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  • Write true or false and justify your answer. PQRS is a rectangle inscribed in a quadrant of a circle of radius 13 cm. A is any point on PQ.If PS = 5 cm,then ar (PAS) = 30 cm^2.

Write true or false and justify your answer. PQRS is a rectangle inscribed in a quadrant of a circle of radius 13 cm . A is any point on PQ. If $\mathrm{PS}=5 \mathrm{~cm}$,then ar $(\mathrm{PAS})=30 \mathrm{~cm}^2$.

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Solution.

It is given that $P Q R S$ is a rectangle inscribed in a quadrant of a circle of radius 13 cm .

$A$ is any point on $PQ$, therefore, $\mathrm{PA}<\mathrm{PQ}$.

It is given that $A$ is any point on $PQ$, therefore $\mathrm{PA}<\mathrm{PQ}$.

$
\mathrm{PS}=5 \mathrm{~cm}, \mathrm{PR}=13 \mathrm{~cm}
$

In $\triangle \mathrm{PSR}$, using Pythagoras theorem,

$
\begin{aligned}
& P S^2+S R^2=P R^2 \\
& 5^2+S R^2=13^2 \\
& S R^2=169-25=144
\end{aligned}
$

$\mathrm{SR}=12 \mathrm{~cm}=\mathrm{PQ}$ (opposite sides of a rectangle are equal)
The area of a triangle is given as $\frac{1}{2} \times$ base $\times$ height
Now,

$
\operatorname{ar}(\triangle P Q R)=\frac{1}{2} \times P Q \times Q R=\frac{1}{2} \times 12 \times 5=30 \mathrm{~cm}^2
$

$\begin{aligned} & \text { As, } P A<P Q \\ & \operatorname{ar}(\triangle P A S)<\operatorname{ar}(\triangle P Q R) \\ & \text { or } \operatorname{ar}(\triangle P A S)<30 \mathrm{~cm}^2 \\ & \text { But it is given that } \operatorname{ar}(P A S)=30 \mathrm{~cm}^2\end{aligned}$

Hence, the given statement is false.

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infoexpert26

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