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 △ABC
Since The Height or median of the equilateral triangle
(AD)=√3X∕2
And we know that centroid divides the median into 2:1 ratio
i.e OA:OD=2:1
⇒OA:AD=2:3
⇒y:√‌3x∕2=2:3
⇒2y∕√‌3x=2∕3
⇒3y=√‌3x
⇒√‌3y=x
∴ The correct relation between x
and y is √3y=x .