### Video Transcript

If the coordinates of ๐ด and ๐ต are five, five and negative one, negative four, respectively, find the coordinates of the point ๐ถ that divides ๐ด๐ต internally in the ratio two to one.

So letโs just picture the situation here. We have two points, ๐ด and ๐ต, whose relative positions look something like this. Weโre looking to find the coordinates of a point ๐ถ that divides ๐ด๐ต internally in the ratio two to one. So ๐ถ is a point somewhere along the length of ๐ด๐ต. And the length of the line segment ๐ด๐ถ is twice as long as the length of the segment ๐ต๐ถ.

Another way of phrasing this is that ๐ถ is two-thirds of the way along ๐ด๐ต. As if ๐ด๐ต is divided into three equal parts, then two of them are on one side of ๐ถ and one of them is on the other side.

To answer this question, Iโm going to think about how we get from ๐ด to ๐ต in terms of how the coordinates change. First of all, Iโll consider the horizontal change. At ๐ด, the ๐ฅ-coordinate is five. And at ๐ต, itโs negative one. So this is a change of negative six. We move six units to the left.

Next, letโs consider the vertical change. At ๐ด, the ๐ฆ-coordinate is five. And at ๐ต, itโs negative four. So this is a change of negative nine. We move nine units down. Letโs 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, ๐ถ divides this line in the ratio two to one. So to get from ๐ด to ๐ถ, 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 ๐ถ, we can therefore apply this transformation to the coordinates of ๐ด. If weโre moving four units to the left, we need to subtract four from the ๐ฅ-coordinate. So we have five minus four. And if weโre moving six units down, we need to subtract six from the ๐ฆ-coordinate. So we have five minus six. This gives the coordinates of ๐ถ, which are one, negative one.