# Geo 11.4 Chords, Secants and Tangents

## Geometry

It is some scallion bomb holder. We're looking at a section 11.4 chord seeking. Tangents, the last section for circles. I'm going to start you off with a little contest, trivia, shall we say? Check this out with this. How about if we win, they have to get rid of the rooster. Oh, that's interesting. If you win, give up the. But if we win, we get your apartment. Oh. Deal. Well, that's interesting. We're going to see how that works out for Joey and Chandler later. We're going to start by learning a theorem about two chords on the inside of a circle. All right, so the theorem says that the measure of an angle formed by two chords, the intersect on the inside of the circle. All right, so it's basically you draw an X on the inside of a circle. There's one. There's the other. The measure of that angle is equal to half the sum of the measures of the intercept at arcs. So what does that mean? That means that, I'm going to call this angle X in the circle on the left. X would equal one half the measures of this arc here and this are here because those are the two arcs that are intercepted, so in this case it is 70 plus 50. So X is going to equal half of, we'll say one 20, call it 60 degrees, and we're done with that one. All right, can we look at the next example? What do we have? Here's the angle, one 25, so you have to look at where the arcs are intercepted. So what do we got? We have this arc here. And we have this arc down here. So remember, the angle formed at the chords is half the measure. All right, so I'm going to write it out that way. Let's do the angle, which is one 25. One 25 is half the measure of the sum. So you have to add those two together. So we're going to get 8 X plus one. That's the first angle. Second one is one 85, so we're going to add them all together. Let's combine some like terms. Let's get 8 X plus one 86. Can we distribute everybody loves a distribute? All right, we're going to get four X plus 93. If I subtract 93 from each side, what do we get 32? We're going to get 32 equals four X and X equals 8. All right, so hopefully the algebra is not a problem for you. Because we're going through this pretty quickly. Make sure you pause the video if you need to so you can get this all written down. Let's try two by yourself, pause the video and we'll see how you do. Go. All right, hopefully we're honestly working through these problems all by yourself. And the first one, let's set it up. So one 15 equals one half of 100 plus X or X plus a hundred. I then distribute. You get 50 here, subtract 50 from each side. How do you get rid of a half X times two each side, you get one 30 for that one. So that one's pretty easy if you know your algebra. Second one, ten X minus two is equal to half of one 56 plus 60. That simplifies to two 16. You can multiply that by half and get one O 8, add 12 to both sides, one 20, divide by ten to get X equals 12. So this is one of the more easy theorems that you're going to come across. If the two chords intersect on the inside of the circle, it's equal to half the sum of the intercepted arcs. Easy enough. So now we need to define what a secant line is. Now before we've seen tangent lines, remember tangent, that was section one. Here's a tangent line. It's a line that's on the exterior of a circle that only touches in one place. A secant line touches in two places. So the definition is a secant line is a line that intersects the circle in two points. It goes right through the circle. That's a secant line. This is a secant line. If it's a line, that's a secant line. That's a secant line. So those are all secant lines. The next theorem deals with secant lines or tangents, let's take a read at it. The measure of an angle formed by two secants to tangents or a secant and tangent drawn from an external point. So at some point outside the circle, is equal to half the difference. All right, so in the last theorem you had to add the two arcs. This is half the difference of the measures of the intercepted arcs. So let's see how that works. All right, so what do we have here? In the first example, we have two secants. They go through the circle, and they both intersect twice. So remember the measure of this angle out here at this external point, I'm going to call it X I don't like question mark. So X is going to equal one half the difference. You need to subtract and you have to subtract the smaller angle from the larger angle. So in this case, you can't end up with negatives. If you end up with negatives, you've done messed up. So X is going to equal one half, what do we get 90 for this? What's half a 90? Let's call it 45 and be done with this problem. Everybody get it? Let's do the next one. I don't know what this is out here. I'm going to put a little X here we have a tangent and a secant line. All right? It's the same theorem. It's hard to screw this up. So 58, the angle on the outside, 58 is going to equal one half the difference. So which one of these is larger? The X is larger. You can tell by looking at it. So it's going to be one half X minus 89. You always want to subtract the smaller arc. All right? So now what can we do? Here's a little trick. I'm going to multiply both sides by two, and the reason why I don't want to deal with that half. I just hate hat. So this is going to give me one 16. And that half and that too will cancel. So I'll just have equals X -89. Now if I add 89 to each side, we can