Question

Let $p,q,r$be the sides opposite to the angles $P,Q,R$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

Explanation for the correct answer:

Finding the type of triangle.

We know that in $△ABC$
$$\begin{gathered} \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R \\ \therefore r^2\sin P\sin Q=pq \end{gathered}$$
$\Rightarrow r^2\cdot \frac{p}{2R_1}\cdot \frac{q}{2R_1}=pq,$ where $R_1$​ is circumradius of $△PQR$
$$\begin{gathered} \Rightarrow r^{{}_2}=4{R_1}^2 \\ \Rightarrow r=2R_1 \end{gathered}$$ so,

$$\begin{gathered} \Rightarrow 2R_1\sin R=2R_1 \\ \Rightarrow R_1=90° \end{gathered}$$
$\therefore △PQR$ is right angled.

Hence the correct option is (D).