Examples of Section Formula Given Midpoitn

In coordinate geometry, Section formula is 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 [edit]

Internal division with section formula

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

${\displaystyle P=\left({\dfrac {mx_{2}+nx_{1}}{m+n}},{\dfrac {my_{2}+ny_{1}}{m+n}}\right)}$ ^[6]

The ratio m:n can also be written as ${\displaystyle m/n:1}$ , or ${\displaystyle k:1}$ , where ${\displaystyle k=m/n}$ . So, the coordinates of point ${\displaystyle P}$ dividing the line segment joining the points ${\displaystyle \mathrm {A} (x_{1},y_{1})}$ and ${\displaystyle \mathrm {B} (x_{2},y_{2})}$ are:

${\displaystyle \left({\dfrac {mx_{2}+nx_{1}}{m+n}},{\dfrac {my_{2}+ny_{1}}{m+n}}\right)}$

${\displaystyle =\left({\frac {{\frac {m}{n}}x_{2}+x_{1}}{{\frac {m}{n}}+1}},{\frac {{\frac {m}{n}}y_{2}+y_{1}}{{\frac {m}{n}}+1}}\right)}$

${\displaystyle =\left({\frac {kx_{2}+x_{1}}{k+1}},{\frac {ky_{2}+y_{1}}{k+1}}\right)}$ ^{[4] [5]}

Similarly, the ratio can also be written as ${\displaystyle k:k-1}$ , and the coordinates of P are ${\displaystyle ((1-k)x_{1}+kx_{2},(1-k)y_{1}+ky_{2})}$ .^[1]

Proof [edit]

This section is empty. You can help by adding to it. (August 2021)

External Divisions [edit]

External division with section formula

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

${\displaystyle P=\left({\dfrac {mx_{2}-nx_{1}}{m-n}},{\dfrac {my_{2}-ny_{1}}{m-n}}\right)}$ ^[6]

Proof [edit]

This section needs expansion. You can help by adding to it. (October 2020)

Midpoint formula [edit]

The midpoint of a line segment divides it internally in the ratio ${\textstyle 1:1}$ . Applying the Section formula for internal division:^{[4] [5]}

$${\displaystyle P=\left({\dfrac {x_{1}+x_{2}}{2}},{\dfrac {y_{1}+y_{2}}{2}}\right)}$$

Derivation [edit]

${\displaystyle P=\left({\dfrac {mx_{2}+nx_{1}}{m+n}},{\dfrac {my_{2}+ny_{1}}{m+n}}\right)}$

${\displaystyle =\left({\frac {1\cdot x_{1}+1\cdot x_{2}}{1+1}},{\frac {1\cdot y_{1}+1\cdot y_{2}}{1+1}}\right)}$

${\displaystyle =\left({\dfrac {x_{1}+x_{2}}{2}},{\dfrac {y_{1}+y_{2}}{2}}\right)}$

In 3-Dimensions [edit]

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 ${\displaystyle AP:PB=m:n}$ . Then the section formulae give the coordinates of P as

${\displaystyle \left({\frac {mx_{2}+nx_{1}}{m+n}},{\frac {my_{2}+ny_{1}}{m+n}},{\frac {mz_{2}+nz_{1}}{m+n}}\right)}$ ^[7]

If, instead, P is a point on the line such that ${\displaystyle AP:PB=k:1-k}$ , its coordinates are ${\displaystyle ((1-k)x_{1}+kx_{2},(1-k)y_{1}+ky_{2},(1-k)z_{1}+kz_{2})}$ .^[7]

In vectors [edit]

The position vector of a point P dividing the line segment joining the points A and B whose position vectors are ${\displaystyle {\vec {a}}}$ and ${\displaystyle {\vec {b}}}$

1. in the ratio ${\displaystyle m:n}$ internally, is given by ${\displaystyle {\frac {n{\vec {a}}+m{\vec {b}}}{m+n}}}$ ^{[8] [1]}
2. in the ratio ${\displaystyle m:n}$ externally, is given by ${\displaystyle {\frac {m{\vec {b}}-n{\vec {a}}}{m-n}}}$ ^[8]

See also [edit]

• Cross-section Formula

• Distance Formula
• Midpoint Formula

References [edit]

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/acref-9780199679591-e-2546, ISBN978-0-19-967959-1 , 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
4. ^ ^{a b c} Aggarwal, R.S. Secondary School Mathematics for Class 10. Bharti Bhawan Publishers & Distributors (1 January 2020). ISBN978-9388704519.
5. ^ ^{a b c} Sharma, R.D. Mathematics for Class 10. Dhanpat Rai Publication (1 January 2020). ISBN978-8194192640.
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/acref-9780199679591-e-2547, ISBN978-0-19-967959-1 , retrieved 2020-10-30
8. ^ ^{a b} https://ncert.nic.in/ncerts/l/leep210.pdf

External links [edit]

section-formula by GeoGebra

Examples of Section Formula Given Midpoitn

Source: https://en.wikipedia.org/wiki/Section_formula

Share this post