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Watch Sal figure out the coordinates of a point between two other points that give a certain ratio. Created by Sal Khan.
Video transcript
A, B and C are collinear, and B is between A and C. The ratio of AB to AC is 2 to 5. If is A is at negative 6 comma 9, and B is at 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 is at negative 6 comma 9. B, let's see, it's less negative in the horizontal direction, it's lower in the vertical direction. So we could put B right over here. B is at the point negative 2 comma 3. And that C, it's going to be collinear, so we're going to go along the same line, let me draw that line right now. So it's going to be collinear and they tell us that the ratio of AB to AC is 2 to 5. So B is going to be 2/5 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's x-coordinate, it's at negative 6. B's x-coordinate is at negative 2. So it's this change right over here. This is the horizontal change that we care about. Now what is that? Well if you start at negative 6 and you go to negative 2, 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 C is going to be 2 to 5. So let's call that change, this entire change, let's call this x. So we could say that the ratio between 4 and x is equal to-- and notice this is the horizontal change from A to B just if you look on the horizontal axis-- so the ratio of that, which is 4, 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 2/5. Now to solve for x, a fun thing might be to just take the reciprocal of both sides. So x/4 is equal to 5/2. We can 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? Well here, we go from 9 to 3, we've gone down 6. Another way you could say it is, 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 negative 6 was positive 4. 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, we're going to have to have that same ratio. So our change in y between A and C, 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 6, to the ratio between our change in y from A to C, negative 6 to y, is once again going to be equal to 2 to 5. Once again, we could take the reciprocal of both sides, y over negative 6 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. So the coordinates for point C are at-- oh, sorry 9 minus 15 is going put us at negative 6. I almost made a careless mistake. 9 minus 15 is negative 6. I was wondering, this seems very low to be at 0.6. So 4 comma negative 6 is the coordinates of point C.