Question

In a tringle, the lengths of the two larger sides are $10$ and
$9$ respectively. If the angles are in $A.P.,$ then the
length of the third side can be

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

Answer 1

Answer: (A), (D)

$5+\sqrt{6}$
$5-\sqrt{6}$

Ans. $\left(a\right)$ and $\left(d\right)$.
As in part $\left(b\right),\angle B={60}^o$ Also $a=10,b=9$.
$\therefore \cos B=(a^2+c^2-b^2)/2ac$
$\Rightarrow \cos {60}^o=(100+c^2-81)\left(/2ac\right)$
$\Rightarrow 1/2=(19+c^2)/\left(20c\right)\Rightarrow 10c=19+c^2$
or $c^2-10c+19=0$
$\therefore c=\frac{10\pm \sqrt{(100-76)}}{2}=5\pm \sqrt{6}$.
Since both the value of $c$ are positive and less then both $a$ and $b$, we have the answer $\left(a\right)$