Question

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

Answer 1

It is given that A is any point on PQ, therefore $PA<PQ.$

Now, $ar\left(\Delta PQR\right)=\frac{1}{2}\times base\times height$

$PS=QR=5cm$ as it's a rectangle

considering $rt.△PQR$

$P{Q}^2=P{R}^2-Q{R}^2={13}^2-5^2=12cm$

Now, $ar\left(\Delta PQR\right)=\frac{1}{2}\times PA\times PS=\frac{1}{2}\times PA\times QR$

Now, $ar\left(\Delta PQR\right)=\frac{1}{2}\times PQ\times QR=\frac{1}{2}\times 12\times 5=30c{m}^2$

$[\because$ PQRS is a rectangle $\therefore RQ=SP=5cm]$

As $PA<PQ(=12cm)$

So $ar\left(\Delta PAS\right)<ar\left(\Delta PQR\right)$

Or $ar\left(\Delta PAS\right)<30c{m}^2\left[ar\right(\Delta PQR)=30c{m}^2]$

Hence, the given statement is false.