### Video Transcript

Consider points π΄ two, three and π΅ negative four, negative three. 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 π΄π΅.

So the first thing Iβve drawn is Iβve drawn the points π΄ and π΅, and then also what weβve drawn here is the line segment π΄π΅. But now what we need to do is think about where the point πΆ is. Well, weβre told that itβs on the ray π΅π΄ but not on the line segment π΄π΅. But what does the ray π΅π΄ mean? Well, if we know that πΆ is on the ray π΅π΄, this means that from π΅ going towards π΄, πΆ is gonna lie along this line somewhere. However, we know that itβs not on the segment π΄π΅, so we know that itβs not in our line segment π΄π΅. So that means itβs not where the blue area is, so we know that it must extend beyond π΄.

So what Iβve drawn here is a sketch of the scenario, just looking at the idea of a line. So weβre told that π΄ to πΆ is equal to two π΄π΅. So therefore, we know that our line is gonna be in the ratio one to two. And thatβs if we have the points π΅, π΄, and πΆ in that order. Now, the way we can approach this is to look at our π₯-coordinates and then look at our π¦-coordinates. So we start by looking at our π₯-coordinates. So what weβve done is labeled our coordinates π₯ sub two, π¦ sub two and π₯ sub one, π¦ sub one. Iβve just done it in this way around because weβre going from π΅ to π΄.

So to find the difference between our π₯-coordinates, what weβre gonna do is π₯ sub two minus π₯ sub one, which is gonna be equal two minus negative four. Well, if we subtract a negative, itβs the same as adding a positive, so we get two plus four which is equal to six. So we know that the difference between π΅ and π΄ in the π₯-coordinate is six units. So now if we consider our ratio, which is one to two, we can see that the difference between the π₯-coordinates between π΄ and πΆ is gonna be twice that of π΅ to π΄, so we make two multiplied by six. So therefore, itβs going to be 12.

So therefore, if we want to find out π₯ which weβre calling the π₯-coordinate of πΆ, then this is going to be the π₯-coordinate of π΄ which is two plus 12 because thatβs the difference between them, which is going to be equal to 14. Okay, great, so now weβre gonna look at the π¦-coordinates. So now what we want to do is find the difference between the π¦-coordinates. Itβs gonna be π¦ sub two minus π¦ sub one, which is gonna be three minus negative three. Well, if you subtract a negative, itβs the same as adding a positive like we said before.

So once again, we can see the difference this time between the π¦-coordinates is six. So therefore, using the same rationale as before, thatβs our ratio one to two, we can see that, therefore, the difference between the π¦-coordinates of π΄ and πΆ must be double that. So itβs gonna be 12. So therefore, the π¦-coordinate of πΆ is going to be three, which was the π¦-coordinate of π΄ plus 12 which is gonna be equal to 15. So therefore, we can say that the coordinates of point πΆ given that πΆ is on the ray π΅π΄ but not on the segment π΄ to π΅ and π΄πΆ is equal to two π΄π΅ are 14, 15.