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 \(\left({x_1,y_1,z_1}\right)\) and \(\left({x_2,y_2,z_2}\right)\) respectively. If the lambda number λ is known, then the coordinates \(\left({x,y,z}\right)\) of point M are given by
where \(\lambda \ne -1.\)
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
Example 1.
Divide a rod \(36\text{ cm}\) long in a ratio of \(3:4:5.\)
Solution.
Let's denote the division points as \(K\left({x_K}\right)\) and \(M\left({x_M}\right).\) The boundaries of the rod are defined by points \(A\left({0}\right)\) and \(B\left({36}\right).\)
The point K divides the rod into \(2\) parts in a ratio of \(\lambda_K = \frac{3}{4+5} = \frac{1}{3}.\) The coordinate of this point is
The point M divides the rod in a ratio of \(\lambda_M = \frac{3 + 4}{5} = \frac{7}{5}.\) Its coordinate is
The points \(K\left({9}\right),\) \(M\left({21}\right)\) divide the rod into \(3\) parts with a length of \(9, 12,\) and \(15\text{ cm}.\)
Example 2.
The line segment bounded by points \(A\left({-3,1,-2}\right)\) and \(B\left({6,7,1}\right)\) is divided into three equal parts. Determine the coordinates of the division points.
Solution.
Let the division points be \(K\left({x_K,y_K,z_K}\right)\) and \(M\left({x_M,y_M,z_M}\right).\) If point K is located at a distance of \(\frac{1}{3}AB\) from point A, then \(\lambda_K = \frac{1}{2}\) for this point. Respectively, for the point M we obtain \(\lambda_M = 2.\)
Calculate the coordinates of point K:
Similarly, we find the coordinates of point M:
So the division points have coordinates \(K\left({0,3,-1}\right),\) \(M\left({3,5,0}\right).\)
Example 3.
Points \(K\left({2,3}\right),\) \(L\left({6,4}\right)\) and \(M\left({5,2}\right)\) are the midpoints of the sides of a triangle. Find the coordinates of its vertices.
Solution.
Let \(A\left({x_A,y_A}\right),\) \(B\left({x_B,y_B}\right),\) \(C\left({x_C,y_C}\right)\) 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 \(x_A,\) \(x_B,\) \(x_C.\)
We express \(x_A\) from the first equation and substitute in the third equation:
Add the second and third equation to eliminate \(x_B:\)
Now we know \(x_C\) and can find \(x_B\) and \(x_A:\)
Similarly, one can determine the \(y-\)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: \(A\left({1,1}\right),\) \(B\left({3,5}\right),\) \(C\left({9,3}\right).\)
Example 4.
Given the coordinates of two vertices of a triangle \(A\left({2,-1}\right),\) \(B\left({-3,5}\right)\) and the point of intersection of medians \(M\left({1,1}\right),\) find the coordinates of the vertex C.
Solution.
In a triangle, the point of intersection of its medians (centroid) divides each median in the ratio \(2:1.\) 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 \(x_K, y_K:\)
Find now the coordinates of the vertex C.
Hence
Thus the vertex C has coordinates \(\left({4,-1}\right).\)