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An equilateral triangle is inscribed in a circle. If the perimeter of

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An equilateral triangle is inscribed in a circle. If the perimeter of [#permalink] New post   06 Sep 2021, 10:00
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An equilateral triangle is inscribed in a circle. If the perimeter of the triangle is z inches and the area of the circle is y square inches, which of the following equations must be true?


A) \(9z^2-\pi y=0\)

B) \(3z^2-\pi y=0\)

C) \(\pi z^2-3y=0\)

D) \(\pi z^2-9y=0\)

E) \(\pi z^2-27y=0\)
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E
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An equilateral triangle is inscribed in a circle. If the perimeter of [#permalink] New post   30 Nov 2021, 06:25
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Side of an equilateral triangle (a) inscribed in a circle with radius r is given by a = r√𝟑
[ Watch this video to know how ]

=> Perimeter of triangle = 3a = 3 * √𝟑 r = z (given) => r = \(\frac{z }{ 3 √𝟑}\)

Area of circle with radius r = \(\pi r^2\) = y (given)
=> \(\pi\) * \((\frac{z }{ 3 √𝟑})^2\) = y (substituting value of r from (1) )
=> \(\pi\) * \(\frac{z^2}{9*3}\) = y
=> \(\pi\) * \(\frac{z^2}{27}\) = y
=> \(\pi\) * \(z^2\) = 27 * y
=> \(\pi z^2-27y=0\)

So, Answer will be E
Hope it helps!

Watch the following video to Learn Properties of Equilateral Triangle inscribed in a Circle

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Re: An equilateral triangle is inscribed in a circle. If the perimeter of [#permalink] New post   06 Sep 2021, 17:42
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Answer is E) pi *z^2−27y=0

Let s be the side of the triangle and r be the radius of the circle, the relation between them is s = r * sqrt(3).
So z = 3 * s = 3*sqrt(3) * r.
and area = pi * r^2.
All the answer choices are relation between y and z and y already has a pi in it, so in the answer only C,D,E has pi multiplied for z.
Next step is simple, relation between z and y in terms of r and E is the only one that satisfies.

PS: Not sure this is the GRE method, please share if there is a shorter approach.
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Re: An equilateral triangle is inscribed in a circle. If the perimeter of [#permalink] New post   14 Apr 2024, 10:39
Hello from the GRE Prep Club BumpBot!

Thanks to another GRE Prep Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

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Re: An equilateral triangle is inscribed in a circle. If the perimeter of [#permalink]
14 Apr 2024, 10:39
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