Question

A triangle $ABC$ is such that sides $a,b,c$ are in $G.P.$ and $\sin A,\sin B,\sin C$ are in $A.P.,$ then the $△ABC$ is
(where $a,b,c$ are sides (in $\text{units}$) of $△ABC$ opposite to $\angle A,\angle B$ and $\angle C$ respectively)

• A. scalene triangle
• B. None of these
• C. Equilateral triangle
• D. Right angled triangle

Answer 1

The correct option is C Equilateral triangle

Given : Sides $a,b,c$ are in $G.P.$
$\Rightarrow b^2=ac\cdots \left(i\right)$
and $\sin A,\sin B,\sin C$ are in $A.P.,$
$\Rightarrow 2\sin B=\sin A+\sin C$
Using sine rule : $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$
$\Rightarrow 2b=a+c\Rightarrow 4b^2=a^2+c^2+2ac\Rightarrow 4ac=a^2+c^2+2ac\left[\text{from}\right(i\left)\right]\Rightarrow (a-c{)}^2=0\Rightarrow a=c$
$\therefore a=b=c\left[\text{from}\right(i\left)\right]$