Source: http://www.timssvideo.com/us4-secants-and-tangents
Timestamp: 2019-04-21 12:03:07+00:00

Document:
This eighth grade mathematics lesson focuses on the measurement of angles formed by secants and tangents intersecting with a circle. It is the fourth lesson in a six-lesson unit on this topic. The lesson is 45 minutes in duration. This is an advanced eighth grade geometry class. There are 15 students enrolled in the class.
00:00:00 We're doing nine point four today.
00:00:09 If you need to sharpen your pencil, you better do that right this second.
00:00:19 Probably not there, Matt. Oh, goodness.
00:00:35 I hope you guys have your activities ready from the boxes from nine point four.
00:00:40 That was the only other thing that I asked you to do.
00:00:44 Ben, you don't even have your books open, ready to go. What's going on?
00:00:52 I thought you were going to be the star of my show.
00:00:54 I'm definitely the star.
00:01:00 We're gonna go through the activities kind of quickly, 'cause you should have already had them done and ready to go.
00:01:19 Okay? Here's your problem of the day.
00:01:35 A little review from nine point three.
00:01:53 You may already have this in your notes.
00:01:58 You might have to look back, try it from memory or look in your notes.
00:02:17 I wanna know about the inscribed angle theorem, the right angle corollary, and the arc intercept corollary.
00:02:53 Page 582 if anybody's wondering.
00:04:39 Help me out with the inscribed angle theorem. Tell me what- what it is.
00:04:43 In your own words, how would you describe that one?
00:04:51 If an angle's inscribed in a circle and it intercepts part of the circle, then the angle's measure is equal to half of the other angle.
00:04:59 Okay. It ends up coming out to be half of the measure of the arc that it intercepts. Right?
00:05:05 How about right angle corollary?
00:05:09 If an inscribed angle intercepts a semicircle, then the angle is a right angle.
00:05:18 A right angle. Okay?
00:05:20 And arc intercept corollary, Margaret.
00:05:22 When you have two inscribed angles and they intercept the same arc (inaudible).
00:05:37 You can open to nine point four, page 588.
00:05:44 The first thing we're gonna look at is what kind of angles that we're gonna be talking about today.
00:05:51 They're angles formed by secants and tangents. Here's all different cases.
00:06:08 It all depends on where the vertex is... and the type of lines that it- that it includes.
00:06:19 If the vertex is actually on the circle, you have secant and a tangent right here, or two secants here.
00:06:31 The second group is where the vertex is inside the circle, but it's not on the center usually.
00:06:36 Okay. Talking about here.
00:06:38 Last- the last group, the vertex is outside the circle. You have two tangents, two secants over here or a secant and a tangent. Okay?
00:06:46 That's all gonna help you determine how to find the measure.
00:06:51 Are you ready for the activities?
00:06:53 Take that- out, whatever you brought from Friday for homework.
00:07:09 We're not gonna go over each one of these in deep detail, 'cause you should have already had most of them done.
00:07:14 We'll go over the first, probably, row or two, and then we're gonna talk about the general pattern at the end with X.
00:07:24 You have this circle with secant and tangent. The secant and tangent angle is a right angle here. Okay?
00:07:31 The secant contains the center of the circle. What do you think the measure of A V C, angle A V C is?
00:07:37 What'd you come up with? Michaela?
00:07:41 Why do you say 90 degrees?
00:07:43 Because it says that secant and tangent angle is a right angle and it (inaudible).
00:07:51 Okay. So- and the measure of a semicircle here, arc A V is what?
00:07:55 A hundred and eighty degrees.
00:07:57 A hundred and eighty degrees.
00:07:58 So what can you get from that? If the whole arc is 180 and the angle is 90, what does that kind of remind you of?
00:08:08 The right angle corollary.
00:08:09 Okay. Yes. Kind of like the right angle one or where- basically because it's half. Right?
00:08:26 the inscribed angle and the intercepted arc is the fact that it's half of what that arc is.
00:08:33 That was the easy part.
00:08:36 Okay. Look at number two. The secant tangent angle- we're gonna go over- this was the- if it's the tangent and it's a right angle.
00:08:43 Next we're talking about the acute angles. What happens if the angle's acute? It's not a straight.
00:08:48 It's not coming straight through the center, so it's gonna be a little more tricky for you.
00:08:54 Look at arc A V.
