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 .