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.[1] 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.[2][3][4][5]
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 .[1]
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:[4][5]
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 3-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 formulae give the coordinates of P as
If, instead, P is a point on the line such that , its coordinates are .[7]
In vectors
The position vector of a point P dividing the line segment joining the points A and B whose position vectors are and
See also
References
- ^ a b c Clapham, Christopher; Nicholson, James (2014-09-18), "section formulae", The Concise Oxford Dictionary of Mathematics, Oxford University Press, doi:10.1093/acref/9780199679591.001.0001, ISBN 978-0-19-967959-1, retrieved 2020-10-30
- ^ "Section Formula | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-10-16.
- ^ https://ncert.nic.in/ncerts/l/jemh107.pdf [bare URL PDF]
- ^ a b c Aggarwal, R.S. Secondary School Mathematics for Class 10. Bharti Bhawan Publishers & Distributors (1 January 2020). ISBN 978-9388704519.
- ^ a b c Sharma, R.D. Mathematics for Class 10. Dhanpat Rai Publication (1 January 2020). ISBN 978-8194192640.
- ^ a b Loney, S L. The Elements of Coordinate Geometry (Part-1).
- ^ a b Clapham, Christopher; Nicholson, James (2014-09-18), "section formulae", The Concise Oxford Dictionary of Mathematics, Oxford University Press, doi:10.1093/acref/9780199679591.001.0001, ISBN 978-0-19-967959-1, retrieved 2020-10-30
- ^ a b https://ncert.nic.in/ncerts/l/leep210.pdf [bare URL PDF]