Question : In $\Delta ABC$ the straight line parallel to the side $BC$ meets $AB$ and $AC$ at the point $P$ and $Q,$ respectively. If $AP=QC$, the length of $AB$ is $\operatorname{12 units}$ and the length of $AQ$ is $\operatorname{2 units}$, then the length (in units) of $CQ$ is:
Option 1: $4$
Option 2: $6$
Option 3: $8$
Option 4: $10$
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Correct Answer: $4$
Solution :
Given that $AP=QC$, $AB=\operatorname{12 units}$ and $AQ=\operatorname{2 units}$
Let $AP=QC=x\operatorname{ units}$
Since $PQ$ is parallel to $BC$, then,
$\Delta ABC\sim\Delta APQ$
Since the triangles are similar, the ratios of their corresponding sides are equal.
$\frac{AP}{AB}=\frac{AQ}{AC}$
⇒ $\frac{x}{12}=\frac{2}{2+x}$
⇒ $2x+x^2=24$
⇒ $x^2+2x-24=0$
⇒ $(x+6)(x-4)=0$
⇒ $x=4$ ($\because x$ cannot be negative)
Hence the correct answer is $4$.
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