Question

In a right angled triangle, one acute angle is double the other. The hypotenuse is __________ the smallest side.

• A. $\sqrt{2}$ times
• B. three times
• C. double
• D. 4 times

Answer 1

The correct option is C double

Given: A $△PRQ$ in which
$\angle Q={90}^{\circ }$ and $\angle PRQ=2\angle QPR$
Const. Produce $RQ$ to $S$ such that
$RQ=QS.$ Join $PS.$
$QS=QR$ (By construction)
$PQ=PQ$ (Common)
$\angle PQS=\angle PQR$ (Each$={90}^{\circ }$)
$\therefore$ $△PQS\cong △PQR$ (SAS)
$\Rightarrow$ $PS=PR$ and $\angle SPQ=\angle RPQ$ (cpct)
Let $\angle SPQ=x.$ Then $\angle PRQ=2x$ (Given)
Then $\angle SPR=\angle SPQ+\angle RPQ$
$=x+x=2x$
$\Rightarrow$ $\angle spr=\angle prq=SR=PS$ (Sides opp. equal $\angle$s are equal)
$\Rightarrow$ $2QR=PS$ $\{\because SQ=QR\because SR=2QR\}$
$\Rightarrow$ $2QR=PR$ $(\because PS=PR)$
$\Rightarrow$ The hypotenuse $PR$ is double the smallest side $QR.$