00:08:57 It tells you that it's 120 degrees and then it gives you 120 degrees for angle one, here.
00:09:05 How do you know that? How can- how do they justify that? Leah?
00:09:10 Because it's the central angle, and the central, and the arc that they intercept equals the measure.
00:09:15 Good. That's back to the beginning of the chapter where the central angle is gonna be the same as whatever that arc that it intercepts.
00:09:23 Angle two. Thirty degrees. Where do they come up with that one?
00:09:29 Triangle A P V is isosceles, so angle three and angle two have to be congruent.
00:09:43 You subtract 120 and you divide them into 65.
00:09:46 Very good. You get 30 degrees.
00:09:50 So where do we come up with this one? P V C. P V C.
00:10:00 That's the one they left blank in this first- first box.
00:10:10 You're on a roll, let's go.
00:10:12 Yeah, P V C has to be a right angle because it intercepts V C.
00:10:24 Back in the beginning, nine two or nine three. It had to be nine two because we just went over nine three.
00:10:31 Nine two, that- the radius that intercepts that has that same tangent point... that's gonna be 90 degrees.
00:10:41 That wasn't the hard part.
00:10:43 Measure of angle A V C. How would you get that one?
00:10:49 A V C. They say 60 degrees.
00:10:55 If A V C is 90 degrees and angle two is 30 degrees (inaudible).
00:11:02 Okay. And you get 60.
00:11:03 This kind of- this being 60 degrees, how does that relate to back here, the arc?
00:11:10 Matt says it's half. Mike says it's half.
00:11:21 One half the measure on the intercepted arc.
00:11:25 So we got two cases down. We got- if it's a 90 degrees angle with a secant and tangent, we got acute angles.
00:11:35 What do you think this... general statement... here, measure of arc A V- if it's X, what's angle one gonna be?
00:11:50 Okay. What about the measure of angle two? Michaela, describe that for us.
00:11:59 You first have to do 180 minus X (inaudible) and then you divide it by two.
00:12:05 Divide it by two.
00:12:06 Then they'll be the same as three.
00:12:07 These angles are gonna be the same.
00:12:17 Ninety degrees. Okay. That's not gonna- that's not gonna change.
00:12:22 And so the last here- this angle is going to be what?
00:12:25 Half of the arc.
00:12:34 So we know it works for that same intersected arc- intercepted arc theorem, works for 90 degrees, works for acute angles.
00:12:45 Last case is obtuse angles.
00:12:49 This one is a little bit more tricky. You might have a little bit more trouble doing this one.
00:13:02 And they come up with this measure of angle one being 160.
00:13:05 How do you think they got that?
00:13:15 Full circle has to equal 360 degrees, take away the 200.
00:13:25 We take away 200, it's gonna leave you with 160.
00:13:28 Okay. So that gives you angle one.
00:13:30 Measure of angle two here. How do you get that one?
00:13:41 Serum- theorem. It has to be equal to 180.
00:13:44 If angle one is already 160, then angle two and angle three are equal.
00:13:48 Good. That leaves you with 20, which means one of them has to be 10.
00:13:53 Okay. How about P V C?
00:13:56 Anything different about that one than the other example?
00:13:59 No. It's gotta be 90.
00:14:00 I'm just gonna copy that one down right now.
00:14:02 And how about the last one, this whole A V C. The actual obtuse angle. It says it's 100.
00:14:11 If you're following a pattern, what are you getting?
00:14:17 Okay? So, I'm just gonna fill this in now.
00:14:19 Measure of an obtuse secant tangent angle with its vertex on a circle is one half the measure of its intercepted arc.
00:14:29 So let's fill this in.
00:14:32 For measure of angle A V C, how would you relate that back to X for any- for any measure?
00:14:39 Okay. I'm just gonna work my way backwards. We know the 90.
00:14:43 Well, let's go to angle one.
00:14:44 How would you figure out angle one if you know X for A X V?
00:14:48 Three sixty minus X.
00:14:49 Three sixty minus X.
00:14:53 Oh, I was gonna write X over two. Three sixty minus X.
00:14:59 And Katie, how would you do that last one, that angle two?
00:15:03 Angle two would be (inaudible).
00:15:09 This one's a little bit tricky.
00:15:11 The 180 thing works like- I know how to explain it- I'm not sure exactly what the equation is, but I could do it.
