Question

Let $p, q$ and $r$ be the sides opposite to the angles $P, Q$ and $R$, respectively in a $\Delta P Q R$. If $r^{2} \sin P \sin Q=p q$, then the triangle is

• A. equilateral
• B. acute angled but not equilateral
• C. obtuse angled
• D. right angled

Answer 1

Answer: (D)

right angled

Answer 2

We know that in $\Delta A B C$
$\frac{a}{\sin A} =\frac{b}{\sin B}=\frac{c}{\sin C}=2 R$
$\therefore r^{2} \sin P \sin Q=p q$
$\Rightarrow r^{2} \cdot \frac{p}{2 R_{1}} \cdot \frac{q}{2 R_{1}}=p q$,
where $R_{1}$ is circumradius of $\Delta P Q R$.
$\Rightarrow r^{2}=4 R_{1}^{2}$
$\Rightarrow r=2 R_{1}$
$\Rightarrow 2 R_{1} \sin R=2 R_{1}$
$\Rightarrow R_{1}=90^{\circ}$
$\therefore \Delta P Q R$ is right angled.