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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.

9 years ago

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