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Square or equilateral triangle inscribed in a circle: side in terms of radius

This type of question, asked 3 times: CDS 2 2016, CDS 1 2022, CDS 1 2023. All 3 questions below, oldest first.

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  1. Question 1

    CDS 2 2016

    42
    If a square of side xx and an equilateral triangle of side yy are inscribed in a circle, then what is the ratio of xx to yy ?

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  2. Question 2

    CDS 1 2022

    82
    An equilateral triangle of side xx is inscribed in a circle of radius yy. Which one of the following is correct?

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  3. Question 3

    CDS 1 2023

    65
    Let x be the area of a square inscribed in a circle of radius r and y be the area of an equilateral triangle inscribed in the same circle. Which one of the following is correct ?

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