In triangle `A B C ,ifPdotQ ,R` divides sides`B C ,A C ,` and `A B ,` respectively, in the ratio `k :1(inor d e r)dot` If the ratio `((a r E A P Q R)/(a r e a A B C))` IS `1/3,t h e nk` is equal to `1/3` (b) 2 (c) 3 (d) none of these

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`