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in triangle OAB , E is the mid-point of OB and D is a point on AB such that AD:DB=2:1. If OD and AE intrsects at P, determine the ratio OP:PD using vector methods.
given triangle OAB
sorry i cant construct the triangle and note: A, B,D,E,P represent vectors
D=2(B)+(A)1/2+1
=2(B)+(A)/3
E=1(O)+1(B)/1+1
=B/2
given the point of interesection of OD and AE is P
let the OP:PD is 1:m and the ratio AP:PE is 1:n
therefore P=1(D)+m(O)/1+m
=D/1+m
=2(B)+(A)/3(1+m)
and also P=n(A)+1(B/2)/1+n
=2n(A)+B/2(1+n)
equating values of P
2(B)+(A)/3(1+m)=2n(A)+B/2(1+n)
2(B)/3(1+m)+(A)/3(1+m)=2n(A)/2(1+n)+(B)/2(1+n)
comparing values of A and B we get
2/3(1+m)=1/2(1+n)---->I
1/3(1+m)=2n/2(1+n)----->II
divide II/I we get
1/2=2n
n=1/4
substitute n=1/4 in I we get m=2/3
OP:PD is 1:2/3
=3:2
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