### Video Transcript

If the coordinates of π΄ and π΅ are five, five and negative one, negative four, respectively, find the coordinates of the point πΆ that divides π΄π΅ internally in the ratio two to one.

So letβs just picture the situation here. We have two points, π΄ and π΅, whose relative positions look something like this. Weβre looking to find the coordinates of a point πΆ that divides π΄π΅ internally in the ratio two to one. So πΆ is a point somewhere along the length of π΄π΅. And the length of the line segment π΄πΆ is twice as long as the length of the segment π΅πΆ.

Another way of phrasing this is that πΆ is two-thirds of the way along π΄π΅. As if π΄π΅ is divided into three equal parts, then two of them are on one side of πΆ and one of them is on the other side.

To answer this question, Iβm going to think about how we get from π΄ to π΅ in terms of how the coordinates change. First of all, Iβll consider the horizontal change. At π΄, the π₯-coordinate is five. And at π΅, itβs negative one. So this is a change of negative six. We move six units to the left.

Next, letβs consider the vertical change. At π΄, the π¦-coordinate is five. And at π΅, itβs negative four. So this is a change of negative nine. We move nine units down. Letβs think about this ratio then, which has three equal parts. Each horizontal and vertical move will be one-third of the total horizontal and vertical move.

Dividing negative six by three, we have negative two. And dividing negative nine by three, we have negative three. So each part of this ratio is two units to the left and three units down. Remember, πΆ divides this line in the ratio two to one. So to get from π΄ to πΆ, we actually move two parts of the ratio. Therefore, we need to move four units to the left and six units down.

To find the coordinates of πΆ, we can therefore apply this transformation to the coordinates of π΄. If weβre moving four units to the left, we need to subtract four from the π₯-coordinate. So we have five minus four. And if weβre moving six units down, we need to subtract six from the π¦-coordinate. So we have five minus six. This gives the coordinates of πΆ, which are one, negative one.