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A square is made by joining the mid-points of the sides of the larger square. There is circle inscribed in the smaller square and an equilateral triangle inscribed in the circle. Find the ratio of the side of larger square to the side of the equilateral triangle? 
  • a)
    (2√2)/3 
  • b)
    √2/√3
  • c)
      (4√2)/√3 
  • d)
    (2√2)/√3
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
A square is made by joining the mid-points of the sides of the larger ...
Let the side length of the larger square be $s$, and let the side length of the smaller square be $x$. The diagonal of the smaller square is the side length of the larger square, so by Pythagoras, $x^2+x^2=s^2$, or $2x^2=s^2$. Taking the square root of both sides, we have $x=\frac{s}{\sqrt{2}}$.

The circle inscribed in the smaller square has its center at the center of the square, and its radius is half the side length of the square, so the radius is $\frac{x}{2}=\frac{s}{2\sqrt{2}}$.

The equilateral triangle inscribed in the circle has its circumradius equal to the radius of the circle. The formula for the circumradius of an equilateral triangle with side length $a$ is $R=\frac{a}{\sqrt{3}}$. In this case, the radius of the circle is $\frac{s}{2\sqrt{2}}$, so the side length of the equilateral triangle is $a=R\sqrt{3}=\frac{s}{2\sqrt{2}}\sqrt{3}=\frac{s\sqrt{3}}{2\sqrt{2}}=\frac{s\sqrt{6}}{4}$.

Therefore, the ratio of the side length of the larger square to the side length of the equilateral triangle is $\frac{s}{\frac{s\sqrt{6}}{4}}=\frac{4}{\sqrt{6}}=\boxed{\frac{2\sqrt{6}}{3}}$.
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Community Answer
A square is made by joining the mid-points of the sides of the larger ...
Let the side of larger square = 2a
Side of smaller square = √(a²+a²) = √a a
Radius of circle = (√a a)/2 = a/√a
Let side of triangle be ‘x’ Radius of circumcircle of triangle = x/√3
Therefore; x/√3 = a/√2 a/x = √2/√3 2a/x = (2√2)/√3
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A square is made by joining the mid-points of the sides of the larger square. There is circle inscribed in the smaller square and an equilateral triangle inscribed in the circle. Find the ratio of the side of larger square to the side of the equilateral triangle?a)(2√2)/3b)√2/√3c) (4√2)/√3d)(2√2)/√3Correct answer is option 'D'. Can you explain this answer?
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