### Video Transcript

Consider π΄-coordinate negative one, negative two and π΅-coordinate negative seven, seven. Find the coordinates of πΆ, given that πΆ is on the ray π΄π΅, but not on the segment π΄π΅ and π΄πΆ equals two πΆπ΅.

So letβs start this question by considering our two coordinates. If itβs helpful, we could plot these two points on a coordinate grid. But sometimes, itβs sufficient just to plot their relative position to each other. Here, weβre told that thereβs a ray π΄π΅. That means that the line starts at π΄, goes through π΅, and continues indefinitely. Weβre told that thereβs a coordinate πΆ, which is on the ray π΄π΅, but not on the segment π΄π΅, which means that πΆ isnβt between π΄ and π΅. So we can draw it on the line beyond π΄π΅. Weβre told that π΄πΆ equals two πΆπ΅. This means that if we said π₯ was the length of πΆπ΅, then π΄πΆ must be two times that, giving us two π₯. At this point, we donβt know the length π₯ or the length πΆπ΅. But given we know the coordinates of π΄ and π΅, we could work out the length π΄π΅, which would also be of a length π₯.

An alternative way to consider this would be to think that the ratio of π΄π΅ to π΅πΆ would be one to one. So letβs now consider our length π΄π΅. And we can do this by considering how we go from π΄ to π΅ with our coordinates. If we look at the π₯-coordinate of point π΄ as negative one and we go to our π₯-coordinate of π΅, which is negative seven. This means that we would move negative six horizontally. If we consider our π¦-values, then we have negative two on our π΄-coordinate up to seven on our π΅-coordinate, which represents a move of nine vertically. So now as we know that the ratio of π΄π΅ to π΅πΆ is one to one. This means that the journey from π΅ to πΆ must also be negative six horizontally and nine vertically. So to find our π₯-value in coordinate πΆ, we go from the π₯-value in π΅, which is negative seven. And we subtract six since we had a negative six movement horizontally. So our π₯-value would be negative 13. For our π¦-values then, our π¦-value of π΅ was at seven. And we must add nine since we move nine vertically. And since seven add nine is 16, this gives us the π¦-value of 16.

So the coordinate of πΆ is negative 13, 16.