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(D) The correct option is $(d)$.$I$. Since $	riangle BCX$ and $	riangle BCY$ share the same base $BC$ and lie between the same parallel lines $XY$ a

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Both $I$ and $II$

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(D) The correct option is $(d)$.$I$. Since $	riangle BCX$ and $	riangle BCY$ share the same base $BC$ and lie between the same parallel lines $XY$ and $BC$,their areas are equal. Thus,$[BCX] = [BCY]$ is true.$II$. Using the area formula $	ext{Area} = \frac{1}{2} ab \sin C$:$[ACX] = \frac{1}{2} (AX)(AC) \sin A$$[ABY] = \frac{1}{2} (AY)(AB) \sin A$$[ACX] \cdot [ABY] = \left( \frac{1}{2} (AX)(AC) \sin A ight) \cdot \left( \frac{1}{2} (AY)(AB) \sin A ight)$$= \left( \frac{1}{2} (AX)(AY) \sin A ight) \cdot \left( \frac{1}{2} (AB)(AC) \sin A ight)$$= [AXY] \cdot [ABC]$Thus,$II$ is also true.Therefore,both $I$ and $II$ are true.

In a $\triangle ABC$,points $X$ and $Y$ are on $AB$ and $AC$,respectively,such that $XY$ is parallel to $BC$. Which of the two following equalities always hold? (Here $[PQR]$ denotes the area of $\triangle PQR$).
$I$. $[BCX] = [BCY]$
$II$. $[ACX] \cdot [ABY] = [AXY] \cdot [ABC]$

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In a $\triangle ABC$,points $X$ and $Y$ are on $AB$ and $AC$,respectively,such that $XY$ is parallel to $BC$. Which of the two following equalities always hold? (Here $[PQR]$ denotes the area of $\triangle PQR$).$I$. $[BCX] = [BCY]$$II$. $[ACX] \cdot [ABY] = [AXY] \cdot [ABC]$