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.
Internal divisions
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 .
Proof Triangles .
External divisions
If a point P (lying on the extension of AB) divides AB in the ratio m:n then
Proof Triangles (Let C and D be two points where A & P and B & P intersect respectively). Therefore ∠ACP = ∠BDP
Midpoint formula
The midpoint of a line segment divides it internally in the ratio . Applying the Section formula for internal division:
Derivation
Centroid
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:
In three dimensions Let A and B be two points with Cartesian coordinates (x1, y1, z1) and (x2, y2, z2) 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 .
In vectors The position vector of a point P dividing the line segment joining the points A and B whose position vectors are and
in the ratio internally, is given by in the ratio externally, is given by
See also Cross-section Formula Distance Formula Midpoint Formula Projective harmonic conjugate
References
External links section-formula by GeoGebra