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Current time:0:00Total duration:6:58

Finding arc measures with equations

CCSS.Math:

Video transcript

so we're told Circle P is below this is circle P right over here what is the arc measure of Arc BC in degrees so this is point B this is Point C let me pick a different color so you can see the arc and since they only gave us two letters we really want to find the minor arc so we want to find the shorter arc between B and C so the major arc would be the long way around and if they wanted to specify the major arc they would have had to give us three letters to force us to go the long way around so B if they said arc BAC or B DC that would go the long way around but since they just gave us just B and C we assume it's going to be the minor arc so we want to find that arc measure right over there now the arc measure is going to be the exact same measure in degrees as the measure of the angle that the central angle that intercepts that arc so it's going to be the same thing as the measure of this central angle which is 4 K plus 159 degrees so we can figure out what K is we're going to know what this central angle measure is and then that's going to be the same thing as this arc measure so how do we figure that out well what might jump out at you is that this angle angle BP C that we care about is vertical to angle APD these are vertical angles and so and vertical angles are going to have the same measure they're going to have there they are they are they're going to be congruent so let's set these two measures equal to each other so we know that for K plus 159 is going to be equal to 2k plus 153 so let's get all of our K terms on the left-hand side and all of the all of the non K terms on the right-hand side so let's subtract 2k from both sides so we can subtract 2k from both sides and let's subtract well let me just do that first I don't want to skip steps and so I got rid of the K's on the right hand side so it's just going to have be left with the 153 and on the left-hand side for K minus 2 K is 2k and I still have plus 159 now let's get rid of this 159 on the left and side so let's subtract it but if I do it on the left hand side I need to do it on the right hand side as well so subtract 159 from both sides and I'm left with 2 K is equal to 153 minus 159 is negative 6 so K is equal to just divide both sides by 2 K is going to be equal to negative 3 now you might be tempted to say Oh negative 3 but that's not where we're trying we're not just trying to solve for K we're trying to figure out this angle measure which is going to be the same as the arc measure that we care about and that's just expressed in terms of K so it's 4 times K + 159 so that's going to be 4 times negative 3 plus 159 well what's that going to be 4 times negative 3 is negative 12 plus 159 is going to be 147 so the this angle right here has a measure of 147 degrees and you can calculate that's the same thing as over here knit 2 times negative 3 is negative 6 plus 153 is 147 degrees these two are the same and so 147 degrees this angle measure is the same as the measure of Arc BC let's do one more of these circle P is below what is the arc measure of b/c in degrees now since it once again they only gave us two letters we can assume it as the minor arc so we care about we care about BC we care about this right over here and what so what is the measure of this arc is going to be the same thing as the measure of the central angle that intercepts that arc and that measure is going to be the sum of these two angles so it's going to be so it's going to be 4y plus 6 plus 7 y minus 7 so what's that 4y plus 7y the Y we can combine the Y terms is going to be 11 y and then 6 minus 7 is going to be negative 1 so it's going to be 11 Y minus 1 and how do we figure that out how do we figure out what Y is we need to figure out why is in order what 11 y -1 is well we know let me write this down so the angle that we care about is 11 y -1 11 y -1 we know that that angle plus this big angle that I'm going to show in blue then if we add them together that it's going to be 360 degrees because we would have gone all the way around the circle so we know that 11 y -1 plus 20 Y minus 11 is going to be equal to 360 degrees and so now we can just solve for y what is let me get some new colors involved what is 11 y plus 20y well that's going to be 31 Y and then if I have negative 1 and negative 11 that's going to be negative let me do this in a different color so that's going to be that's going to be negative 1 negative 11 that's negative 12 and that's going to be equal to 360 degrees 360 so let's see we could add 12 to both sides to get rid of that negative 12 right over there and that's going to leave us with 31 y is equal to 372 and so if we divide both sides by 31 it looks like 12 yep it'll go exactly 12 times so Y is equal to 12 which is equal to 12 and remember we weren't trying to solve for y we were trying to solve for xi y minus 1 so what is 11 times 12 we know that Y is 12 11 times 12 minus 1z 11 times 12 is 102 120 21 and then 121 minus 1 is going to be all right now it's going to sex discredit via and my multiplication tables are off it's it's been a long day 11 times 12 is going to be a hundred and and 132 132 minus 1 is going to be 131 and it's going to be in degrees so 131 degrees that's the measure of this angle which is going to be the measure of minor arc BC