If $r:R:r_1=2:5:12,$ then $\angle A=$

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${45}^{\circ }$
4. ${90}^{\circ }$

The correct option is D ${90}^{\circ }$
Given, $r:R:r_1=2:5:12$
$\Rightarrow r=2k,R=5k,r_1=12k$
As we know that
$\frac{r}{r_1}=1-\frac{a}{s}$
$\frac{2}{12}=1-\frac{a}{s}\Rightarrow \frac{a}{s}=\frac{5}{6}$
Also, we have
$\frac{R}{r_1}=\frac{a}{2\sin A}\times \frac{1}{s\tan \frac{A}{2}}$
$\Rightarrow \frac{5}{12}=\frac{a}{2\sin A}\times \frac{1}{s\tan \frac{A}{2}}$
$\Rightarrow \sin A\cdot \tan \frac{A}{2}=1$
$\Rightarrow 2\sin \frac{A}{2}\cos \frac{A}{2}\cdot \frac{\sin \frac{A}{2}}{\cos \frac{A}{2}}=1$
$\Rightarrow 2{\left(\sin \frac{A}{2}\right)}^2=1$
$\Rightarrow \sin \frac{A}{2}=\frac{1}{\sqrt{2}}$
$\Rightarrow \frac{A}{2}={45}^{\circ }$
$\Rightarrow A={90}^{\circ }$