Arithmetic Progression In Triangles

Solution:

Let the angles of the triangle be $a,b$ and $c$

We know,

$\begin{aligned}\large a+b+c&=180° \hspace{1cm}&\color{#3D99F6}\text{(Angle Sum Property)}\\ \large a+c&=2b \hspace{1 cm} &\color{#3D99F6}\text{(Arithmetic Mean)}\\ \large b&=60°\end{aligned}$

We know that the greatest side has the greatest opposite angle.As $60°$ is the second greatest angle,it must be opposite to the second longest side which is $22$ cm.

Let us assume the third side to be $x$ .

By applying Law of Cosines,

$\color{#D61F06}{\large\begin{aligned} x^2+24^2-48x\cos{60°}&=22^2\\ x^2-24x+92 &=0\end{aligned}}$

By applying Quadratic Formula,

$\color{#20A900}{\large x=\dfrac{24±\sqrt{208}}{2}\\ \large x=12±2\sqrt{13}\\ \large\therefore a+b+c=12+2+13=\boxed{27}}$