do that plus 89. K two 16, two O 6, two O 5 equals X see what it there in my head. See what I did. So the measure of this angle here or this arc, sorry, is 205°. Easy enough, two tangents now. Look, they don't even give you anything. How are you supposed to figure that out? 50°, they don't even give you anything, but this little arc right here, let's change that color. So what do we have? We're going to call that X well if that's X, what's this arc right here? The long way around. Well, the two arcs make up the whole circle. So if they make up the whole circle, the two arcs add to 360°. So if I call this arc angle X, then I can call this arc out here 360 minus X I don't know what it is, but it's 360° minus X so let's use those two measures. Look, we have an X and we have 360 minus X and let's write an equation and see if we can solve it. All right, so here we go. 50 is going to equal one half the difference. So let's make sure we do this properly. The difference would be 360 minus X and then you subtract X minus X again. Everybody see what I did there. This part is here. And this smaller arc is there. All right, that's the difference of the two arcs. I've just subtracting them. So what do we get? 50 equals one half. I'm just going to combine like terms first. Negative X and a negative X is negative two X so we have minus two X I'll distribute the half, fine. I'll distribute the half. I like to multiply both sides by two, but here, you know, whatever. So I distribute the half I get 50 equals one 80 minus X if I subtract one 80 from each side, see how this one's a little more algebra. What are we going to get there? I'm going to get negative one 30 equals negative X we divide by negative one, and we're going to get one 30. So X is going to equal one 30. What is the other angle here? That would have to be what? You subtract one 30 from it. So this, the answer to the question is 130°. But the other arc, if they do ask us, is 360 minus one 30, which would be 230°, which is larger than one 30, so everything works out. That works awesome. Why don't you try the next three all by yourself. Pause the video, work them out, be honest. All right, so I didn't do the third one yet because I want to emphasize something that I think some of you might have a trouble with. So we'll do that one together. But for the first one, you have to find this arc first. All right, so 360 minus two 50 is one ten. You've put in the equation. The angle equals half the difference. You work it all out. You get X equals 6. That one's easy enough. Next one. What do we have here? 18 -6 minus one 14. So that's what I put. That is the difference right here. That's what the angle is equal to half the difference of the two arcs so when you work it all out, you get X equals 11. All right, now the third one, what I'm going to do is I'm going to clear the board. So pause the video if you need to copy these down a little bit. Hopefully you worked it out yourself. Let's clear the board. So 7 X plus 7 is going to equal half the difference. All right, so the larger angle is one 39 minus the smaller angle, but you can't just write -16 X plus 5 like this. Because that's only going to subtract the 16 X. It's not going to subtract the whole angle. So you need to include parentheses. So this is the tricky part. I wanted to emphasize. If you're doing one 39 minus this whole angle, make sure you subtract the entire angle and you're going to need parentheses there. So this is kind of ugly. So let's simplify it. I'm going to, let's see. Let's just leave everything and distribute that negative. That negative's got to go to both. So it's -16 X and -5. That's going to make a big difference here. So 7 X plus 7 is going to equal one half. What do we get one 39 -5? That's going to be one 34. -16 X I'm going to distribute that half now. I like that because they're both even so that'll work out nicely for us. So what do we get here? 65, 6, 7. So we get 67 and then -8 X so how do we solve this? This is a very, very simple algebra one problem. Let's add 8 X to both sides, put that right there. Draw a line 15 X plus 7 equals 67 if I subtract 7 from both sides, 15 X will equal 60 X equal four. So we're done with that one. All right, so I just wanted to emphasize where that last one, then when you subtract, you have to subtract the entire angle, so put that in parentheses. I think we need some more trivia here, help us out through this long lesson. Here we go. What was Monica's nickname when she was a field hockey goalie? Big fat goalie. Correct. Rachel claims this is her favorite movie. Dangerous liaisons. Correct. Her actual favorite movie is. Weekend at Bernie. Correct. Monica categorizes her towels. How many categories are there? Okay. Everyday use, fancy, guest, fancy guest. Two seconds. 11 unbelievable. 