### Video Transcript

Given points π΄ three, negative two and π΅ negative two, four, find the coordinates of point πΆ, which divides the line π΄π΅ externally in the ratio four to three.

Weβve been given the coordinates of two points π΄ and π΅ and told that a third point πΆ divides π΄π΅ externally in the ratio four to three. This means that point πΆ is not on the line segment joining points π΄ and π΅ but on the continuation of this line, so point πΆ is somewhere over here.

Now letβs think about what it means if point πΆ divides π΄π΅ externally in the ratio four to three. Well, it means that the ratio of the length of π΄πΆ to the length of π΅πΆ is four to three. We can approach this problem in two ways: formally using the section formula with external division and then a more informal logical approach.

Letβs consider the section formula first. This states that for distinct points π΄ with the coordinates π₯ one, π¦ one and π΅ with coordinates π₯ two, π¦ two, if the point π, which does not lie on the line segment π΄π΅, divides the line π΄π΅ such that the ratio of the length of π΄π to the length of ππ΅ is π to π, then π has coordinates ππ₯ two minus ππ₯ one over π minus π, ππ¦ two minus ππ¦ one over π minus π.

Letβs see if we can work out the values of π₯ one, π¦ one, π₯ two, π¦ two, π, and π for this problem. π΄ is the point π₯ one, π¦ one, and π΅ is the point π₯ two, π¦ two. We then have the statement that the ratio of the length of π΄π to the length of ππ΅ is π to π. Well, in this case, that would be the ratio the length of π΄πΆ to the length of πΆπ΅. And weβve already written down that the ratio of the length of π΄πΆ to the length of π΅πΆ, which is the same line segment but just traveling in the opposite direction, is four to three. So this tells us that the value of π is four and the value of π is three.

We can now apply the section formula with external division to determine the coordinates of point πΆ. For the π₯-coordinate of point πΆ, we have π multiplied by π₯ two, thatβs four multiplied by negative two, minus π multiplied by π₯ one, thatβs three multiplied by three, over π minus π, which is four minus three. And then for the π¦-coordinate, π multiplied by π¦ two is four multiplied by four, minus π multiplied by π¦ one, thatβs three multiplied by negative two, over π minus π, which is four minus three.

Simplifying, we have negative eight minus nine over one for the π₯-coordinate of πΆ and 16 plus six over one for the π¦-coordinate, which further simplifies to the point with coordinates negative 17, 22.

So weβve used the formal approach to this question, but letβs also think about a slightly less formal approach that we could take. If the ratio of the length of the line segment π΄πΆ to the length of the line segment π΅πΆ is four to three, then the ratio of the length of π΄π΅ to π΅πΆ is one to three. We can then think about how far we move and in what direction to get from point π΄ to point π΅. Well, we need to move five units to the left so that the π₯-coordinate changes from the three to negative two and six units up so that the π¦-coordinate changes from negative two to four.

As π΅πΆ is three times as long as π΄π΅, we have to move three times these distances in the same direction to get from point π΅ to point πΆ. So we have to move 15 units to the left and 18 units up. We can then work out the coordinates of point πΆ by subtracting 15 from the π₯-coordinate of point π΅ and adding 18 to the π¦-coordinate. That gives negative two minus 15 for the π₯-coordinate of point πΆ and four plus 18 for the π¦-coordinate, which once again gives the point with coordinates negative 17, 22.

So, using the section formula with external division and a slightly more informal approach, weβve shown that the coordinates of point πΆ, which divides the line π΄π΅ externally in the ratio four to three, are negative 17, 22.