Question

In a $\Delta ABC$, the lengths of the two larger sides are $10$ and $9$ units, respectively. If the angles are in AP, then the length of the third side can be

• A. $5 \pm \sqrt{6}$
• B. $3 \sqrt{3}$
• C. $5$
• D. None of these

Answer 1

The Correct Option is A

Let $A, B$ and $C$ be the three angles of $\Delta A B C$
and Let $a =10$ and $b =9$
It is given that the angles are in AP.
$\therefore 2 B=A+C$ on adding $B$ both the sides,
we get $3 B=A+B+C$
$\Rightarrow 3 B=180^{\circ}$
$\Rightarrow B=60^{\circ}$
Now, we know $\cos B=\frac{a^{2}+c^{2}-b^{2}}{2 a c}$
$\Rightarrow \cos 60^{\circ}=\frac{10^{2}+c^{2}-9^{2}}{2 \times 10 \times c}$
$\Rightarrow \frac{1}{2}=\frac{100+c^{2}-81}{2 \times 10 \times c}$
$\Rightarrow \frac{1}{2}=\frac{100+c^{2}-81}{20 c}$
$\Rightarrow c^{2}-10 c+19=0$
$\Rightarrow c=5 \pm \sqrt{6}$