Section Formula

We will proof the definition of section formula.

Section of a Line Segment

Let AB be a line segment joining the points A and B. Let P be any point on the line segment such that AP : PB = λ : 1

Then, we can say that P divides internally AB is the ratio λ : 1.

Note: If AP : PB = m : n then AP : PB = $\frac{m}{n}$ : 1 (since m : n = $\frac{m}{n}$ : $\frac{n}{n}$. So, any section by P can be expressed as AP : PB = λ : 1

Definition of section formula: The coordinates (x, y) of a point P divides the line segment joining A (x$_{1}$, y$_{1}$) and B (x$_{2}$, y$_{2}$) internally in the ratio m : n (i.e., $\frac{AP}{PB}$ = $\frac{m}{n}$) are given by

x = ($\frac{mx_{2} + nx_{1}}{m + n}$, y = $\frac{my_{2} + ny_{1}}{m + n}$)

Proof:

Let X’OX and YOY’ are the co-ordinate axes.

Let A (x$_{1}$, y$_{1}$) and B (x$_{2}$, y$_{2}$) be the end points of the given line segment AB.

Let P(x, y) be the point which divides AB in the ratio m : n.

Then, $\frac{AP}{PB}$ = $\frac{m}{n}$)

We want to find the coordinates (x, y) of P.

Draw AL ⊥ OX; BM ⊥ OX; PN ⊥ OX; AR ⊥ PN; and PS ⊥ BM

AL = y$_{1}$, OL = x$_{1}$, BM = y$_{2}$, OM = x$_{2}$, PN = y and ON = x.

By geometry,

AR = LN = ON – OL = (x - x$_{1}$);

PS = NM = OM – ON = (x$_{2}$ -  x);

PR = PN – RN = PN – AL = (y - y$_{1}$)

BS = BM – SM = BM – PN = (y$_{2}$ - y)

Clearly, we see that triangle ARP and triangle PSB are similar and, therefore, their sides are proportional.

Thus, $\frac{AP}{PB}$ = $\frac{AR}{PS}$ = $\frac{PR}{BS}$

⟹ $\frac{m}{n}$ = $\frac{x - x_{1}}{x_{2} - x}$ = $\frac{y - y_{1}}{y_{2} - y}$

⟹ $\frac{m}{n}$ = $\frac{x - x_{1}}{x_{2} - x}$ and $\frac{m}{n}$ = $\frac{y - y_{1}}{y_{2} - y}$

⟹ (m + n)x = (mx$_{2}$ + nx$_{1}$) and (m + n)y = (my$_{2}$ + ny$_{1}$)

⟹ x = ($\frac{mx_{2} + nx_{1}}{m + n}$ and y = $\frac{my_{2} + ny_{1}}{m + n}$)

Therefore, the co-ordinates of P are  ($\frac{mx_{2} + nx_{1}}{m + n}$,  $\frac{my_{2} + ny_{1}}{m + n}$).

● Distance and Section Formulae

• Distance Formula
• Distance Properties in some Geometrical Figures
• Conditions of Collinearity of Three Points
• Problems on Distance Formula
• Distance of a Point from the Origin
• Distance Formula in Geometry
• Section Formula
• Midpoint Formula
• Centroid of a Triangle
• Worksheet on Distance Formula
• Worksheet on Collinearity of Three Points
• Worksheet on Finding the Centroid of a Triangle
• Worksheet on Section Formula

10th Grade Math

From Section Formula to HOME