Question

if the radius of the circumcircle of an equilateral triangle is 6 cm find the radius of its incircle

Answer 1

Answer:

Draw a triangle of side 2 a with R being the distance from the center of the triangle to a vertex and r the distance from the midpoint of a side to the center of the triangle.

Then     r^2 + a^2 = R^2

let h = height of the triangle

(h - r)^2 = r^2 + a^2

h^2 - 2 r h + r^2 = r^2 + a^2

h^2 - 2 r h = a^2

Also a^2 + h^2 = 4 a^2   and h = (3)^1/2 a

So     3 a^2 - 2 (3)^1/2 a r = a^2  

2 (3)^1/2 a r = 2 a^2     and a = (3)^1/2  r

From the very first equation r^2 + 3 r^2 = R^2   or R = 2 r

So the inscribed circle has radius of 3 cm.