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If the coordinates of 𝐴 and 𝐵 are (5, 5) and (−1, −4) respectively, find the coordinates of the point 𝐶 that divides 𝐴𝐵 internally in the ratio 2 : 1.
If the coordinates of 𝐴 and 𝐵 are five, five and negative one, negative four, respectively, find the coordinates of the point 𝐶 that divides 𝐴𝐵 internally in the ratio two to one.
So let’s just picture the situation here. We have two points, 𝐴 and 𝐵, whose relative positions look something like this. We’re looking to find the coordinates of a point 𝐶 that divides 𝐴𝐵 internally in the ratio two to one. So 𝐶 is a point somewhere along the length of 𝐴𝐵. And the length of the line segment 𝐴𝐶 is twice as long as the length of the segment 𝐵𝐶.
Another way of phrasing this is that 𝐶 is two-thirds of the way along 𝐴𝐵. As if 𝐴𝐵 is divided into three equal parts, then two of them are on one side of 𝐶 and one of them is on the other side.
To answer this question, I’m going to think about how we get from 𝐴 to 𝐵 in terms of how the coordinates change. First of all, I’ll consider the horizontal change. At 𝐴, the 𝑥-coordinate is five. And at 𝐵, it’s negative one. So this is a change of negative six. We move six units to the left.
Next, let’s consider the vertical change. At 𝐴, the 𝑦-coordinate is five. And at 𝐵, it’s negative four. So this is a change of negative nine. We move nine units down. Let’s 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, 𝐶 divides this line in the ratio two to one. So to get from 𝐴 to 𝐶, 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 𝐶, we can therefore apply this transformation to the coordinates of 𝐴. If we’re moving four units to the left, we need to subtract four from the 𝑥-coordinate. So we have five minus four. And if we’re moving six units down, we need to subtract six from the 𝑦-coordinate. So we have five minus six. This gives the coordinates of 𝐶, which are one, negative one.
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