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Finding arc measures with equations

CCSS Math: HSG.C.B.5

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 wanna 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've had to give us three letters to force us to go the long way around, so if they said arc BAC or BDC, 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 wanna 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, 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 4k + 159 degrees. So if we can figure out what K is, we're gonna 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 BPC that we care about, is vertical to angle APD. These are vertical angles, and vertical angles are going to have the same measure, they are, they're going to be congruent. So let's set these two measures equal to each other. So we know that 4k + 159 is going to be equal to 2k + 153, so let's get all of our K terms on the left-hand side, and 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 gonna be left with the 153. And on the left-hand side, 4k - 2k is 2k, and I still have + 159. Now let's get rid of this 159 on the left-hand 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 2k is equal to 153 - 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 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 - 3 + 159 well what's that going to be? 4 times -3 is -12. Plus 159 is going to be 147. So this angle right over here has a measure of 147 degrees and you can calculate, that's the same thing as over here. 2 times -3 is -6, plus 153 is 147 degrees, these two are the same, and so 147 degrees. This angle measures 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 BC in degrees? Now since once again they only gave us two letters, we can assume it is the minor arc. So we care about BC, we care about this right over here. And 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 4y + 6 + 7y - 7. 4y + 7y, we can combine the y terms, is going to be 11y. And then 6 - 7 is going to be negative 1. So it's going to be 11y - 1, and how do we figure that out? Or how do we figure out what Y is? We need to figure out what Y is in order to figure out what 11y - 1 is. Well, we know, let me write this down. So the angle that we care about is 11y - 1, 11y - 1. We know that that angle, plus this big angle that I'm going to show in blue, that if we add them together that it's going to be 360 degrees, 'cause we would've gone all the way around the circle. So we know that 11y - 1 + 20y - 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 11y + 20y? Well that's going to be 31y, and then if I have - 1 and -11 that's going to be negative, let me do this in a different color, so that's going to be, - 1 and -11, that's -12, and that's going to be equal to 360 degrees, 360. So let's see, we can add 12 to both sides to get rid of that - 12 right over there, and that's going to leave us with 31y 31y 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 11y - 1, so what is 11 times 12? We know that Y is 12. 11 times 12 - 1, let's see. 11 times 12 is 121. And then 121 - 1 is going to be, oh sorry no, my multiplication tables are off, it's been a long day. 11 times 12 is going to be 132, 132 - 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.