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

Section of a Line Segment

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






10th Grade Math

From Section Formula to HOME




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Word Problems on Dividing Money | Solving Money Division Word Problems

    Feb 13, 25 10:29 AM

    Word Problems on Dividing Money
    Read the questions given in the word problems on dividing money. We need to understand the statement and divide the amount of money as ordinary numbers with two digit numbers. 1. Ron buys 15 pens for…

    Read More

  2. Addition and Subtraction of Money | Examples | Worksheet With Answers

    Feb 13, 25 09:02 AM

    Add Money Method
    In Addition and Subtraction of Money we will learn how to add money and how to subtract money.

    Read More

  3. Worksheet on Division of Money | Word Problems on Division of Money

    Feb 13, 25 03:53 AM

    Division of Money Worksheet
    Practice the questions given in the worksheet on division of money. This sheet provides different types of questions on dividing the amount of money by a number; finding the quotient

    Read More

  4. Worksheet on Multiplication of Money | Word Problems | Answers

    Feb 13, 25 03:17 AM

    Worksheet on Multiplication of Money
    Practice the questions given in the worksheet on multiplication of money. This sheet provides different types of questions on multiplying the amount of money by a number; arrange in columns the amount…

    Read More

  5. Division of Money | Worked-out Examples | Divide the Amounts of Money

    Feb 13, 25 12:16 AM

    Divide Money
    In division of money we will learn how to divide the amounts of money by a number. We carryout division with money the same way as in decimal numbers. We put decimal point in the quotient after two pl…

    Read More