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Current time:0:00Total duration:5:32

Video transcript

find the distance between the point negative two comma negative four at this point right here and the line y is equal to negative one-third x plus two that's this line right over here now to do it we just need to figure out the a perpendicular line to this blue line to Y is equal to negative one-third X plus two that contains this point right over here that contains this point so we need to figure out we need to figure out the equation of this line and then we need to figure out where do these two lines intersect and then we need to find the distance between these two points of intersection and we have the shortest distance between this point and this line right over here so the first step is to figure out what is the slope of this perpendicular line well the slope of a perpendicular line is going to be the negative inverse of the slope of this blue line so the negative inverse of one-third of negative 1/3 is going to be positive 3 so this line right over here is going to have a slope of 3 so it's going to have the form Y is equal to 3x plus B where B is its y-intercept it looks just eyeballing it here that it's going to be pretty close to 2 but let's verify that so to figure out what B actually is let's substitute this point right over here we know that not only is this line slope 3 but it hat this point has to sit on it so this point has to satisfy this equation so when X is negative 2 y is negative 4 or we have negative 4 is equal to 3 times negative 2 plus B let me write the negative 2 in there 3 times negative 2 plus B and now we can solve for B we get negative 4 is equal to negative 6 plus B add and add 6 to both sides you get 2 is equal to B or B is equal to 2 so we will write the y-intercept for the second line is at 2 so we immediately can eyeball and or we can't verify where they both intersect they both intersect the y-axis at at what at y equals 2 for both of these when X is equal to 0 Y is equal to 2 if it wasn't so obvious we could set these two equation equal to each other we could say look we have 3x plus 2 we know that this is now 3x plus 2 because b is - when does 3x plus 2 equal equal negative 1 3 X plus 2 well let's see if we subtract 2 from both sides when does 3x equal negative 1/3 X well there's a couple of things that we could do right over here we could add 1/3 X to both sides and then we will get 3 so we'll get 3 and 1/3 which is the same thing as 10 thirds X is equal to 0 and if you multiply both sides by 3/10 3/10 you get you get X is equal to 0 so these two lines intersect when X is equal to 0 for both of them when X is equal to 0 Y is equal to 2 X is equal to 0 Y is equal to 2 but you could have eyeballed it here you could look at you could have seen that both of their y-intercepts which happens when X is equal to 0 Y is equal to 2 so this point right over here is the point 0 2 we already know that this point right over here is the point negative 2 comma negative 4 and now we need to need to find the distance between these two points and the distance formula really is just an application of the Pythagorean theorem we just need to find the distance in the change in the Y direction and the change in the X direction so let's do that separately so in the Y direction in the Y Direction what is what is this distance right over here so we went from Y is equal to negative 4 to Y is equal to 2 this distance right over here is 6 and what is this distance right over here well we go from x equals negative 2 to x equals 0 so this distance right over here is 2 so the distance between these two points is really just a hypotenuse of a right triangle that has sides 6 and 2 so we could say if we call this distance D we could say that the distance square is equal to an all I'm really doing here is restating the distance form the distance formula tells you all this you know why - - you know y2 - y1 which is 6 squared but that's just the Pythagorean theorem that's just saying 6 squared 6 squared plus x2 minus x1 which is 0 minus negative 2 which is positive 2 squared is going to be equal to the distance squared but we see that's just the Pythagorean theorem but anyway let's solve for the distance so the distance squared is going to be equal to 36 plus 4 which is 40 and now we can let's see 40 it's that this is 2 the distance squared or so I could say the distance is equal to the square root of 40 square root of 40 is the same thing as the square root of 4 times 10 and so that's the same so the distance is equal to 2 if we factor out the 4 square root of 4 times square root of 10 2 is the square root of 4 2 square roots of 10 and we're done