.(a) In APQR, ∠PQR= 90°. If its area is 216cm\(^2\) and |PQ|:|QR| is 3:4, find |PR|. (b) The present ages of a man and his son are 47 years and 17 years res...

Answer Details

(a) In \(\triangle PQR\), \(\angle PQR=90^{\circ}\), so \(PQ\) and \(QR\) are the perpendicular sides. Let \(PQ=3k,\ QR=4k\).

\[\text{Area}=\tfrac12\,PQ\cdot QR=216\Rightarrow\tfrac12(3k)(4k)=216\Rightarrow 6k^{2}=216\Rightarrow k^{2}=36\Rightarrow k=6.\]

So \(PQ=18\text{ cm},\ QR=24\text{ cm}\). By Pythagoras:

\[|PR|=\sqrt{18^{2}+24^{2}}=\sqrt{324+576}=\sqrt{900}=30\text{ cm}.\]

(b) Let the man's age be twice his son's after \(x\) years:

\[47+x=2(17+x)\Rightarrow 47+x=34+2x\Rightarrow x=13.\]

In 13 years' time the man will be twice as old as his son (60 and 30).