00:15:20 So tell me what to do and we'll try to put it in X for you.
00:15:23 All right. You have to get the measure of angle A P V.
00:15:32 That's 360 minus X.
00:15:39 (inaudible) have to be equal so you have divide whatever the original angle is.
00:15:44 Like it- I don't know. I'm sorry.
00:15:47 You subtract A P V from 180 and then divide it by two.
00:15:54 That's what I would do anyway.
00:15:56 So, 180 minus what we get from angle one. I'm gonna put measure of angle one right here.
00:16:01 Minus measure of angle one, and then you take that and divide it by... two.
00:16:12 I couldn't fit all that in that little, little box.
00:16:15 So you get the theorem. This kind of- all these build upon each other, you get the 90, the acute, and the obtuse angles.
00:16:28 It could be a chord and a tangent, you're gonna come up with what?
00:16:34 If you have an angle that intersects with a secant and a tangent all in the circle?
00:16:41 Half of that arc.
00:16:48 You have any questions on that?
00:16:51 I'm gonna wait to give you an example for that one until we get to the other side.
00:17:06 I guess I don't have one for this page, so we're gonna have to do this, just out loud together.
00:17:13 So look at the top for activity two.
00:17:28 You have the intersection of two secants inside the circle, and they want to figure out how you can get those angle measures.
00:17:37 So look at the first one across for number- table two.
00:17:42 You have the measure of arc A C gives you 160.
00:17:47 Then you have B D. That gives you 40 degrees.
00:17:50 How about- where did they get measure of angle one, do you think?
00:17:56 It says 80 degrees. Where do you think they got that?
00:18:03 Arc A C equals 160, and angle one is an inscribed angle, so it's half of A C.
00:18:14 And since A C is 160, half is 80.
00:18:20 That's what we actually just talked about reviewing from nine point three.
00:18:25 So, how about angle two?
00:18:29 B D is 40 (inaudible), so angle of that, so it's half.
00:18:39 How about A V C?
00:18:45 Similar to what we were doing- been doing, but it's got a slightly different twist on it.
00:18:48 Where do you think they could have gotten the 100 degrees if you're looking at that circle?
00:18:53 A V C is the angle we're looking at.
00:19:04 We know arc A C and we know arc B D.
00:19:09 Where could they have gotten that 100 degrees?
00:19:17 There's two ways you can get it. The first way is to add the measures of angle one and angle two together.
00:19:22 And the second way is to add the measure of the two arcs together and divide it by two.
00:19:27 So Michaela says there's two ways.
00:19:28 She thinks you can do 160 plus 40 and you get 200, divide it in half, and that gives you 100.
00:19:36 Or she got 80 plus 20 equals 100.
00:19:42 Okay? Do we have anything to back up either one of those more than the other?
00:19:47 Or do you want to just do the next row and see what we come up with?
00:19:51 Let's do the next row. So the next one says if A C is 180 degrees, B D is 70 degrees, what about angle one?
00:20:08 Those are the inscribed angles.
00:20:16 If A C is 180, what does that angle one have to be?
00:20:22 Ninety degrees. Okay? Half.
00:20:23 How about B D?
00:20:29 Seventy degrees. Seventy degrees. What do you think? Inscribed angle.
00:20:35 So what does that leave us with for A V C?
00:20:42 With Michaela's first method, it would be- what did we come up with? Ninety and 35 added together. What would that give us?
00:20:52 What about the other way? If we do 180 plus 70, what does that give us?
00:21:03 One twenty-five. They both work.
00:21:13 What do you think?
00:21:20 First angle is half the A C.
00:21:29 That for measure of angle one has to be half of the arc A C.
00:21:40 Measure of angle one has to be half of A C. Do we agree with that?
00:21:53 B D. X two.
00:21:56 So what do you think about measure of A V C? Katie?
00:22:15 Okay. Say that again.
00:22:24 All right. That you add the two angles at the top, and subtract that from one- that from 180 because it makes a triangle.
00:22:28 So you subtract that from 180 and that gives you what the other angle is.
00:22:33 Okay. You're talking about A V C or D V B?
00:22:36 A V C. The first one.
00:22:40 And it works for the other one, too.
00:22:44 Is that a lot to do for a general rule?
00:22:47 Maybe, but it works.
00:22:50 So now we have maybe three different ways to find it?
00:22:53 But that actually- that actually proves that- what Michaela said works.