11 is correct. Pulls a number out of nowhere. Sometimes I think that's how bruss got through college. All right, so the first theorem, if two sequence are drawn from an external point to a circle, the products of the lengths of one secant segment and its external segment equals the products of the lengths of the other secant segment in its external segment. Say what? All right, so what does that mean? That means that if you take the entire secant times the external segment to the part on the outside, it's going to equal any other secant times its external. As long as they're coming from the same point. So the way I can write this out, using variables, is we have a times the entire secant. A plus B is going to equal C times C plus D all right? So we're going to see what that looks like with numbers here in a few minutes. Let's look at the next theorem if a tangent and a second are drawn from an external point to a circle. It's basically the same theorem. One of them is a tangent now. It's not we have a tangent, not two sequence. We have one secant and a tangent. So what does that look like? The products of the lengths of one secant segment that set external segment. So here's the secant segment. And its external, so a times a plus B again is going to equal the square of the length of the tangent. So here that equals C squared. So let's see how that's applied to different problems. First problem, we have two secants, so I need to find, we have the external pieces, the 8 and the 7, and we need to find the length of the entire secant. So for the first one, it's going to be 8 times the entire secant, which is X plus 8, that's going to be equal to 7 times the entire secant, which would be 7 plus 9. So hopefully you can go through this. We get a to X, I'll distribute here plus 64 is going to equal, what do we get? Well, you can distribute if you want to. But why don't we just add those together? 7 times 16. So what is 7 times 16? 7 times 16 is, what do we get? 7 times 8 is 56. So one 12 would be 7 times 16. See what I did there? Subtract 64 from each side. Okay, if I subtract 64, I get 48. And then X here would equal 6. Easy enough, huh? Okay, second example here. Remember we have to do when we have a tangent and a secant line. So let's look at the secant first. It's the length of the external segment times the entire secant. All right, so that's going to give us 27. Times what do we have here? X plus 27. X plus 27 is going to equal the length of the tangent squared. So that's going to equal 36 squared. So now it's just a matter of doing a little math here. What do we got distributive property? 27 X plus 27 times 27. 7 29. Equals 1296 sorry I had to do that in my head here. Get a little confused. We get 27 X subtract 7 29 from each side. We get 5, 67, divide by 27, X is going to equal 21 for that one. Easy enough, okay, the next two are for you. Pause the video, do the next two all by yourself. Go. All right, let's focus in on these two right here. Hopefully you did okay. For the first one, we have a tangent and a secant. So remember, it's the external. Let's get out that highlighter here. The external part times the entire sequence. So it's 16 times X plus 16 equals the tangent squared. The other piece squared equals 20 squared. Work it all out. You should get X equals 9. And for the last one, you have two sequence. It's the external part times the entire secant. So that's ten times X plus ten equals 9 times external times the entire, which would be 20. Okay, you can just add those together, work out a little algebra. You get X equals 8 for that one. So hey, those are those examples. Let's do a little more Friends. Joey had an imaginary childhood friend. His name was Maurice. Correct. His profession was. Space cowboy. Correct. What is Chandler Bing's job? Oh, gosh. I'm starting to need this so you lose the game. It's something to do with trans funding. Oh, oh, he's a transplant. Trans monster. That's not even a word. Oh, no. Okay, last theorem here, theorem number 5. This is so easy that Sully and bruss can do it. And I'm serious. You got to be pretty easy. Two chords intersect inside a circle, the products of the lengths of the segments of one chord is equal to the products of the lengths of the other chords. So we have a times B that's the product, is going to be equal to C times D, it is that simple. You take the two pieces, you multiply it, you multiply the same, it has to be the same chord. But the different parts that are being intersected here, where you have this segment and that segment, those two segments being multiplied, the product of that will equal the product of the other two. So how easy is this to work out? Well, you're going to be given a circle like this. You just have to multiply. And I always make sure that I do it so that you have the coefficient, the variable. So 12 times X equals ten times 6. Now I'll write it out just so we know what's going on, but you're going to get 12 X equals. What do you get 60? X equals 5. I told you it was easy. And I was Miley serious. That is serious. Let's do the next one. Ten X equals, what do you get? 16 times 15. So 15 times 16 is going to give us a two 40. Divide by ten to 5 by ten. We get X equals 24. Seriously, that easy, Billy Ray Sirius. Last one, do it yourself. Ready, go. This was so easy. I don't even have to work it out. 8 X equals ten times 12, ten times 12 is one 20. I do one 20 divided by 8. I get 15. It is that simple. Hey, that's it for circles. There's a lot of stuff. You just got to practice it so you know when to use what theorem. So on and so forth. Let's end it with some of that Friends trivia. This is mister Kelly bomb holder remember. It's nice to be important. It's more important to be nice. You know what? You are mean boys who are just being mean. Hey, don't get mad at us. No one forced you to raise the stakes. That is not true. She did. She forced me. We'd still be living here if you hadn't gotten the question wrong. Well, it was a stupid, unfair question. Don't blame the questions. Would you all stop yelling in our apartment? You are ruining moving day for us.