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10 May 2021, 01:30
Kudos
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
61% (02:27) correct
39%
(02:54)
wrong
based on 93
sessions
History
Date
Time
Result
Not Attempted Yet
ABC is an equilateral triangle of area 3, and arc DE is centered at C. If E is the midpoint of AC, what is the area of the shaded region?
A. \(3-\frac{\sqrt{3}\pi}{2}\)
B. \(3-\frac{\pi}{\sqrt{3}}\)
C. \(3-\frac{\pi}{2}\)
D. \(3-\frac{\pi}{2\sqrt{3}}\)
E. \(3-\frac{\pi}{6}\)
Attachment:
1.png [ 8 KiB | Viewed 3672 times ]
11 May 2021, 01:15
Kudos
Bookmarks
Bunuel
The area of equilateral triangle is \(side^2*\frac{\sqrt{3}}{4}\).
Given that \(side^2*\frac{\sqrt{3}}{4}=3\) --> \(side=2\sqrt[4]{3}\).
Since E is the midpoint of AC, then CE = radius = half of the side = \(\sqrt[4]{3}\).
Next, as ABC is an equilateral triangle, then angle C is 60 degrees, thus sector CED is 1/6 of the area of the whole circle --> \(area=\pi*(\sqrt[4]{3})^2*\frac{1}{6}=\frac{\pi\sqrt{3}}{6}=\frac{\pi}{2\sqrt{3}}\).
The area of the shaded region = \(3-\frac{\pi}{2\sqrt{3}}\).
Answer: D.
General Discussion
10 May 2021, 03:03
Kudos
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Bunuel
Area of triangle = \(\frac{\sqrt{3}}{4}*a^2\) = 1
a= side
a= 2* \(3^1/4\)
Radius of the arc=\( \frac{a}{2}\) ( we are given E is the mid point of AC )
Are of the arc=\( 1/6 * Pi * 3^1/2\)
Area of shaded region= 3- area of the arc
=3-\( \frac{pi}{6 }* 3^1/2\)
=3-\(\frac{pi}{2}*\sqrt{3}\)
D is the right answer
