Question

If the angles $A,B,C$ of a triangle are in A.P. and sides $a,b,c$ are in G.P. then $a^2,b^2,c^2$ are in

• A. A.P.
• B. H.P.
• C. G.P.
• D. None of these

Answer 1

The correct option is A A.P.

Since the angles $A,B,C$ are in $A.P.$, therefore

$2B=A+C\Rightarrow 3B=A+B+C\Rightarrow 3B={180}^0$

$\Rightarrow B={60}^0\Rightarrow \cos B=\frac{1}{2}$

$\Rightarrow \frac{c^2+a^2-b^2}{2ac}=\frac{1}{2}\Rightarrow c^2+a^2-b^2=ac$

$\Rightarrow c^2+a^2-b^2=b^2$ $[\because a,b,c$ are in G.P $\therefore b^2=ac]$

$\Rightarrow c^2+a^2=2b^2$

$\Rightarrow a^2,b^2,c^2$ are in A.P.