Circle theorems — Basic example

what is the length of arc ABC so arc ABC is this arc right over here and the reason why they have to give us three points cuz if they just said arc AC it would still be ambiguous because there's two arc AC s it could be this arc right over here or it could be this arc over here and so when they tell us B we know that we're going from A to C through B so that's why this kind of by giving us a third letter it resolves the ambiguity on which arc we're talking about so what is the length of that arc well the first thing we could do is well let's just think about the entire what's the what's the circumference of this circle and then we could say well what fraction of this of the entire circumference is this arc and a big hint here is if we weren't if we were to go all the way around the circle that would be 360 degrees but we're only going 135 of those 360 degrees but let's think first think about the circumference so the circumference of a circle is equal to 2 is equal to 2 pi times the radius of the circle and they give us the radius is 3 inches so the circumference is going to be 2 pi times 3 inches so times 3 inches or 2 times 3 is 6 times pi so it's going to be 6 PI 6 pi inches now and actually if you look at the if you look at the choices if you look at the choices over here the entire circumference of the circle is 6 PI inches and remember this pi is 3.14159 so this is going to be something in the you know it's gonna be a little bit over 18 or 19 inches and so you could immediately if the entire circumference is a little over 18 or 19 inches this is only a fraction of that entire circumference so actually if you even we're looking at the choices you'd say well hey these are way these are way too large but I should let's see if we can get to if we can get to the exact right answer so the entire circumference is 6 PI inches but this arc length isn't the entire circumference it only goes 135 degrees out of 360 degrees so we could say so it is 130 five 360th of the entire circumference remember if we were to go all the way around that's 360 degrees while this angle right over here is 135 degrees the angle that is subtended by this arc so this arc length is going to be 135 out of 3 135 360 it's of the entire circumference so x times 6 pi 6 pi inches so let's see if we can simplify this a little bit so let's see 300 we could divide the numerator and the denominator by 6 6 divided by 6 is 130 360 divided by 6 is 60 and let's see if we divide 135 by 5 that is going to be 27 yeah 100 divided by 5 is 20 and then 35 divided by 5 is 7 so this is gonna be 27 and then this would be 12 if we divide 60 by 5 and let's see we can then divide 26 we can divide the numerator and denominator by 3 27 divided by 3 is 9 and 12 divided by 3 is 4 so we're left with 9 fourths times pi inches so 9 fourths times PI inches or we could multiply the numerator times the pie we would say 9 PI over 4 inches you know when you look at the choices now that is this one that is this one right over there