Section formula (Point that divides a line in given ratio)

Given two coordinates (x1, y1) and (x2, y2), and m and n, find the co-ordinates that divides the line joining (x1, y1) and (x2, y2) in the ratio m : n

👁 Section formula problems

Examples:

Input : x1 = 1, y1 = 0, x2 = 2 y2 = 5,
 m = 1, n = 1
Output : (1.5, 2.5)
Explanation: co-ordinates (1.5, 2.5)
divides the line in ratio 1 : 1

Input : x1 = 2, y1 = 4, x2 = 4, y2 = 6,
 m = 2, n = 3
Output : (2.8, 4.8)
Explanation: (2.8, 4.8) divides the line
in the ratio 2:3

The section formula tells us the coordinates of the point which divides a given line segment into two parts such that their lengths are in the ratio m : n

👁 Section formula

Output: 

(2.8, 4.8)

Time Complexity: O(1) 
Auxiliary Space: O(1)

How does this work?

👁 Section formula working

From our diagram, we can see,
PS = x – x1 and RT = x2 – x

We are given,

PR/QR = m/n

Using similarity, we can write
RS/QT = PS/RT = PR/QR

Therefore, we can write
 PS/RR = m/n
 (x - x1) / (x2 - x) = m/n

From above, we get
 x = (mx2 + nx1) / (m + n)

Similarly, we can solve for y.

References:
http://doubleroot.in/lessons/coordinate-geometry-basics/section-formula/#.WjYXQvbhU8o