Question

In an equilateral triangle the circumradius is n times inradius then 'n' is equal to

Answer 1

the ans is 2. in equilateral triangle incentre,circumcentre and centroid are the same point .We know that centroid divides a line in the ratio 2:1,

Answer 2

we know from properties of triangle.
circumradius of any triangle of side length a , b and c is given by $R=\frac{abc}{4A}$
where A is area of that triangle.

again , inradius of the triangle is given by $r=\frac{A}{S}$, where S is semiperimeter of that triangle.

in case of equilateral triangle ,
a = b = c = k
then, $R=\frac{k^3}{4\frac{\sqrt{3}}{4}k^2}=\frac{k}{\sqrt{3}}$

and $r=\frac{\frac{\sqrt{3}}{4}k^2}{\frac{3k}{2}}=\frac{k}{2\sqrt{3}}$

so, ratio of R and r = $\frac{\frac{k}{\sqrt{3}}}{\frac{k}{2\sqrt{3}}}=2:1$