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....
GMAT Set 7 Q8
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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 : x√3 : 2x.
This ratio can also be represented as follows:
x/2 : (x/2)√3 : x.
In the figure above:
AC = 2(AD) = 2(r/2)√3 = r√3.
BD = r + r/2 = 3r/2.
Since the area of ∆ABC is 9√3, we get:
(1/2)(AC)(BD) = 9√3
(1/2)(r√3)(3r/2) = 9√3
3r²/4 = 9
3r² = 36
r² = 12.
Thus:
Area of circle = πr² = 12π.
The correct answer is C.
In a 30-60-90 triangle, the sides are in the following ratio:
x : x√3 : 2x.
This ratio can also be represented as follows:
x/2 : (x/2)√3 : x.
An equilateral triangle that has an area of 9√3 is inscribed in a circle. What is the area of the circle?
A. 6Ï€
B. 9Ï€
C. 12Ï€
D. 9π√3
E. 18π√3
In the figure above:
AC = 2(AD) = 2(r/2)√3 = r√3.
BD = r + r/2 = 3r/2.
Since the area of ∆ABC is 9√3, we get:
(1/2)(AC)(BD) = 9√3
(1/2)(r√3)(3r/2) = 9√3
3r²/4 = 9
3r² = 36
r² = 12.
Thus:
Area of circle = πr² = 12π.
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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Find length of one side via pythagoras = 6Abhijit 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....
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