00:22:59 Which one there? She did two.
00:23:02 It proves that both of them work.
00:23:03 Both of them got the same thing.
00:23:04 Both of them got the same answer as the other one that Katie just said, and that really proves it.
00:23:08 The 125 and the 100 that we already went over?
00:23:14 Because basically just adding angle one and angle two together is simplifying adding A C and B D- B D together and divide it by two.
00:23:25 You do 180 minus 125, you get 55.
00:23:28 Well then, if you do 180 minus 55 you get 125 again.
00:23:34 It's longer, but it proves it.
00:23:35 Okay. Well, so let's look at the theorem that it wants us to fill in.
00:23:39 The measures of an angle formed by two secants or chords that intersect the interior of the circle is what?
00:23:48 Blank the blank of the measures.
00:23:55 Which one of those methods would fit into that? We have three different ways, which one would fit into there?
00:24:02 Half the sum of the two arcs.
00:24:05 Okay. So if you take the sum of the two arcs and you divide them in half, I think that's the one that fits best in there.
00:24:11 Any of your methods were pretty good.
00:24:14 Okay. Michaela had the two that I was thinking of and then Katie had a third one that was really good.
00:24:21 The measures end up being half of whatever those two sums are.
00:24:31 Because not all the examples are gonna be as specific as these are to follow through with your other- with the third way to do it.
00:24:40 Do you understand what I mean?
00:24:42 You could probably do it yourself on every example, but that might be time consuming.
00:24:47 All right. Let's look at the very last one.
00:24:50 You guys are doing really good with these patterns.
00:24:55 B D, measure of arc B D is 200 degrees. Now we have the vertex of two secants outside the circle.
00:25:04 Okay. We had it on the circle. We had it inside the circle. Now we're dealing with the outside of the circle.
00:25:09 A lot of you guys had this question last week when you were talking about that test.
00:25:14 You have measure of arc B D. That's at 200 degrees.
00:25:18 And then you have A C, the smaller one right before they intersect at 40 degrees.
00:25:22 Then you have the measure of angle one. They give you one hundred. Where did they get that?
00:25:30 Measure of angle one. Kind of following right along. Melissa?
00:25:33 (inaudible) intercepted arc of- for angle of B D has to be half.
00:25:37 Good. So it has to be half. Good.
00:25:40 Two. Measure of angle two. They say 20 degrees. What do you think?
00:25:48 The same reason as for measured angle.
00:25:50 Good. You guys are following right along.
00:25:51 The same reason. Okay? The other one's 40. This one's half. It's gotta be 20.
00:25:56 And they get A V C.
00:25:57 Back to this measure angle A V C.
00:25:59 If it's outside the circle. What do you think they- how do you think they got that 80 degrees?
00:26:17 And then you do that plus adding the two, and then subtract that from 181, and then you have A V C.
00:26:33 So he said V C B, if you do that minus angle one, which was 100, that leaves you with what?
00:26:42 That leaves you with 80.
00:26:55 Eighty. That works. Right? That works.
00:26:58 Or you can do A C is the (inaudible) outside, the angle is outside the circle.
00:27:08 Which one- A C what? The arc?
00:27:11 A V C. Angle A V C.
00:27:18 The arc's 40 degrees, and since angle A V C is 80, maybe it's 40 times- times two, (inaudible).
00:27:26 Maybe it's 40 times two? Do you have something to back that up?
00:27:31 It also intercepts arc B D.
00:27:37 But, if you do B D which equals 200 plus A C (inaudible) 240.
00:27:46 At- no, wait. Do 200 minus 40 and then you get 160.
00:27:52 Divide that by two and that equals 80.
00:27:55 Okay. 'Cause we're trying to stick with this general pattern of it ending up being- what's it end up being every time?
00:28:08 Half- you're scaring me- it ends up being half of whatever- whatever pattern you're following. Okay.
00:28:17 The last one we added the two arcs together because it was inside the circle.
00:28:22 Now we have a different case.
00:28:23 It's still gonna be half, but Leah says it might be half of subtracting them because now you're outside the circle. Okay.
00:28:31 So if you're- if you have your- what side's it on?
00:28:34 If you have your B D and then you're a C arc here, the angle's way over here. Subtracting them might make sense.
00:28:42 Do you see what I mean? Subtracting them might make a little bit more sense.
