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
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.
Oct 08, 24 10:53 AM
Oct 07, 24 04:07 PM
Oct 07, 24 03:29 PM
Oct 07, 24 03:13 PM
Oct 07, 24 12:01 PM
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.