Write True or False and justify your answer: PQRS is a rectangle … - Vidyadip

Question

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

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Answer

True Solution: Given, $P S=5 cm$
radius of circle $=S Q=13 cm$
In right angled $\triangle SPQ , SQ ^2=P Q^2+ PS ^2(13)^2= PQ ^2+(5)^2$
$ \Rightarrow P Q^2 169-25=144 $
$\Rightarrow P Q=12 cm$ [taking positive square root, because length is always positive]
Now, area of $\triangle APS =\frac{1}{2} \times$ Base $\times$ Height $\frac{1}{2} \times PS \times PQ$
$=\frac{1}{2} \times 5 \times 12=30 cm^2$ So, given statement is true, if $A$ coincides $Q .$

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