Question

The sides of a triangle are in the ratio $1:\sqrt{3}:2$. Then the angles are in the ratio

• A. 1 : 2 : 3
• B. 1 : 2 : 4
• C. 1 : 4 : 5
• D. 1 : 3 : 5

Answer 1

Answer: (A)

We have,
Ratio of sides of triangle are $1:\sqrt{3}:2$.
Let the sides are $k,\sqrt{3}k,2k$

Since, this triangle is a right angle triangle
$(2k{)}^2=(\sqrt{3}k{)}^2+(k{)}^2=(2k{)}^2$
$\therefore \sin A=\frac{\sqrt{3}k}{2k}=\frac{\sqrt{3}}{2}\Rightarrow A=60^{\circ }$
$\Rightarrow \sin C=\frac{k}{2k}=\frac{1}{2}\Rightarrow C=30^{\circ }$
$\Rightarrow B=90^{\circ }$
$\therefore$ Ratio of angles are $30^{\circ }:60^{\circ }:90^{\circ }$
$\Rightarrow 1:2:3$