00:28:46 And now Ben's idea still worked, but it might be difficult for you.
00:28:52 You're trying to get something quick and easy that's a gen- you're following this pattern. The triangle sum works. Right?
00:28:58 But that might take you a little extra work every time.
00:29:02 If you're only given the measure of two arcs that it intercepts, it's gonna be, probably, a lot easier just to subtract them.
00:29:12 Boom. Get your answer.
00:29:14 So, go for it Raul. Read that theorem for me and fill in the blanks.
00:29:20 The measure of an angle formed by two secants that intersect the exterior of a circle is half the sum of the measures of the intercepted arcs.
00:29:28 Half of the sum?
00:29:42 Difference. We're subtracting them this time.
00:29:44 I thought I said that.
00:29:45 You did say that. I was trying to see if Raul would get that.
00:29:49 I thought that angle one was 100 degrees. Right?
00:29:55 And then in order to find A B C, you had- oh, yes.
00:29:57 That was Ben's way of the triangle- the triangle sum. We were trying to get down to that same general pattern of it being half.
00:30:06 Okay. So if you- if you subtract them, if you get the difference, that's gonna be a quicker way to do it.
00:30:14 'Cause you might not always have that triangle working for you there.
00:30:19 That's a little- like you're taking- you're doing it- but you're taking an extra step.
00:30:23 You see what I mean?
00:30:24 You're going an extra step. Basically they just want to get the general pattern. It's gonna be a lot easier for you if you subtract them.
00:30:33 So you do it by half. Divide it in half.
00:30:37 All right. I have an example for you.
00:30:42 This is... what we just did with a little twist on it.
00:30:57 This is my example with a little twist on it.
00:31:29 you have to find the measure of arc B D.
00:32:32 I like how you wrote that like that.
00:32:48 How did you get your answer there?
00:32:50 I used a theorem.
00:32:52 What did you do?
00:32:55 I just used that theorem.
00:32:57 How, though? How did you use it?
00:32:59 I don't know. I did something.
00:33:25 Okay. See now I like how you're filling in those blanks.
00:33:28 Write it out, show me what you did.
00:33:41 You want to go write that up there for me?
00:33:46 Take your book with you if you need it.
00:34:04 Some people had to draw a picture. That's good. Draw your own picture. Some people like to draw it and write an equation.
00:34:12 Either way you do it, you gotta show me what you did.
00:34:17 Ms. Lancour, should we solve it?
00:34:18 Yep, I want you to solve it.
00:34:29 Leah did hers with an equation?
00:34:42 She followed the subtraction, 130 minus X divided by two should give you 60 degrees, she says.
00:34:53 She moved the two over and got 120. One thirty minus X equals 120. So she filled in her blank with the X.
00:35:01 X has to be 10 degrees. One thirty minus 10 degrees is 120 degrees.
00:35:10 Anybody else want to tell a different way that you did it?
00:35:22 You all did it the exact same way as Leah did it?
00:35:25 That is absolutely amazing. That is absolutely amazing. Nobody can- Okay. I'll let it go.
00:35:33 Okay. That one was the- that one was the- the nice one.
00:35:40 That one was the nice one.
00:35:42 I like nice ones.
00:35:46 Now I don't have to show my work.
00:35:48 No, you gotta show the work.
00:35:55 You have to actually do it.
00:36:33 Okay. That one's good. Good start on that one.
00:36:43 What are you gonna do first? Which one are you gonna work on first? The R S U?
00:36:50 R S U looks like the vertex is on the circle.
00:36:53 What can you do for that?
00:36:58 Sometimes it helps if you draw a picture. It helps me to draw a picture.
00:37:21 Raul, what are you doing over here?
00:37:24 Well, I don't have any (inaudible).
00:37:25 Which angle are you doing first?
00:37:27 Well, I don't- I haven't (inaudible) problem, but I'm doing U T before (inaudible) anything else. Makes it simple for me.
00:37:34 Okay. Why do you think that one's easier? Why is that?
00:37:38 'Cause then I feel like the question would be R S U. There's R S U. Oh, that would be half of that. (inaudible) that.
00:37:45 Oh, I see. You're finding this arc first.
00:37:47 Yeah, this one first.
00:37:48 Oh! Okay. I like that. Not bad.
00:38:00 How'd you get this one, Melissa?
