Question

Let the centroid of an equilateral triangle ABC be at the origin. Let one of the sides of the equilateral triangle be along the straight line x+y=3. If R and r be the radius of circumcircle and incircle respectively of ΔABC, then (R + r) is equal to :

• A. 2$\sqrt{2}$
• B. 9/$\sqrt{2}$
• C. 7$\sqrt{2}$
• D. 3$\sqrt{2}$

Hint:

In any equilateral triangle, Circumradius ($R$), Inradius ($r$), and altitude ($h$) are related as $R=2r$ and $h = R+r = 3r$.

Answer 1

The Correct Option is B

Step 1: For an equilateral triangle, the centroid coincides with the incenter. The inradius $r$ is the perpendicular distance from the origin $(0,0)$ to the side $x+y-3=0$. $r = \left| \frac{0+0-3}{\sqrt{1^2+1^2}} \right| = \frac{3}{\sqrt{2}}$.
Step 2: In an equilateral triangle, $R = 2r$. $R = 2(\frac{3}{\sqrt{2}}) = \frac{6}{\sqrt{2}}$.
Step 3: $R + r = \frac{6}{\sqrt{2}} + \frac{3}{\sqrt{2}} = \frac{9}{\sqrt{2}}$.