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Video transcript

find the point B on segment AC such that the ratio of a B to BC is 3 to 1 I encourage you to pause this video and try this on your own so let's think about what they're asking so if that's point C I'm just going to redraw this line segment just to conceptualize what they're asking for and that's point a they're asking us to find some point B some point B that the distance between C and B so that's this distance right over here so if this distance is X then this is between B and a is going to be three times that it's going to be three times that so this will be 3x that the ratio but the ratio of a B to B C is three to one so that would be the ratio let me write this down it would be a B it looks like an HB it would be a B to B C is going to be equal to three X to X which is the same thing as three to one which is the same thing as three to one if we wanted to write it a slightly different way so how can we think about it you might be tempted to say oh well you could use the distance formula find the distance which by itself isn't completely uncomplicated and then this will be one-fourth of the way because if you think about it this entire distance is going to be 4x this entire distance is going to be let me draw that a little bit neater this entire distance if you have an X plus a 3 X it's going to be 4x so you say well this is one out of the four X's along the way this is going to be 1/4 of the distance between the two points so this is let me write that down this is 1/4 of the way 1/4 of the way between C and B going from C sorry going from C to a B is going to be 1/4 of the way so maybe you try to find the distance and you say well what are all the points that are 1/4 away but it has to be 1/4 of that distance away but then it has to be on that line but that makes it complicated because this is not this is this line is an incline it's not just horizontal it's not just vertical what we can do however is break this problem down into the vertical part the vertical change between a and C and the horizontal between a and C so for example the horizontal change between a and C a is at 9 a is at nine right over here and C is at negative seven C is at negative seven so this distance right over here is nine minus negative seven nine minus negative seven which is equal to 9 plus 7 which is equal to 16 and you see that here 9 plus 7 this total distance is 16 that's the horizontal distance change going from A to C or going from C to a and the vertical change and you could even just count that that's going to be 4 C is that one a is that a is at five going from one to five you've changed vertically 4 so what we can say going from C to B in each direction in the vertical direction and the horizontal direction we're going to go 1/4 of the way so if we go 1/4 in the vertical direction we're going to end up we're going to end up at Y is equal to 2 and if we go so I'm just going starting at C 1/4 of the way 1/4 of 4 is 1 so I've just moved up one so our so our Y is going to be equal to 2 and if we go 1/4 in the horizontal direction 1/4 of 16 is 4 so we go 1 2 3 4 so we end up right over here our X is negative 3 so we end up at that point right over there we end up at this point this is the point negative 3 comma 2 and if you are really careful with your drawing you could have actually just drawn actually you don't have to be that careful since this is graph paper you actually could have just said hey look we would go 1/4 this way and where does that intersect the line hey it intersects the line right over there or you could have said we're going to go 1/4 this way where does that intersect the line and that would have let you figure it out either way so this point right over here is B it is 1/4 of the it is 1/4 of the way between C and a or another way of thinking about the distance between C and B which we haven't even figured out we could do that using the distance formula or the pathetic theorem which it really is we could this distance the distance CB is one third of the distance ba or ba the ratio of a B a B to B C is three to one