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Video transcript

a B and C are collinear and B is between a and C the ratio of a B to a C is 2 to 5 if a is at negative 6 comma 9 and B is that negative 2 comma 3 what are the coordinates of point C and I encourage you to now pause this video and try this on your own so let's try to visualize this so a a is that negative 6 comma 9 negative 6 comma 9 B let's see it's less negative in the horizontal direction it's lower in the vertical direction so maybe we could put B right over here B is at the point negative 2 comma 3 and at C it's going to be collinear so we're going to go along the same line so let me draw that line right now so it's going to be it's going to be cold in here it's going to be co-linear and it tell us the ratio of a B to a C is 2 to 5 so B is going to be two-fifths of the way so let's say C is I'm just trying to eyeball it right now let's put C right over here and we don't know C's coordinates well the way we could think about it is to break it up into horizontal change in coordinate and vertical change in coordinate and apply the same ratio so for example what is the horizontal change in coordinates going from A to B well let's draw that going from A to B so this is a x' x-coordinate it's at negative 6 B's x-coordinate is at negative 2 so it's this just change right over here this is the horizontal change that we care about now what is that well if you go if you start at negative 6 and you go to negative 2 you have you have increased you have increased by 4 another way of thinking about it is negative 2 minus negative 6 is the same thing as negative 2 plus 6 which is going to be 4 now the ratio between this change and the change of the x-coordinate between a and is going to be two to five so let's call that change let's call that change this entire change let's call this let's call this let's call this X so we could say that the ratio between four and X is equal to notice this is a change the horizontal change between from A to B just if you look on the horizontal axis so the ratio of that which is four to the horizontal change between a and C well that's going to have to be the same ratio so it's going to be two over five now to solve for X a fun thing might be to just to find take the reciprocal of both sides so x over four is equal to 5 over 2 we could multiply both sides times 4 and we are left with X is equal to 5 times 4 divided by 2 which is equal to 10 so the change in X from A to C is going to be 10 so what does that tell us about C's x-coordinate so we could start with a's x-coordinate which is negative 6 add 10 to it negative 6 plus 10 is 4 so we figured out the x-coordinate now we just have to do the same thing for the Y so what is the change in Y going from A to B going from A to B what is the change in Y well here we go from 9 to 3 we've gone down 6 another way you could say is well 3 minus 9 is negative 6 to find the change you could think I'm just taking the end point and subtracting from that the starting point negative 2 minus 6 I'm sorry negative 2 minus negative 6 was positive 4 3 minus 3 minus 9 is negative 6 or you could just look at it we've gone down 6 so we can write negative 6 here now our change in Y is going that we're going to have to have that same ratio so our change in Y between a and C let's just call that let's just call that distance Y so our change in Y is that's our change in Y and we're going to have to have the same ratios so we could write that the ratio between our change in Y from A to B which is negative six - the ratio between our change in Y from A to C negative six to Y is once again going to be equal to two to five once again we can take the reciprocal of both sides Y over negative six is equal to 5 over 2 multiply both sides times negative 6 and we are left with Y is equal to 5 times negative 6 is negative 30 divided by 2 is negative 15 so our change in Y or I guess our change in our vertical axis which we're calling Y in this case is negative 15 so here if our Y value is 9 and if we were to subtract 15 from that where does that put us well 9 minus 15 is going to put us at 6 9 minus 15 is going to put us at 6 so the coordinates for Point C are at the court of sorry it's 9 minus 15 is going to put us at negative 6 almost made a careless mistake 9 minus 15 is negative 6 I was wondering this seems very low to be at point 6 so 4 comma negative 6 is a coordinates of Point C