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 π΄(3, β2) and π΅(β2, 4), find the coordinates of point πΆ which divides the line π΄π΅ externally in the ratio 4 : 3.

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### Video Transcript

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

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

Now letβs think about what it means if point πΆ divides π΄π΅ externally in the ratio four to three. Well, it means that the ratio of the length of π΄πΆ to the length of π΅πΆ 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.

Letβs 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 ππ΅ is π to π, then π has coordinates ππ₯ two minus ππ₯ one over π minus π, ππ¦ two minus ππ¦ one over π minus π.

Letβs 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 ππ΅ is π to π. Well, in this case, that would be the ratio the length of π΄πΆ to the length of πΆπ΅. And weβve already written down that the ratio of the length of π΄πΆ to the length of π΅πΆ, 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 πΆ. For the π₯-coordinate of point πΆ, we have π multiplied by π₯ two, thatβs four multiplied by negative two, minus π multiplied by π₯ one, thatβs 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, thatβs 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 πΆ and 16 plus six over one for the π¦-coordinate, which further simplifies to the point with coordinates negative 17, 22.

So weβve used the formal approach to this question, but letβs also think about a slightly less formal approach that we could take. If the ratio of the length of the line segment π΄πΆ to the length of the line segment π΅πΆ is four to three, then the ratio of the length of π΄π΅ to π΅πΆ 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΅πΆ 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 πΆ. So we have to move 15 units to the left and 18 units up. We can then work out the coordinates of point πΆ 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 πΆ 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, weβve shown that the coordinates of point πΆ, which divides the line π΄π΅ externally in the ratio four to three, are negative 17, 22.

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