Note: For this problem I will use ~ as a square root sign, sorry for the inconvenience.
An equilateral triangle that has an area of 9~3 is inscribe in a circle. What is the area of the circle?
A. 6pi
B. 9pi
C. 12pi
D. 9pi~3
E. 18pi~3
Thanks
Inscribed in a circle
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Nice sign convention. I will just use sqrt for clarityalex.gellatly wrote:Note: For this problem I will use ~ as a square root sign, sorry for the inconvenience.
An equilateral triangle that has an area of 9~3 is inscribe in a circle. What is the area of the circle?
A. 6pi
B. 9pi
C. 12pi
D. 9pi~3
E. 18pi~3
Thanks
We know that if side of an equilateral triangle is a, then.
Height = sqrt(3)*a/2.
Area = sqrt(3)*a^2/4.
Also, When an equilateral triangle is inscribed in a circle, the median (which is the same as the height) is divided into 1:2 ratio. so for the circle r = 2/3*sqrt(3)/2 = a/sqrt(3)
So for the circle we have:
r= a/sqrt(3)
Area of circle = pi*r^2 = pi* a^2/3.
Now we are given that for the triangle area = 9*sqrt(3).
So we have sqrt(3)*a^2/4 = 9* sqrt(3) => a^2 = 4*9.
So finally, Area of circle = pi* a^2/3 = pi*4*9/3 = 12pi.
Hence C is the right answer.
Let me know if this helps
Last edited by eagleeye on Thu Jun 28, 2012 6:41 pm, edited 1 time in total.
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Refer to the following figure,alex.gellatly wrote:An equilateral triangle that has an area of 9√3 is inscribe in a circle. What is the area of the circle?
If length of each side of the triangle is a, then area of the triangle is given by (√3/4)a²
Hence, (√3/4)a² = 9√3 ---> a = √(9*4) = 6
Therefore, AB = BC = AC = 6
Hence, AD = Height of the triangle = (2*Area)/(Base) = 2*9√3/6 = 3√3
Now, O is the centroid of the triangle, i.e. where the medians of the triangles intersect each other at 2:1 ratio.
Hence, radius of the circle = OA = 2/3 of AD = (2/3)*3√3 = 2√3
Hence, area of the circle = π(2√3)² = 12π
The correct answer is C.
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I have one doubt here. Here Equilateral triangle is considered as 30 : 60 :90 . And taken as x:xsqrt3 :2x .. Why is it so?? Equilateral triangle will be 60 : 60 : 60 . right??
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ABC is a equilateral triangle with all angles equal to 60°.Jeevanantham wrote:I have one doubt here. Here Equilateral triangle is considered as 30 : 60 :90 . And taken as x:xsqrt3 :2x .. Why is it so?? Equilateral triangle will be 60 : 60 : 60 . right??
Whereas ABD and ACD are right angle triangle with angles 30°, 60° and 90°
Hope that helps.
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