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In the circle shown in the figure, the length of the arc ACB is 3 time

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Bunuel
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In the circle shown in the figure, the length of the arc ACB is 3 time [#permalink] New post   18 Jul 2019, 22:14
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In the circle shown in the figure, the length of the arc ACB is 3 times the length of the arc AB. What is the length of the line segment AB ?


A. \(\sqrt{2}\)

B. \(\sqrt{3}\)

C. 2

D. \(3\sqrt{2}\)

E. \(\sqrt{3}\)

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itoyj
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Re: In the circle shown in the figure, the length of the arc ACB is 3 time [#permalink] New post   18 Jul 2019, 22:24
Arc ACB is 3 times Arc AB
complete angle is 360
So 360= 3x+x
X= 90
So angle AOB =90
Applying Pythagorean theorem OA2+OB2= AB2
Hence AB= 3√2

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Re: In the circle shown in the figure, the length of the arc ACB is 3 time [#permalink] New post   18 Jul 2019, 22:24
Bunuel wrote:

In the circle shown in the figure, the length of the arc ACB is 3 times the length of the arc AB. What is the length of the line segment AB ?


A. \(\sqrt{2}\)

B. \(\sqrt{3}\)

C. 2

D. \(3\sqrt{2}\)

E. \(\sqrt{3}\)

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Attachment:
2019-07-19_0910.png


Since the length of the arc ACB is 3 times the length of the arc AB
=> Angle extending by line segment AB on center = \(\frac{360}{(1+3)}\) = 90

The length of the line segment AB = \(3 *\sqrt{2}\)
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Re: In the circle shown in the figure, the length of the arc ACB is 3 time [#permalink] New post   19 Jul 2019, 04:22
Bunuel wrote:

In the circle shown in the figure, the length of the arc ACB is 3 times the length of the arc AB. What is the length of the line segment AB ?


A. \(\sqrt{2}\)

B. \(\sqrt{3}\)

C. 2

D. \(3\sqrt{2}\)

E. \(\sqrt{3}\)

Show Spoiler
Attachment:
2019-07-19_0910.png


Given: In the circle shown in the figure, the length of the arc ACB is 3 times the length of the arc AB.
Asked: What is the length of the line segment AB ?

Length of arc AB = 1/(1+3) * Circumference of the circle = 1/4 * Circumference of the circle
Angle AOB = 360/4 = 90

Since AO = OB = Radius = 3
AB = \(\sqrt{3^2+3^2} = 3\sqrt{2}\)

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Re: In the circle shown in the figure, the length of the arc ACB is 3 time [#permalink] New post   25 Jul 2019, 04:37
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Solution


To find:
    • A circle with center O
    • The length of the arc ACB is 3 times the length of the arc AB
    • Radius = OC = 3

To find:
    • The length of the line segment AB

Approach and Working Out:
    • Circumference of the circle = length of arc ACB + length of arc AB = 3 * length of arc AB + length of arc AB = 4 * length of arc AB
      o This implies, angle AOB = 360/4 = 90 degrees

    • Now, in triangle AOB, OA = OB = radius = 3
      o Thus, \(AB^2 = OA^2 + OB^2 = 3^2 + 3^2 = 18\)
      o Therefore, AB = 3√2

Hence, the correct answer is Option E.

Answer: E

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Re: In the circle shown in the figure, the length of the arc ACB is 3 time [#permalink]
25 Jul 2019, 04:37
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