In coordinate geometry, the Section formula is a formula used to find the ratio in which a line segment is divided by a point internally or externally. It is used to find out the centroid, incenter and excenters of a triangle. In physics, it is used to find the center of mass of systems, equilibrium points, etc.
If point P (lying on AB) divides the line segment AB joining the points and in the ratio m:n, then
The ratio m:n can also be written as , or , where . So, the coordinates of point dividing the line segment joining the points and are:
Similarly, the ratio can also be written as , and the coordinates of P are .
Triangles .
If a point P (lying on the extension of AB) divides AB in the ratio m:n then
Triangles (Let C and D be two points where A & P and B & P intersect respectively). Therefore â ACP = â BDP
The midpoint of a line segment divides it internally in the ratio . Applying the Section formula for internal division:
The centroid of a triangle is the intersection of the medians and divides each median in the ratio . Let the vertices of the triangle be , and . So, a median from point A will intersect BC at . Using the section formula, the centroid becomes:
Let A and B be two points with Cartesian coordinates (x<sub>1</sub>, y<sub>1</sub>, z<sub>1</sub>) and (x<sub>2</sub>, y<sub>2</sub>, z<sub>2</sub>) and P be a point on the line through A and B. If . Then the section formula gives the coordinates of P as
If, instead, P is a point on the line such that , its coordinates are .
The position vector of a point P dividing the line segment joining the points A and B whose position vectors are and