Question

The angles of a triangle are in the ratio $2:4:3,$ then the smallest angle(in degrees) is , while the largest angle(in radians) is

• A. $\frac{\pi }{3}$
• B. ${40}^{\circ }$
• C. $\frac{4\pi }{9}$
• D. ${20}^{\circ }$

Answer 1

Answer: (B), (C)

$\frac{4\pi }{9}$

Let the angles of a triangle $\left(\text{say}\Delta ABC\right)$ be $2\alpha ,3\alpha \&4\alpha$ as shown.

$\Rightarrow 2\alpha +3\alpha +4\alpha ={180}^{\circ }$
$\Rightarrow 9\alpha ={180}^{\circ }\Rightarrow \alpha ={20}^{\circ }$
Hence, the angles are:
$\angle B={40}^{\circ },\angle C={60}^{\circ },\angle A={80}^{\circ }$

Now, the smallest angle $=\angle B={40}^{\circ }$
and the largest angle $=\angle A={80}^{\circ }$
Now, using the relation:
$\frac{D}{{180}^{\circ }}=\frac{R}{\pi }$

$\Rightarrow R=\pi \times \frac{D}{{180}^{\circ }}$

$\Rightarrow R=\pi \times \frac{{80}^{\circ }}{{180}^{\circ }}=\frac{4\pi }{9}\text{radians}$