UPSC CDS 1 2022 Math Paper

Given: Side of equilateral triangle $=x$ unit The radius of the circle (OA or OB or OC) $=y$ unit Concept Used: For an equilateral triangle All the centers i.e circumcentre, orthocentre, centroid, incentre lie on the same point Centroid divides median in the ratio of $2:1.$ Calculation: According to the question $ABC$ is an equilateral triangle So, The center of the circle is also the centroid of $\triangle ABC$ Since The Height or median of the equilateral triangle $(AD)=\frac{\sqrt{3X}}{2}$ And we know that centroid divides the median into $2:1$ ratio i.e $OA:OD=2:1$ $\Rightarrow OA:AD=2:3$ $\Rightarrow y:\frac{\sqrt{3x}}{2}=2:3$ $\Rightarrow \frac{2y}{\sqrt{3x}}=\frac{2}{3}$ $\Rightarrow 3y=\sqrt{3x}$ $\Rightarrow \sqrt{3}y=x$ $\therefore$ The correct relation between $x$ and $y$ is $\sqrt{3}y=x$ .