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What is the circumference of the circle above with center O?

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What is the circumference of the circle above with center O? [#permalink]  26 Dec 2012, 03:59
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79% (01:49) correct 21% (00:55) wrong based on 215 sessions
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What is the circumference of the circle above with center O?

(1) The perimeter of triangle OXZ is 20 + 10\sqrt{2}.
(2) The length of arc XYZ is 5\pi.
[Reveal] Spoiler: OA
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Re: What is the circumference of the circle above with center O? [#permalink]  26 Dec 2012, 04:05
Expert's post

What is the circumference of the circle above with center O?

(1) The perimeter of triangle OXZ is 20 + 10\sqrt{2}. Since <O=90 degrees and OX=OZ=r, then XZ=r\sqrt{2}. Thus we have that r+r+r\sqrt{2}=20 + 10\sqrt{2} --> we can find r, thus we can find the circumference. Sufficient.

(2) The length of arc XYZ is 5\pi. Again, since <O=90 degrees, then the length of arc XYZ is 1/4 of the circumference (360 degrees), thus the circumference = 20\pi. Sufficient.

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Re: What is the circumference of the circle above with center O? [#permalink]  12 Jan 2013, 07:33
Bunuel wrote:

What is the circumference of the circle above with center O?

(1) The perimeter of triangle OXZ is 20 + 10\sqrt{2}. Since <O=90 degrees and OX=OZ=r, then XZ=r\sqrt{2}. Thus we have that r+r+r\sqrt{2}=20 + 10\sqrt{2} --> we can find r, thus we can find the circumference. Sufficient.

(2) The length of arc XYZ is 5\pi. Again, since <O=90 degrees, then the length of arc XYZ is 1/4 of the circumference (360 degrees), thus the circumference = 20\pi. Sufficient.

Is there a formula for this explanation?
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Re: What is the circumference of the circle above with center O? [#permalink]  13 Jan 2013, 02:29
Expert's post
fozzzy wrote:
Bunuel wrote:

What is the circumference of the circle above with center O?

(1) The perimeter of triangle OXZ is 20 + 10\sqrt{2}. Since <O=90 degrees and OX=OZ=r, then XZ=r\sqrt{2}. Thus we have that r+r+r\sqrt{2}=20 + 10\sqrt{2} --> we can find r, thus we can find the circumference. Sufficient.

(2) The length of arc XYZ is 5\pi. Again, since <O=90 degrees, then the length of arc XYZ is 1/4 of the circumference (360 degrees), thus the circumference = 20\pi. Sufficient.

Is there a formula for this explanation?

Yes, check here: math-circles-87957.html (see Arc Length under Arcs and Sectors chapter)
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Re: What is the circumference of the circle above with center O?   [#permalink] 13 Jan 2013, 02:29
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