Question Video: Finding the Coordinates of a Point That Divides a Line Segment Externally in a Given Ratio | Nagwa Question Video: Finding the Coordinates of a Point That Divides a Line Segment Externally in a Given Ratio | Nagwa

Question Video: Finding the Coordinates of a Point That Divides a Line Segment Externally in a Given Ratio Mathematics • First Year of Secondary School

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Given points 𝐴(3, βˆ’2) and 𝐡(βˆ’2, 4), find the coordinates of point 𝐢 which divides the line 𝐴𝐡 externally in the ratio 4 : 3.

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Video Transcript

Given points 𝐴 three, negative two and 𝐡 negative two, four, find the coordinates of point 𝐢, which divides the line 𝐴𝐡 externally in the ratio four to three.

We’ve been given the coordinates of two points 𝐴 and 𝐡 and told that a third point 𝐢 divides 𝐴𝐡 externally in the ratio four to three. This means that point 𝐢 is not on the line segment joining points 𝐴 and 𝐡 but on the continuation of this line, so point 𝐢 is somewhere over here.

Now let’s think about what it means if point 𝐢 divides 𝐴𝐡 externally in the ratio four to three. Well, it means that the ratio of the length of 𝐴𝐢 to the length of 𝐡𝐢 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.

Let’s consider the section formula first. This states that for distinct points 𝐴 with the coordinates π‘₯ one, 𝑦 one and 𝐡 with coordinates π‘₯ two, 𝑦 two, if the point 𝑃, which does not lie on the line segment 𝐴𝐡, divides the line 𝐴𝐡 such that the ratio of the length of 𝐴𝑃 to the length of 𝑃𝐡 is π‘š to 𝑛, then 𝑃 has coordinates π‘šπ‘₯ two minus 𝑛π‘₯ one over π‘š minus 𝑛, π‘šπ‘¦ two minus 𝑛𝑦 one over π‘š minus 𝑛.

Let’s see if we can work out the values of π‘₯ one, 𝑦 one, π‘₯ two, 𝑦 two, π‘š, and 𝑛 for this problem. 𝐴 is the point π‘₯ one, 𝑦 one, and 𝐡 is the point π‘₯ two, 𝑦 two. We then have the statement that the ratio of the length of 𝐴𝑃 to the length of 𝑃𝐡 is π‘š to 𝑛. Well, in this case, that would be the ratio the length of 𝐴𝐢 to the length of 𝐢𝐡. And we’ve already written down that the ratio of the length of 𝐴𝐢 to the length of 𝐡𝐢, 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‘š is four and the value of 𝑛 is three.

We can now apply the section formula with external division to determine the coordinates of point 𝐢. For the π‘₯-coordinate of point 𝐢, we have π‘š multiplied by π‘₯ two, that’s four multiplied by negative two, minus 𝑛 multiplied by π‘₯ one, that’s three multiplied by three, over π‘š minus 𝑛, which is four minus three. And then for the 𝑦-coordinate, π‘š multiplied by 𝑦 two is four multiplied by four, minus 𝑛 multiplied by 𝑦 one, that’s three multiplied by negative two, over π‘š minus 𝑛, which is four minus three.

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

So we’ve used the formal approach to this question, but let’s also think about a slightly less formal approach that we could take. If the ratio of the length of the line segment 𝐴𝐢 to the length of the line segment 𝐡𝐢 is four to three, then the ratio of the length of 𝐴𝐡 to 𝐡𝐢 is one to three. We can then think about how far we move and in what direction to get from point 𝐴 to point 𝐡. 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 𝑦-coordinate changes from negative two to four.

As 𝐡𝐢 is three times as long as 𝐴𝐡, we have to move three times these distances in the same direction to get from point 𝐡 to point 𝐢. So we have to move 15 units to the left and 18 units up. We can then work out the coordinates of point 𝐢 by subtracting 15 from the π‘₯-coordinate of point 𝐡 and adding 18 to the 𝑦-coordinate. That gives negative two minus 15 for the π‘₯-coordinate of point 𝐢 and four plus 18 for the 𝑦-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, we’ve shown that the coordinates of point 𝐢, which divides the line 𝐴𝐡 externally in the ratio four to three, are negative 17, 22.

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