In triangle ABC, if P, Q, R divides sides BC, AC, and AB, respectively, in the raito `k : 1` (in order). If the ratio `(("area "DeltaPQR)/("area " DeltaABC)) " is " (1)/(3)`, then k is equal to

Options

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

Answer

B

Explanation

Area of triangle

Area of `Delta ARQ = (b)/((k+1)) xx (kc)/((k+1)) xx (1)/(2) sinA = (kDelta)/((k + 1)^(2))`
Similarly, area `(DeltaBPR) = " Area of " (DeltaPCQ) = (kDelta)/((k+1)^(2))`
Now, `(DeltaPQR)/(DeltaABC) = (1)/(3)`
or `(Delta-(3k)/((k + 1)^(2)) Delta)/(Delta) = (1)/(3)`
or `2k^(2) - 5k + 2 = 0`
or `k = (1)/(2), 2`