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

alex.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 Nice sign convention. I will just use sqrt for clarity 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

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? Refer to the following figure, 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.

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??

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?? ABC is a equilateral triangle with all angles equal to 60°. Whereas ABD and ACD are right angle triangle with angles 30°, 60° and 90° Hope that helps.