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Difficulty:
65%
(hard)
Question Stats:
48% (02:52) correct
52%
(02:45)
wrong
based on 27
sessions
History
Date
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Find the ratio of area of shaded circle to the area of Equilateral triangle
A. \(π : 36\)
B. \(2π : 27\)
C. \(π : 54\)
D. \(π : 27\sqrt{3}\)
E. \(π : 108\)
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Posts: 15
The answer will be D. It is pretty hard to explain as well as to type. If you want me to explain, kindly let me know.
Hi,
I don't think it is hard to explain if one just applies the theorems and let's the user read through them if he wants to really understand Gauravsirohi.
Let a be the length of a side. A solution could look like this:
We know that for a circle inscribed in an EL, the radius is (1/3)*h=(root(3)/6)*a. We know:
diameter_inscribed_circle=2r, h_EL_triangle=3r -> h_small_triangle=3r-2r=r
Now imagine that you draw a base which touches the small circle right at it's lowest point. We have created another EL triangle (the smaller triangle has same angles and therefore proportions as the big one) and we also now that the height of that triangle is r.
We apply again what we know for an circle inscribed in an EL, just this time for our small circle:
r_small_circle=(1/3)*h_small_circle=(root(3)/18)*a
The question asks for the ratio of the areas. For an EL triangle, area is given by (root(3)/4)*a^2, so we have:
(A_small_circle)/(Area_EL_triangle)= (pi*(root(3)/18)*a)^2) / ((root(3)/4)*a^2)=12pi/(root(3)*18^2)=pi/(27*root(3), which is (D)
I don't think it is hard to explain if one just applies the theorems and let's the user read through them if he wants to really understand Gauravsirohi.
Let a be the length of a side. A solution could look like this:
We know that for a circle inscribed in an EL, the radius is (1/3)*h=(root(3)/6)*a. We know:
diameter_inscribed_circle=2r, h_EL_triangle=3r -> h_small_triangle=3r-2r=r
Now imagine that you draw a base which touches the small circle right at it's lowest point. We have created another EL triangle (the smaller triangle has same angles and therefore proportions as the big one) and we also now that the height of that triangle is r.
We apply again what we know for an circle inscribed in an EL, just this time for our small circle:
r_small_circle=(1/3)*h_small_circle=(root(3)/18)*a
The question asks for the ratio of the areas. For an EL triangle, area is given by (root(3)/4)*a^2, so we have:
(A_small_circle)/(Area_EL_triangle)= (pi*(root(3)/18)*a)^2) / ((root(3)/4)*a^2)=12pi/(root(3)*18^2)=pi/(27*root(3), which is (D)
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