Question

Let $p,q$ and $r$ be the sides opposites to the angles $P,Q$ and $R$, respectively in a $△PQR$.

If $r^2\sin P\sin Q=pq$, then the triangle is?

• A. Equilateral
• B. Acute angled but not equilateral
• C. Obtuse angled
• D. Right angled

Answer 1

The correct option is D Right angled

We know that in $△ABC$
$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$

$\therefore r^2\sin P\sin Q=pq$

$\Rightarrow r^2\cdot \frac{p}{2R_1}\cdot \frac{q}{2R_1}=pq$, where $R_1$ is circumradius of $△PQR$

$\Rightarrow r^2=4R_1^2$

$\Rightarrow r=2R_1$

$\Rightarrow 2R_1\sin R=2R_1$

$\Rightarrow R_1={90}^{\circ }$

$\therefore △PQR$ is right angled.