GMAT Set 7 Q8

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Abhijit K: Master | Next Rank: 500 Posts; Posts: 110; Joined: Mon Feb 09, 2015 1:52 amFollowed by: 4 members

by Abhijit K » Thu Feb 19, 2015 1:49 am

An equilateral triangle that has an area of 9root3 is inscribed in a circle. What is the area
of the circle?
A. 6*3.14
B. 9*3.14
C. 12*3.14
D. 9*3.14*root3
E. 18*3.14*root3

Approach please....

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Source: Beat The GMAT — Problem Solving | Thu Feb 19, 2015 1:49 am

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Thu Feb 19, 2015 7:16 am

An equilateral triangle can be divided into 30-60-90 triangles:

In a 30-60-90 triangle, the sides are in the following ratio:
x : xâˆš3 : 2x.
This ratio can also be represented as follows:
x/2 : (x/2)âˆš3 : x.

An equilateral triangle that has an area of 9âˆš3 is inscribed in a circle. What is the area of the circle?

A. 6Ï€
B. 9Ï€
C. 12Ï€
D. 9Ï€âˆš3
E. 18Ï€âˆš3

In the figure above:
AC = 2(AD) = 2(r/2)âˆš3 = râˆš3.
BD = r + r/2 = 3r/2.

Since the area of âˆ†ABC is 9âˆš3, we get:
(1/2)(AC)(BD) = 9âˆš3
(1/2)(râˆš3)(3r/2) = 9âˆš3
3rÂ²/4 = 9
3rÂ² = 36
rÂ² = 12.

Thus:
Area of circle = Ï€rÂ² = 12Ï€.

The correct answer is C.

Last edited by GMATGuruNY on Thu Feb 19, 2015 7:31 am, edited 1 time in total.

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regor60: Master | Next Rank: 500 Posts; Posts: 416; Joined: Thu Oct 15, 2009 11:52 am; Thanked: 27 times

by regor60 » Thu Feb 19, 2015 7:16 am

Abhijit K wrote:An equilateral triangle that has an area of 9root3 is inscribed in a circle. What is the area
of the circle?
A. 6*3.14
B. 9*3.14
C. 12*3.14
D. 9*3.14*root3
E. 18*3.14*root3

Approach please....

Find length of one side via pythagoras = 6

since each of the angles are the same, they subtend 120 degrees of the circle

that is equivalent to a triangle centered at center of circle with an angle of 120

long leg = 6 as previously determined

two other legs = radius, so 30 degrees each other interior angle

sin 120/6 = sin 30/R,etc.

12pi

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