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13 Aug 2018, 04:46
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D
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In the figure above, an equilateral triangle is inscribed in a circle. If the arc bounded by adjacent corners of the triangle is between \(4\pi\) and \(6\pi\) long, which of the following could be the diameter of the circle?
(A) 6.5
(B) 9
(C) 11.9
(D) 15
(E) 23.5
Attachment:
circle.jpg [ 20.08 KiB | Viewed 12624 times ]
13 Aug 2018, 08:48
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Bunuel
We know that, each of the three arcs corresponds to 120 degree central angle. (Since the measure of central angle is twice the measure of angle at circumference).
hence each arc bounded by adjacent corners of the triangle corresponds to one-third of the circumference of circle.
Given, \(4\pi\)<\(\frac{2\pi*r}{3}\)<\(6\pi\)
Or, 4*3<2r<6*3
Or, 12<d<18
Among the options, 15 is in the range of diameter.
Ans. (D)
31 Mar 2020, 09:54
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the arc bounded by adjacent corners of the triangle is between 4π and 6π long.
so min value of the arc is 4π.
=> min value of the circumference of the circle is 3*4π = 12π ( 3 is being multiplied because the equilateral triangle divides the circle in equal three parts )
let, diameter of the circle is D .
.˙. min value of πD is 12π
or, min value of D is 12
Similarly, max value of the arc is 6π.
max value of the circumference of the circle is 3*6π = 18π
max value of πD is 18π
or, max value of D is 18
so, from the answer choices D could be 15 which lies between 12 and 18.
correct answer is D
so min value of the arc is 4π.
=> min value of the circumference of the circle is 3*4π = 12π ( 3 is being multiplied because the equilateral triangle divides the circle in equal three parts )
let, diameter of the circle is D .
.˙. min value of πD is 12π
or, min value of D is 12
Similarly, max value of the arc is 6π.
max value of the circumference of the circle is 3*6π = 18π
max value of πD is 18π
or, max value of D is 18
so, from the answer choices D could be 15 which lies between 12 and 18.
correct answer is D
30 Nov 2021, 07:14
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Attachment:
temp-3.JPG [ 30.88 KiB | Viewed 8117 times ]
Since the Arc made by two adjacent corners of the triangle is making 120˚ angle at the center so length of the arc is given by \(\frac{120˚}{360˚} *2 \pi r\)
[ Watch this video to know how ]
=> Arc length = \(\frac{1}{3} *2 \pi r\) and it is between \(4\pi\) and \(6\pi\)
=> \(4\pi\) <\(\frac{1}{3} *2 \pi r\) < \(6\pi\)
=> 4 < \(\frac{d}{3}\) < 6 (as Diameter (d) = 2r)
=> 4*3 < d < 3*6
=> 12 < d < 18
Only Option D (15) is in this range
So, Answer will be D
Hope it helps!
Watch the following video to Learn Properties of Equilateral Triangle inscribed in a Circle
