going to trace that one and then you're going to copy it down and compare it down to the angle below so take a second and Trace that and then see how much of the big angle the central angle it fills up what you should see is that perfectly it cuts it in half so as you can see two of those inscribed angles is perfect for the central angle so really what this theorem this is our way of kind of investigating the theorem the theorem says that an inscribed angle is always half as big as a central angle or another way of looking at it the central angle is twice as big as the inscribed so we're going to write our formula now and it's just exactly what we said if we take a central angle and an inscribed angle so that if that angle's beta then the central angle would be twice as big it doesn't matter how you draw it the central angle would be twice as big okay so that means that any inscribed angle as well works so for example if I go ahead and I draw one that connects here here that is also half as big okay any inscribed angle and we're going to look at a theorem about that in a second as well okay so as it says half of the large angle X CZ is equal to XYZ or as we said if you take double the small one you'll get the big one so let's take a look at some actual practice so we're going to go ahead now to page four of our notes and you may need to fill in a couple pieces on there so what we're trying to do here is we're trying to find the the value of x what we know is that the central angle is X on there and the in scribed angle is 18 so even though it's not one right on top of the other one we will be doubling the inscribed angle to get the central angle looking at our second example go ahead and pause it if you need to all right so this time we have the central angle remember the central angle is twice as big as the smaller angle so what that means is we're going to end up with 142 divided by two which is 71 what I would hope you guys would do here even though we'd say not to look just at the angle as a reference obviously that angle X on the bottom one is smaller than 142 the only mistake I would see you making would be to accidentally double it so instead of cutting it in half you might say oh it's 284 I want you to think if that makes sense does that angle look like a 284 degree angle use your common sense and make it work for you instead of against you okay the next thing we're going to do is we're going to go ahead and look at what happens if we have two inscribed angles that take the same points so if I connect basically what I want you to do is I want you to pick any points x and z on the bottom half of the circle you could literally pick them anywhere but if you end up making a diameter or something it's just going to confuse you so pick any two points and try to keep them both in the bottom half of the circle but pick different ones than I did now pick any point on Top y you're going to connect those okay now you're going to pick another point w anywhere on top as well and you're going to connect that as well what I I want you to do is I want you to go ahead and Trace one of them Trace X YZ the one that I have in yellow trace the first one but it doesn't really matter you're going to trace either angle and you're going to compare it to the other one just take a second to do that hopefully what you noticed here is that simply put this is another theorem any inscribed angles that inscribe the same Arc are equal meaning looking at the two points that make our Arc okay we have our Arc right here okay our Arc any angle that I draw that inscribes those will be equal meaning I can go anywhere on the circle that I would like anywhere all of these angles should be as long as I did my lines correctly and I did them very quickly but all those angles are equal and you can kind of look and see that it's very interesting especially as you get over on the side here and you're thinking man how could that possibly be it's so much wider well no it's not really okay anywhere you go even though it seems like you're going to make one of them really really wide well the other one has to be very narrow to make it work so as you can see from this picture they're always the same and here's why just a really quick proof well they both have the same central angle so that means that XYZ is doubled to equal xcz the same thing is true for the other one double xwz must also be xcz well you set them equal to each other divide by two and you've got that they're equal to each other so it's it's a very easy proof and it seems like we can prove it sort of just by looking at it as well but it might not be something you would have expected okay so we're going to move to the next page of our notes now we're going to talk about a couple more theorems okay the first one is called the inscribed angle on a diameter theorem so what I want you guys to do is on your paper you have the line that's drawn horizontally I want you guys to go ahead um and pick and your paper should look like this I want you to pick any point anywhere in the circle that's not already X or Z it needs to be a new point but any point and call that point Y here's where I put mine but call any point you want call it Y and then connect the triangle we know that it's an inscribed angle but I want you to go ahead and um take a look at that angle and see what you notice okay what you should have noticed is that it's a 90° angle okay as you know the inscribed angle is always half the the uh Arc so if you look at a 180° Arc which is half a circle well half of 180° is 90° it makes sense but it's kind of interesting how you can put a point absolutely anywhere in the circle and if you connect it to a diameter in a triangle it makes a right triangle so thinking about some of the examples you could use now you've got right triangles you could use soaa Pythagorean theorem all those things will come into play Okay so let's take a look at the theorem the theorem says a triangle inscribed in a circle is a right triangle if and only if one side of the triangle is the diameter okay what that means is if you take a diameter and you inscribe it you get a right triangle conversely if you have a right triangle and all three points are on the circle the hypotenuse must be the diameter so it's kind of a you can go either way with it as we've said before with if and only if proofs okay so the next theorem we're going to look at is about quadrilaterals inscribed in a circle remember inscribed means all the points are on the circle so we're going to pick four points QR s and t the only thing I want to make sure you do is I want to make sure you go in order so it would should go like if you're looking at the mouse Q then R then s then T so in a circle it goes QR s t uh clockwise these are the four I picked just pick any four random ones but it is important that if you look at it and we connect it like we're going to do that t and Q are cross each other and SNR only for what we're going to look at in a second so pause it pick any four random ones please pick different ones and me pick any four random ones and we'll go from there you're also going to need a protractor so if you don't have your protractor with you when you pause it go grab your protractor all right so what I want you to do specifically is take your protractor and measure these angles first look at Q and T and see how they compare to each other then look at S and R and see how they compare to each other we're trying to find a rule and a pattern that can help us in these problems later on so take a few minutes to do that pause it while you do so and then hit play when you're done all right so what you should have noticed is that the angles across from each other add up to 180 so what the theorem says is if a quadrilateral is inscribed in a circle then opposite angles are supplementary okay meaning Q + T since these are opposite from each other shoot a line straight out you hit T etc those are add up to 180 as well as SNR to 180 notice this proof is not an if and only if proof because just having two angles that add up to 180 does not guarantee you have a quadrilateral so it's an example of a non if and one if and only if proof okay so that's the end of the first part of our Circle theorems lessons um hopefully you guys can use this to review the things we've done in class as well as make sure you understand these theorems e