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If an arc with a length of 12π is 3/4 of the circumference of a circle

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If an arc with a length of 12π is 3/4 of the circumference of a circle  [#permalink]

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New post 10 Jan 2019, 02:23
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If an arc with a length of 12π is 3/4 of the circumference of a circle, what is the shortest distance between the endpoints of the arc?


A. 4

B. \(4\sqrt{2}\)

C. 8

D. \(8\sqrt{2}\)

E. 16

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If an arc with a length of 12π is 3/4 of the circumference of a circle  [#permalink]

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New post 10 Jan 2019, 03:42
Bunuel wrote:
If an arc with a length of 12π is 3/4 of the circumference of a circle, what is the shortest distance between the endpoints of the arc?


A. 4

B. \(4\sqrt{2}\)

C. 8

D. \(8\sqrt{2}\)

E. 16


(3/4)2πr = 12π

i.e. r = 8

The distance between endpoints of arc = Hypotenuse of a right triangle with legs of length equal to radius i.e. 8

Hypotenuse of Triangle = 8√2

Answer: Option D
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Re: If an arc with a length of 12π is 3/4 of the circumference of a circle  [#permalink]

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New post 10 Jan 2019, 04:16
Bunuel wrote:
If an arc with a length of 12π is 3/4 of the circumference of a circle, what is the shortest distance between the endpoints of the arc?


A. 4

B. \(4\sqrt{2}\)

C. 8

D. \(8\sqrt{2}\)

E. 16



given
3/4 * 2*pi* r = 12 pi
r = 8
shortest distance between two endpoints of the arc would be hypotenuse formed along two points on the circle ie radius ..
8sqrt2
IMO D
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Re: If an arc with a length of 12π is 3/4 of the circumference of a circle  [#permalink]

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New post 10 Jan 2019, 04:55
My answer is D

Circumference of circle = 16n

So the radius =8

Now the shortest distance between the two ends can be a straight line ie the hypotenuse of the triangle with the same radius as 2 sides.
Therefore the shortest possible distance is 8sqrt2.
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If an arc with a length of 12π is 3/4 of the circumference of a circle  [#permalink]

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New post 10 Jan 2019, 06:22

Solution


Given:
    • Length of an arc = \(12ᴨ = \frac{3}{4}\) * circumference of the circle

To find:
    • The shortest distance between the endpoints of the arc

Approach and Working:
Length of arc = angle subtended at center/360 * circumference of the circle
    • Implies, angle subtended at center/360 = \(\frac{3}{4}\)
    • Thus, the angle subtented at center = \(\frac{3}{4} * 360 = 270\) degrees
    • The remaining angle = 360 – 270 = 90

Circumference = \(\frac{4}{3} * length of arc = \frac{4}{3} * 12ᴨ = 16ᴨ = 2ᴨr\)
    • Implies, r = 8

So, the \(shortest distance^2 = r^2 + r^2 = 64 + 64\)

Therefore, shortest distance = √128 = 8√2

Hence, the correct answer is Option D

Answer: D

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If an arc with a length of 12π is 3/4 of the circumference of a circle  [#permalink]

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New post 13 Jan 2019, 16:36
>> The remaining angle = 360 – 270 = 90

why are we subtracting from 360? should not be 270 is the angle subtend by the arc?



EgmatQuantExpert wrote:

Solution


Given:
    • Length of an arc = \(12ᴨ = \frac{3}{4}\) * circumference of the circle

To find:
    • The shortest distance between the endpoints of the arc

Approach and Working:
Length of arc = angle subtended at center/360 * circumference of the circle
    • Implies, angle subtended at center/360 = \(\frac{3}{4}\)
    • Thus, the angle subtented at center = \(\frac{3}{4} * 360 = 270\) degrees
    • The remaining angle = 360 – 270 = 90

Circumference = \(\frac{4}{3} * length of arc = \frac{4}{3} * 12ᴨ = 16ᴨ = 2ᴨr\)
    • Implies, r = 8

So, the \(shortest distance^2 = r^2 + r^2 = 64 + 64\)

Therefore, shortest distance = √128 = 8√2

Hence, the correct answer is Option D

Answer: D

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Re: If an arc with a length of 12π is 3/4 of the circumference of a circle  [#permalink]

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New post 13 Jan 2019, 21:03
sunita123 wrote:
>> The remaining angle = 360 – 270 = 90

why are we subtracting from 360? should not be 270 is the angle subtend by the arc?


Image

Hi sunita123,

According to the question, we need to find the length AB in the above figure.

And, we have got the obtuse angle AOB = 270
    Implies, acute angle AOB = 360 - 270 = 90

Now, from triangle AOB, we can find the length of AB using this information.

Regards,
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Re: If an arc with a length of 12π is 3/4 of the circumference of a circle &nbs [#permalink] 13 Jan 2019, 21:03
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