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If a circle with area 36 is inscribed in an equilateral triangle, wha
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29 Jun 2023, 01:14
29 Jun 2023, 01:14
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If a circle with area \(36 \pi\) is inscribed in an equilateral triangle, what is the area of the triangle?
A. \(27 \sqrt{3}\)
B. \(54 \sqrt{3} \)
C. \(108 \sqrt{3}\)
D. \(144 \sqrt{3}\)
E. \(432 \sqrt{3} \)
Post A Detailed Correct Solution For The Above Questions And Get Kudos.
Question From Our Project: Free GRE Prep Club Tests in Exchange for 20 Kudos
\(\Longrightarrow\) GRE - Quant Daily Topic-wise Challenge
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A. \(27 \sqrt{3}\)
B. \(54 \sqrt{3} \)
C. \(108 \sqrt{3}\)
D. \(144 \sqrt{3}\)
E. \(432 \sqrt{3} \)
Post A Detailed Correct Solution For The Above Questions And Get Kudos.
Question From Our Project: Free GRE Prep Club Tests in Exchange for 20 Kudos
\(\Longrightarrow\) GRE - Quant Daily Topic-wise Challenge
\(\Longrightarrow\) GRE GEOMETRY
Re: If a circle with area 36 is inscribed in an equilateral triangle, wha
[#permalink]
16 Jul 2023, 10:12
16 Jul 2023, 10:12
Expert Reply
OE
Area of the circle = \(𝜋𝑟^2\) (r is the radius of the circle)
Given that area of circle = 36π
Hence, r = 6.
We know, when a circle is inscribed in an equilateral triangle, centroid of triangle is the
center of the circle, and centroid divided the median in the ratio 2:1.
i.e. AO: OD = 2:1
GRe triangle circle.jpg [ 29.57 KiB | Viewed 1289 times ]
Now, we know OD = 6, therefore AO = 12
Hence, median = height (of the equilateral triangle) = 6 + 12 = 18
Also, the height of an equilateral triangle \(= \dfrac{\sqrt{3}}{2} \times side=18\)
\(side = \dfrac{36}{\sqrt{3}}\)
Therefore, area of the triangle \(= \dfrac{3}{4} \times side^2= \dfrac{\sqrt{3}}{4} \times \dfrac{36}{\sqrt{3}} \times \dfrac{36}{\sqrt{3}}= 108 \sqrt{3}\)
C is the answer
Area of the circle = \(𝜋𝑟^2\) (r is the radius of the circle)
Given that area of circle = 36π
Hence, r = 6.
We know, when a circle is inscribed in an equilateral triangle, centroid of triangle is the
center of the circle, and centroid divided the median in the ratio 2:1.
i.e. AO: OD = 2:1
Attachment:
GRe triangle circle.jpg [ 29.57 KiB | Viewed 1289 times ]
Now, we know OD = 6, therefore AO = 12
Hence, median = height (of the equilateral triangle) = 6 + 12 = 18
Also, the height of an equilateral triangle \(= \dfrac{\sqrt{3}}{2} \times side=18\)
\(side = \dfrac{36}{\sqrt{3}}\)
Therefore, area of the triangle \(= \dfrac{3}{4} \times side^2= \dfrac{\sqrt{3}}{4} \times \dfrac{36}{\sqrt{3}} \times \dfrac{36}{\sqrt{3}}= 108 \sqrt{3}\)
C is the answer
gmatclubot
Re: If a circle with area 36 is inscribed in an equilateral triangle, wha [#permalink]
16 Jul 2023, 10:12
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