Question

In an equilateral triangle, ( circumradius ) : ( inradius ) : ( exradius ) is equal to

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

Answer 1

Answer: (C)

2 : 1 : 3

Given triangle is an equilateral triangle

Therefore, $A=B=C=\frac{\pi }{3}$

$R:r:r_1=R:4R\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}:4R\sin \frac{A}{2}\cos \frac{B}{2}\cos \frac{C}{2}$

$\Rightarrow R:r:r_1=1:4{\sin }^3\frac{\pi }{6}:\sin \frac{\pi }{6}{\cos }^2\frac{\pi }{6}=1:\frac{4}{8}:\frac{4\times 3}{2\times 4}$

$\Rightarrow R:r:r_1=2:1:3$
Ans: C