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CCSS Math: HSG.C.B.5

Video transcript

- [Voiceover] What I want to talk about in this video, is the notion of Arc Measure, when we're dealing with circles. As we'll see, sometimes when you see something like arc measure, you might think it's the length of an arc, but arc length is actually a different idea. So we will compare these two things. Arc length to arc measure. So arc measure, all that is is just a fancy way of saying, if I have a circle right over here, this is my best attempt at drawing a circle. I have a circle here. The center of the circle, let's call that point O, and let me put some other points over here. So let's say that this is point A, let's say this is point B, and let's say this is point C right over here. And let's say that I have, let's say the central angle, right over here, cause it includes the center of the circle, so the central angle, angle AOB. Let's say it has a measure of 120 degrees. And if someone were to say, what is the measure of arc AB? So, let me write that down. The measure. So, if someone were to say what is the measure of arc AB, and they'd write it like this, so that's referring to arc AB right over here. It's the minor arc, so there's two ways to connect AB, you could connect it right over here, this is the shorter distance, or you can go the other way around, which would be what you'd consider the major arc. Now, if someone's referring to the major arc, they would say arc ACB. So when you're given just two letters, you assume it's the shortest distance between the two. You assume that it is the minor arc. In order to specify the major arc, you would give the third letter, to go the long way around. So the measure of arc AB, and sometimes you'll see it with parenthesis right over here, all this is, this is the same thing as the measure of the central angle that intercepts that arc. Well, the central angle that intercepts that arc has a measure of 120 degrees. So this is just going to be 120 degrees. Now, some of y'all might be saying, well, what about the major arc? Well, let's write that. So if we're talking about arc ACB, so we're going the other way around, so this is major arc. So what is the measure of arc ACB, once again we're using three letters, so that we're specifying the major arc. Well, this angle, this central angle right over here, to go all the way around the circle is 360 degrees. So this is going to be the 360 minus the 120 that we're not including. So 360 degrees minus 120 is going to be, is going to be 240 degrees. So the measure of this angle right over here is 240 degrees, so the measure of this arc, I have to be careful not to say length of that arc, the measure of this arc is going to be the same as the measure of the central angle. It's going to be 240 degrees. These arc measures are going to be the case regardless of the size of the circle, and that's where the difference starts to be from arc measure to arc length. So, I could have two circles, so this circle right over here and that circle right over here, and as long as the central angle that intercepts the arc has the same degree measure, so let's say that that degree measure is the same as, these are central angles, so we're assuming the vertex of the angle is the center of the circle. As long as these two are the same, these two central angles have the same degree measure, then the arc measures, then the corresponding arc measures are going to be the same. But clearly, these two arc lengths are different. The arc length is not going to depend only on the measure of the central angle, the arc length is going to depend on the size of the actual circle. Arc measure is only dependent on the measure of the central angle that intercepts that arc. So your maximum arc measure is going to be 360 degrees. Your minimum arc measure is going to be zero degrees. It's measured in degrees, not in units of length that arc length would be measured in. So, let me write this down. This only depends... So, this is what's going to drive this is the measure of central angle, central angle, that intercepts the arc. That intercepts, intercepts the arc. When you talk about arc length, yes, it's going to be dependent on the angle, but it's also dependent, it's going to be dependent on the measure of that central angle plus the size of the circle. Size of the circle. You're actually talking about a length now, when you're talking about arc length. While here, you're talking about a degree measure.