Question

In a triangle $ABC$, if $a{\cos }^2\frac{C}{2}+c{\cos }^2\frac{A}{2}=\frac{3b}{2}$, then the sides of triangle are in

• A. an arithmetic progression
• B. a geometric progression
• C. a harmonic progression
• D. an arithmetico-geometric progression

Answer 1

The correct option is A an arithmetic progression

$a{\cos }^2\frac{C}{2}+c{\cos }^2\frac{A}{2}=\frac{3b}{2}$

$\Rightarrow a\left(\frac{1+\cos C}{2}\right)+c\left(\frac{1+\cos A}{2}\right)=\frac{3b}{2}$

$\Rightarrow \frac{a}{2}+\frac{a\cos C}{2}+\frac{c}{2}+\frac{c\cos A}{2}=\frac{3b}{2}$

$\Rightarrow \frac{a+c}{2}+\frac{b}{2}=\frac{3b}{2}$

$\Rightarrow a+c+b=3b$

$\Rightarrow a+c=2b$

Thus, the sides are in arithmetic progression.