Question : $P Q R S$ is a rectangle. $T$ is a point on $P Q$ such that $R T Q$ is an isosceles triangle and $P T=5 \mathrm{QT}$. If the area of triangle RTQ is $12 \sqrt{3}$ sq.cm, then the area of the rectangle PQRS is:
Option 1: $144 \sqrt{3}$ sq.cm
Option 2: $142$ sq. cm
Option 3: $134 \sqrt{3}$ sq. cm
Option 4: $142\sqrt{3}$ sq. cm
Correct Answer: $144 \sqrt{3}$ sq.cm
Solution :
Given: $PT = 5QT$
$PT : QT = 5 : 1$
Let the ratio of $PT : QT$ be $5x : x$.
⇒ $PQ = PT + QT$
⇒ $PQ = 5x + x$
⇒ $PQ = 6x$
Area of $\triangle RTQ = \frac{1}{2} \times base \times height$
⇒ $\frac{1}{2}\times x \times RQ = 12\sqrt3$
⇒ $x \times RQ = 24\sqrt3 $
Area of rectangle PQRS = $PQ \times RQ$
= $6x \times RQ$
= $6 \times 24\sqrt3 $
= $144\sqrt3 $
Hence, the correct answer is $144\sqrt3$ sq. cm
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