Circle theorems — Harder example

0:00- [Instructor] A circle has a circumference of 10 pi feet. An arc, x, in this circle has a central angle of 260 degrees. What is the length of x? So let's just visualize what's going on here. So we have a circle. I can't draw a circle that well, but you get the point.

0:18So that's our circle. This is the center of our circle. It has a circumference of 10 pi feet, so if we were to go all the way around the circle, this, it has a circumference of 10 pi feet. Now you have an arc, x. You have an arc, x, in the circle that has a central angle of 260 degrees.

0:39So let's think about, so 260 degrees, so if you go straight up, that would, let's see, if you go this far it'd be 90 degrees, 180 degrees. If you were to go all the way here you'd get to 270, so it's gonna be right around, and we just approximate, right around there.

0:56So that would be be a central angle of 260 degrees. And this is the arc. This is arc. Let me do this in a different color. Let me do it in purple. So this is arc x right over here. And we wanna figure out its length. Well think about it this way. Its central angle is 260 degrees.

1:20What's the central angle of the circumference? Well the circumference is going all the way around. So if you're going all the way around, that is 360 degrees. So the fraction that this, the arc x is, the length of x is of the entire circumference, that's gonna be the same fraction that its central angle is of 360 degrees.

1:45So once again the entire circumference is 10 pi feet. 10 pi feet. That's the circumference. Now the fraction of that circumference that's going to be arc x, that's going to be the fraction that 200. That's gonna be the same fraction of the central angle of that arc relative to the central angle if you were to go all the way around, which would be 360 degrees.

2:07So it'll be 260/360 of the circumference. So what's that going to be? We'll let's see. We can simplify this a little bit. We could, if we divide the numerator and the denominator by 10, 10 divided by 10 is one. And let me make it. Actually, let me just write it this way.

2:26So we could write this as 260 times 10 pi is going to be 2600 pi over 360. And now we just need to simplify. If we divide the numerator and the denominator by 10, this is gonna be 260. This is going to be 36. If we divide the numerator and the denominator by, let's see.

2:49It seems like they are both divisible by four. 260 divided by four is going to be, let's see. 200 divided by four is 50. 60 divided by four is 15. 15 plus 60 it's gonna be 65. So let me do this in a different color. This is gonna be 65 and 36 divided by four is nine.

3:11And just so you see I didn't, you know, I was tryna do it in my head, and it's good to get some practice doing that, but all I did is divided both the numerator and the denominator by four. Four goes into 260. Four goes into 26. Six times four is 24. Subtract, you get a two.

3:27Bring down that zero. Four goes into 20 five times. Five times four is 20. So that's all I did there. So it's 65 pi over nine and it looks like that's about as simplified as I'm gonna get. And that's nice 'cause that's one of the choices right over here.