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

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Internal division with section formula

If point P (lying on AB) divides the line segment AB joining the points   and   in the ratio m:n, then

 [6]

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:

 

 

 [4][5]

Similarly, the ratio can also be written as  , and the coordinates of P are  .[1]

Proof

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Triangles  .

 

External Divisions

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External division with section formula

If a point P (lying on the extension of AB) divides AB in the ratio m:n then

 [6]

Proof

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Triangles   (Let C and D be two points where A & P and B & P intersect respectively). Therefore ∠ACP = ∠BDP

 


Midpoint formula

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The midpoint of a line segment divides it internally in the ratio  . Applying the Section formula for internal division:[4][5]

 

Derivation

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Centroid

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Centroid of a triangle

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

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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

 [7]

If, instead, P is a point on the line such that  , its coordinates are  .[7]

In vectors

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The position vector of a point P dividing the line segment joining the points A and B whose position vectors are   and  

  1. in the ratio   internally, is given by  [8][1]
  2. in the ratio   externally, is given by  [8]

See also

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References

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  1. ^ 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
  2. ^ "Section Formula | Brilliant Math & Science Wiki". brilliant.org. Retrieved 2020-10-16.
  3. ^ https://ncert.nic.in/ncerts/l/jemh107.pdf [bare URL PDF]
  4. ^ a b c Aggarwal, R.S. Secondary School Mathematics for Class 10. Bharti Bhawan Publishers & Distributors (1 January 2020). ISBN 978-9388704519.
  5. ^ a b c Sharma, R.D. Mathematics for Class 10. Dhanpat Rai Publication (1 January 2020). ISBN 978-8194192640.
  6. ^ a b Loney, S L. The Elements of Coordinate Geometry (Part-1).
  7. ^ 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
  8. ^ a b https://ncert.nic.in/ncerts/l/leep210.pdf [bare URL PDF]
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