In a triangle $A B C$ , points $X$ and $Y$ are on $A B$ and $A C$ respectively, such that $X Y$ is parallel to $B C$ . Which of the following always hold? (Here $[P Q R]$ denotes the area of triangle $P Q R$ )
$(I)$ $[B C X] = [B C Y]$
$(I I)$ $[A C X] . [A B Y] = [A X Y] . [A B C]$

Neither

(I)

nor

(I I)

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(I)

only

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(I I)

only

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Both

(I)

and

(I I)

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Solution

The correct option is A Both $(I)$ and $(I I)$
$△ B C X = △ B C Y$ (Obvious)
Same base and same height
Now let $A (\to O), - - \to A B = \to b, - - \to A C = \to c$
So $- - \to A X = λ \to b, - - \to A Y = λ \to c$
$△ A C X = \frac{1}{2} λ ∣ ∣ \to b \times \to c ∣ ∣$
$△ A B Y = \frac{1}{2} λ ∣ ∣ \to b \times \to c ∣ ∣$
$△ A X Y = \frac{1}{2} λ^{2} ∣ ∣ \to b \times \to c ∣ ∣$
$△ A B C = \frac{1}{2} ∣ ∣ \to b \times \to c ∣ ∣$

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