Q.371. Vectors application

Question 371.
Refer to the following figure.

In triangle ABC with two inner line-segments AD and CE, 
BD : DC = 3:1 and AE : EB = 2:1.
Find the ratio of CG : GE and AG : GD .

Answer 371.
Let a and c be the position vectors of A and C with respect to B as null vector.
=> Position vector of D = 3c/4 and that of E = a/3

Let AG : GD = m : 1
=> position vector of G is 1/(m + 1) * (3mc/4 + a) ... ( 1 )

Let EG : CG = n : 1
=> position vector of G is 1/(n + 1) * (nc + a/3) ... ( 2 )

Comparing coefficients of a in ( 1 ) and ( 2 ),
m + 1 = 3(n + 1) => m = 3n + 2 ... ( 3 )
Comparing coefficients of c in ( 1 ) and ( 2 ),
3m/[4(m + 1)] = n/(n + 1) ... ( 4 )

Plugging m = 3n + 2 from ( 3 ) in ( 4 ),
3 (3n + 2)/[4(3n + 3) = n/(n + 1)
=> 3n + 2 = 4n
=> n = 2
and from ( 3 ),
m = 3n + 2 = 8

=> AG : GD = m : 1 = 8 : 1
and EG : CG = n : 1 = 2 : 1.

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