Let the centroid of an equilateral triangle A | vedclass

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(A) The centroid $O$ of the equilateral triangle is at the origin $(0, 0)$.The distance from the centroid to any side of an equilateral triangle is eq

Vedclass · Accepted Answer

$\frac{9}{\sqrt{2}}$

Vedclass · Answer

(A) The centroid $O$ of the equilateral triangle is at the origin $(0, 0)$.The distance from the centroid to any side of an equilateral triangle is equal to the inradius $r$.The equation of the side is $x + y - 3 = 0$.The perpendicular distance from the origin $(0, 0)$ to the line $x + y - 3 = 0$ is given by:$r = \frac{|0 + 0 - 3|}{\sqrt{1^2 + 1^2}} = \frac{3}{\sqrt{2}}$.In an equilateral triangle,the circumradius $R$ is twice the inradius $r$,i.e.,$R = 2r$.Therefore,$R = 2 	imes \frac{3}{\sqrt{2}} = \frac{6}{\sqrt{2}}$.Thus,$R + r = \frac{6}{\sqrt{2}} + \frac{3}{\sqrt{2}} = \frac{9}{\sqrt{2}}$.

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 $r$ are the radius of the circumcircle and incircle respectively of $\Delta ABC$,then $(R + r)$ is equal to ..... .

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