Division of a Line Segment

Division of a Line Segment in a Given Ratio

Let's consider in space two different points A and B and a straight line defined by these points. Let M be any point of the indicated line other than B. The point M divides the line segment AB in a ratio λ which is defined by the formula

Suppose the points A and B have coordinates $$ and $$ respectively. If the lambda number λ is known, then the coordinates $$ of point M are given by

where $$

For positive lambda values the point M lies between the points A and B (Figure 1), and for negative values it lies outside the line segment AB (Figure 2).

Division of a Line Segment in Half

It is obvious that if the point M divides the segment AB in half, then

In this case, the coordinates of the point M are

Solved Problems

Solution.

Let's denote the division points as $$ and $$ The boundaries of the rod are defined by points $$ and $$

The point K divides the rod into $$ parts in a ratio of $$ The coordinate of this point is

The point M divides the rod in a ratio of $$ Its coordinate is

The points $$ $$ divide the rod into $$ parts with a length of $$ and $$

Solution.

Let the division points be $$ and $$ If point K is located at a distance of $$ from point A, then $$ for this point. Respectively, for the point M we obtain $$

Calculate the coordinates of point K:

Similarly, we find the coordinates of point M:

So the division points have coordinates $$ $$

Solution.

Let $$ $$ $$ be the vertices of the triangle.

If K is the midpoint of side AB, L is the midpoint of side BC and M is the midpoint of side AC, then we can write

From the first three equations of the system we can determine the coordinates $$ $$ $$

We express $$ from the first equation and substitute in the third equation:

Add the second and third equation to eliminate $$

Now we know $$ and can find $$ and $$

Similarly, one can determine the $$coordinates. We obtain the following final formulas:

Now you can easily calculate the coordinates of the vertices of the triangle:

So the vertices of the triangle have the following coordinates: $$ $$ $$

Solution.

In a triangle, the point of intersection of its medians (centroid) divides each median in the ratio $$ Hence, we have

where K is the midpoint of side BC.

We denote the coordinates of the points in the figure as follows:

Using the formulas for division of a line segment, we get

Solve these equations for $$

Find now the coordinates of the vertex C.

Hence

Thus the vertex C has coordinates $$