`G` is the centroid of triangle `A B Ca n dA_1a n dB_1` are rthe midpoints of sides `A Ba n dA C ,` respectively. If `"Delta"_1` is the area of quadrilateral `G A_1A B_1a n d"Delta"` is the area of triangle `A B C ,` then `"Delta"//"Delta"_1` is equal to `3/2` b. `3` c. `1/3` d. none of these

Options

A. `3/2`
B. 3
C. `1/3`
D. none of these

Answer

b

Explanation

Let P.V. of A, B and C be `vec0,vecb and vecc`, respectively therefore,
`vecG = (vecb +vecc) /3`
`vecA_(1) = vecb/2, vecB_(1) = vecc/2`
`Delta_(AB_(1)G) = 1/2 |vec(AG)xx vec(AB_(1))|`
`1/2 |(vecb+vecc)/3 xx (vecc/2)|`
`1/12 |vecb xx vecc|`
`Delta_(AA_(1)G) = 1/2 |vec(AG) xx vec(AA_(1))|= 1/2 |(vecb + vecc)/3 xx (vecb/2)|`
`1/12 |vecb xx vecc|`
`Rightarrow Delta_(GA_(1)AB_(1) = 1/6 |vecb xx vecc| = 1/3. 1/2 |vecb xx vecc|`
`1/3 Delta_(ABC)`
`Rightarrow Delta/Delta_(1) = 3 `