Question

In a triangle the length of the two larger sides are $24$ and $22,$ respectively. If the angles are in $A.P.,$ then the third side is

Answer 1

Let the angles be $A,B$ and $C$

As the angles are in $A.P.,$

therefore

$2B=A+C$ ----eq 1

$\Rightarrow 3B=A+B+C$

since sum of angles of a triangle is ${180}^o$

$\Rightarrow B={60}^0$

puting in eq 1

$\Rightarrow A+C={120}^0$

Therefore either $a$ is the smallest side or $c$ is smallest side.

Let $c$ be the smallest side.

Then two larger sides are $a$ and $b$.

let $a=24,b=22$

Now,

$B={60}^0$

applying $cos\theta$ rule

$\Rightarrow \cos B=\frac{1}{2}$

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

$\Rightarrow \frac{c^2+576-484}{48c}=\frac{1}{2}$

$\Rightarrow c=12\pm 2\sqrt{13}$