Question

One of the angles of a triangle is ${30}^{\circ }$. Prove that the length of the side which is opposite to that angle is equal to the radius of the circumscribed circle.

Answer 1

Let the triangle be $ABC$ with angle $\angle ABC={30}^0$. Let $R$ denote the radius of the circumcircle.

we know that $R=\frac{abc}{4A}$ where A is the area of the triangle. Here $a,b,c$ denote the lengths of the side $BC,AC,AB$ respectively.

Also $A=\frac{1}{2}ac\sin \left(B\right)$. Using this in the above expression, we get, $R=\frac{b}{2\sin \left(B\right)}=\frac{b}{2\sin \left({30}^0\right)}=b$