The Point Dividing a Line in a Given Ratio

Understanding The Concept

The concept of Point Dividing a Line in a given Ratio is involved in finding the coordinates of a point that divides a line segment in a certain ratio.
This concept is based on the principle of similar triangles and relates to Section Formula.
The points of consideration are: point P(x1, y1) and point Q(x2, y2). We are looking for a point R that divides PQ in the ratio m:n.
If m and n are the said ratio, the coordinates of the point R(x, y) that divides the line segment PQ into ration m:n are calculated as follows:

x = (m*x2 + n*x1) / (m + n), y = (m*y2 + n*y1) / (m + n)

The formula calculates the weighted average of the coordinates.
The term (m*x2 + n*x1) gets the sum of the product of the x-coordinate of Q and m with the product of the x-coordinate of P and n.
Dividing this sum by (m + n) gives us the x-coordinate of point R.
The same process is repeated to find the y-coordinate of point R.

As an example, find the coordinates of a point R that divides the line segment joining A(3,2) and B(5,6) in the ratio 2:3. Substitute values in the formula: R(x, y) = [(2*5 + 3*3) / (2 + 3), (2*6 + 3*2) / (2 + 3)] = [16/5, 18/5 ] = [3.2, 3.6]
It’s important to note that coordinates could be fractional or decimal.
The line segment can be divided into two regions; either internally or externally. The above formula is for the internal division. In terms of external division, the formula for R(x, y) becomes:

x = (m*x2 - n*x1) / (m - n), y = (m*y2 - n*y1) / (m - n)

The formula used for Point Dividing a Line in a Given Ratio is applicable in numerous problems in coordinates and geometric shapes.
The concept is similar to a weighted average calculation.
It is not just enough to memorize the formulas. Understanding the logic behind them will greatly help in solving complex problems.