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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] New post 26 Dec 2012, 03:59
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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] New post 26 Dec 2012, 04:05
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What is the circumference of the circle above with center O?

(1) The perimeter of triangle OXZ is \(20 + 10\sqrt{2}\). 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.

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

Answer: D.


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

Answer: D.


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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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: What is the circumference of the circle above with center O? [#permalink] New post 02 Jan 2015, 06:56
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Re: What is the circumference of the circle above with center O? [#permalink] New post 02 Jan 2015, 13:06
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Hi All,

Even though this DS question is about Geometry rules, it involves a great Algebra shortcut that you'll see at least once on Test Day.

We're given a circle with an ISOSCELES right triangle within it (with the 3 vertices at the center of the circle and on the circumference), so we have a 45/45/90 right triangle. We're asked for the circumference of the circle.

Fact 1: The perimeter of triangle OXZ = 20 + 10\sqrt{2}

Since the triangle is a 45/45/90, we know the sides are A, A and A\sqrt{2}, thus...

A + A + A\sqrt{2} = 20 + 10\sqrt{2}

We have ONE VARIABLE and ONE equation, so we CAN solve it (and since there are no additional possible answers, re. due to exponents, absolute values, etc. - there will be JUST ONE ANSWER.)
Fact 1 is SUFFICIENT.

Fact 2: Arc XYZ = 5pi

Since this arc is based on a 90 degree angle, we CAN figure out what fraction of the circumference we're dealing with (it's 1/4, but that math is not necessary). In this way we have the SAME shortcut that we had in Fact 1: ONE VARIABLE and ONE equation, so we CAN solve it.
Fact 2 is SUFFICIENT.

Final Answer:
[Reveal] Spoiler:
D


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Re: What is the circumference of the circle above with center O?   [#permalink] 02 Jan 2015, 13:06
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