00:38:03 That, it intercepts arc R U, so it has to be half. (inaudible).
00:38:09 It has to be half. It's the inscribed angle. Okay.
00:38:15 It looks like just about everybody got the first one, R S U. Anybody want to tell me what you did?
00:38:21 Katie, did you get that one? R S U. What did you do?
00:38:25 Since the point's on the circle, I just divided the arc by two.
00:38:37 Half of the arc.
00:38:39 Okay. It has to be half of that intercepted arc. Good.
00:38:56 Where are you, Margaret?
00:38:59 R V U. What did you come up with?
00:39:07 R V U. Well, think about what we just did in the- in the last example.
00:39:13 When it's outside the circle. You know R U, that was given. And you know R S- oh, not R S. S T.
00:39:25 So, you have 140 and you have 30 degrees.
00:39:33 Okay, you started to subtract.
00:39:37 You changing your mind?
00:39:43 So, what are you gonna do with that? You got 110.
00:39:48 Divide it in half. No. Yeah. No.
00:39:51 No, yes, no. Maybe? Yes?
00:39:55 Look back here. What did we just come up with for this one?
00:40:04 Think about it. I'll be back to you.
00:40:12 U S V. U S V. How did you get that one, Leah?
00:40:19 'Cause R S U is half of 140, so it's 70. And that's a line (inaudible) equals 180. One eighty minus 70 equals 110.
00:40:44 Hmm. Show- explain that to me again. Tell me what you just did again.
00:40:54 That's half of that.
00:40:56 And this equals 180, so 180 minus 70 is 110.
00:41:02 Oh, okay. You were doing U S V. Okay. Okay. At first- at- at first I thought you were doing the triangle.
00:41:14 The triangle sum. But you ended up doing... 180 degrees.
00:41:31 I like how you did that different. I wasn't expecting that. I'm sorry, that's why I'm thinking. I wasn't expecting that one.
00:41:42 Okay. We're gonna go on with R V U really quickly. R V U.
00:41:51 This one, R V U. Here. That's kind of like the example we just did, and the very last activity we did. Margaret, go for it.
00:42:03 Minus 30 degrees, and you get 110. So, you divide that in half and get 55 degrees.
00:42:08 Good. One thirty- 140 minus 30 gives you 110. Divided in half... 55 degrees.
00:42:14 Okay, that's the last theorem we just went over. How about U S V? Let's point that one out.
00:42:23 U S V. Going here. I actually got a different way from one person that I w- just really wasn't expecting.
00:42:32 I had it one way in my mind, and somebody else gave me a whole different way to think about it.
00:42:37 I used the triangle sum theorem.
00:42:39 And we just did R V U, so I know that that is 55 degrees.
00:42:45 Okay. So she knows that this one is 55. Okay.
00:42:52 I used the- oh, what's it called? The arc intercept theorem.
00:42:56 To find out that that was 15.
00:42:58 Inscribed angle to get this one. Fifteen, 'cause it's half. Okay. So, so far we have 55 and 15.
00:43:05 I added them together and I got 70, so I did 180 minus (inaudible).
00:43:10 Okay. She did 180 minus 170, got 110 for this angle here. So she ended up with 110 for this by doing triangle sum.
00:43:26 I was on this inscribed angle and getting the triangle sum theorem. It didn't even occur to me.
00:43:33 Angle R S U, we already figured that (inaudible) 70.
00:43:37 Okay. We did this, RSU, which was... 70.
00:43:54 (inaudible) that line, the- the angle of it equals 180.
00:43:58 Okay. It's a straight angle. It has to be 180.
00:44:01 And so you do 180 minus 70. You get 110.
00:44:04 And she got 110.
00:44:06 That's the easy way.
00:44:07 That's an easy way. Well, we didn't even- it didn't- it didn't even click in my head. I'm- I'm sure it didn't- took- I was surprised Leah got that.
00:44:15 And last one. R W S.
00:44:19 Real quick. Don't go. R W S. How would you get that one?
00:44:27 She says 95. You say?
00:44:30 (inaudible) triangle sum again.
00:44:31 You did triangle sum?
00:44:33 I did triangle sum.
00:44:34 I did was to figure out what arc U T equals. It equals 90. You do 100 plus 90 divided by two.
00:44:42 Good. Okay. Thank you, guys.
00:44:48 Study, study, study, study, study, study. Tomorrow.

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