Partitioning a Segment in a Given Ratio

Suppose you have a line segment on the coordinate plane, and you need to find the point on the segment of the way from to .

Let’s first take the easy case where is at the origin and line segment is a horizontal one.

The length of the line is units and the point on the segment of the way from to would be units away from , units away from and would be at .

Consider the case where the segment is not a horizontal or vertical line.

The components of the directed segment are and we need to find the point, say on the segment of the way from to .

Then, the components of the segment are .

Since the initial point of the segment is at origin, the coordinates of the point are given by .

Now let’s do a trickier problem, where neither nor is at the origin.

Use the end points of the segment to write the components of the directed segment.

Now in a similar way, the components of the segment where is a point on the segment of the way from to are .

To find the coordinates of the point add the components of the segment to the coordinates of the initial point .

So, the coordinates of the point are .

Note that the resulting segments, and , have lengths in a ratio of .

In general: what if you need to find a point on a line segment that divides it into two segments with lengths in a ratio ?

Consider the directed line segment with coordinates of the endpoints as and .

Suppose the point divided the segment in the ratio , then the point is of the way from to .

So, generalizing the method we have, the components of the segment are .

Then, the -coordinate of the point is

.

Similarly, the -coordinate is

.

Therefore, the coordinates of the point are .

Example 1:

Find the coordinates of the point that divides the directed line segment with the coordinates of endpoints at and in the ratio ?

Let be the point that divides in the ratio .

Here, and .

Substitute in the formula. The coordinates of are

.

Simplify.

Therefore, the point divides in the ratio .

Example 2:

What are the coordinates of the point that divides the directed line segment in the ratio ?

Let be the point that divides in the ratio .

Here, and .

Substitute in the formula. The coordinates of are

.

Simplify.

Therefore, the point divides in the ratio .

You can note that the Midpoint Formula is a special case of this formula when .