Question

In the given right-angled triangle the angle with $\angle B={30}^{\circ }$, the ratio of the sides AB : AC: BC will be equal to

• A. $2:1:\sqrt{3}$
• B. $1:\sqrt{3}:2$
• C. $\sqrt{3}:2:1$
• D. $1:1:\sqrt{2}$

Answer 1

The correct option is A

$2:1:\sqrt{3}$

$\angle B={30}^{\circ },\angle A={60}^{\circ }$

The angles of the triangle ABC are ${30}^{\circ },{60}^{\circ },{90}^{\circ }$.

$\Rightarrow \sin \left(30\right):\sin \left(60\right):\sin \left(90\right)$

$\Rightarrow \frac{1}{2}:\frac{\sqrt{3}}{2}:1$

$\Rightarrow 1:\sqrt{3}:2$

Therefore, the sides will be in the ratio $1:\sqrt{3}:2$

$\begin{array}{ccccc}{30}^{\circ },& & {60}^{\circ },& & {90}^{\circ }\\ 1& :& \sqrt{3}& :& 2\\ \downarrow & & \downarrow & & \downarrow \\ AC& & BC& & AB\end{array}$

Therefore AB : AC: BC = $2:1:\sqrt{3}$