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# Triangle ACE is equilateral with side lengths of 8. Points B and D are

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Math Expert
Joined: 02 Sep 2009
Posts: 60646
Triangle ACE is equilateral with side lengths of 8. Points B and D are  [#permalink]

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30 Oct 2018, 02:42
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Difficulty:

85% (hard)

Question Stats:

44% (02:47) correct 56% (02:21) wrong based on 40 sessions

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Triangle ACE is equilateral with side lengths of 8. Points B and D are the midpoints of line segments AC and CE respectively. Line segment BD is a diameter of the circle with center F. What is the area of the shaded region?

A. 8√2 − 4π
B. 12√3−2π
C. 12√3 − 4π
D. 16√3 − 2π
E. 16√2 − 4π

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Manager
Joined: 09 Jun 2018
Posts: 183
Location: United States
Schools: Ross '22 (II)
GPA: 3.95
WE: Operations (Transportation)
Re: Triangle ACE is equilateral with side lengths of 8. Points B and D are  [#permalink]

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30 Oct 2018, 12:51
Side of eq. triangle = 8

Area of triangle = $$\sqrt{3}$$*8*8/4 = 16$$\sqrt{3}$$

Property of equilateral triangle is that line segment connecting midpoints of two sides divides the triangle in the ratio of 1:3
So, portion below the line segment joining midpoints = (3/4) * 16$$\sqrt{3}$$ = 12$$\sqrt{3}$$

Diameter of circle = 4, radius = 2
Area = pi*2*2 = 4*pi
Half the area = 2*pi

Area of shaded region = 12$$\sqrt{3}$$ - 2*$$pi$$

Hence Option B!
Manager
Joined: 14 Jun 2018
Posts: 210
Re: Triangle ACE is equilateral with side lengths of 8. Points B and D are  [#permalink]

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30 Oct 2018, 13:25
Area of ACE = √3/4*8*8 = 16√3
Area of BDC = 1/4 * ACE = 4√3
Area of semi circle B-F-D = 1/2 * π * 2 * 2 = 2π
Area of shaded region = 16√3 - 4√3 - 2π = 12 - 2π
Re: Triangle ACE is equilateral with side lengths of 8. Points B and D are   [#permalink] 30 Oct 2018, 13:25
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