A_r = (X_r,Y_r), r=1,2,3………,n is a set of n points. P₁ divides A₁A₂ in the ratio 1:1, P₂ divides P₁A₃ in th ratio 1:2, P₃ divides P₂A₄ in the ratio 1:3 and so on . P_n-1 divides P_n-2A_n in the ratio 1:n-1, Find The Co-ordinates of P_n-1?

Hii vinay !

This is very nice question ....Try to understand this -

A1     P1         A2            P2           A3             P3            A4             P4               A5                  .........Pn-1.......  An

(x1,y1)          (x2,y2)                 (x3,y3)                                                                                                      (xn,yn) 

P1   divides A1 A2 in 1:1

so P1 = [(x1 +x2)/2 ,(y1+y2)/2 ]

P2 divides P1A3 is 1 : 2 ...

so P2 =  [2P1 + A3 ]/3   =   [ (x1+x2+x3)/3 , (y1+y2+y3)/3 ]

Similarly in P3  you will get   [ (x1+x2+x3+x4)/4 , (y1 + y2 + y3 +y4)/4]

And thus Pn-1 = [ (x1+x2+x3........+xn)/n , )y1+y2+y3........+yn)/n ]

Note : simply use the section formula which means if any point (h,k) divide (p,q) and (r,s) in ration a:1

(p.q)...............a.........(h,k)...................1............(r,s)

Then  h  = (a.r + p )/(a+1)   k = ( as + q)/(a+1) 

With regards

yagya

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