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

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Re: What is the circumference of the circle above with center O? [#permalink]
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]
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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.

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Re: What is the circumference of the circle above with center O? [#permalink]
We need to find the value of $$2 \pi R$$
From the figure we can tell that OX=OZ = radius or R. Since these two sides are equal and the other angle is given as $$90^ {\circ}$$, triangle XOZ is an isosceles right triangle with sides $$x:x:x\sqrt{2}$$

Armed with this info, lets look at the statements:

Statement 1: Perimeter = $$20+10\sqrt{2}$$
This straight away gives us the radius as 10..........................Sufficient

Statement 2: Length of arc XYZ = $$5\pi$$
We know that Length of an arc = $$x^{\circ}$$ * Circumference/360
Therefore, $$5\pi$$ = 90 * 2 * $$\pi$$ * R/360
We can find the circumference..........................Sufficient

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Re: What is the circumference of the circle above with center O? [#permalink]
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Attachment:
Circle.png
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 $$5\pi$$.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

By Variable Approach method, we can identify a circle with just 1 varable, which is a radius. Thus D is the answer most likely.
Let $$r$$ be the radius of the above circle.

Condition 1)
$$2 \cdot r + r\sqrt{2} = r \cdot (2 + \sqrt{2} ) = 20 + 10\sqrt{2}$$
Thus $$r = 10$$.
This condition is sufficient.

Condition 2)
The arc XYZ is $$\frac{1}{4} \cdot 2 \pi r = 5 \pi$$.
Thus $$r = 10$$.
This condition is also sufficient.

Therefore D is the answer as expected.

-> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.
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Re: What is the circumference of the circle above with center O? [#permalink]
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1. Traingle is 45,45,90 degree triangle.
So two sides are same and length of radius is 10
Circumference of circle is 20pi
2. Is an arc is is known and angle subtending that arc is known than radius can be calculated.
So Both 1 and 2 are sufficient.

D

Sent from my A0001 using GMAT Club Forum mobile app
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Re: What is the circumference of the circle above with center O? [#permalink]
EMPOWERgmatRichC wrote:
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.

GMAT assassins aren't born, they're made,
Rich

Hello EMPOWERgmatRichC
Thanks for the explanation with kudos.
Could you explain (with example if possible) a bit how the highlighted things change the way?
Thanks__
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Re: What is the circumference of the circle above with center O? [#permalink]
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.

Thanks for the explanation. I just have a query in this explanation. Did you mean $$2πr$$ by the highlighted part? It seems that circumference is $$2πr$$
Thanks__
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Re: What is the circumference of the circle above with center O? [#permalink]
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EMPOWERgmatRichC wrote:
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.

GMAT assassins aren't born, they're made,
Rich

Hello EMPOWERgmatRichC
Thanks for the explanation with kudos.
Could you explain (with example if possible) a bit how the highlighted things change the way?
Thanks__

If Fact 1 had stated:

1) |R - 6| = 4 where R is the radius of circle O (above).

In this scenario, R could be two potential values (R = 10 or R = 2) - and each result would lead to a different answer to the given question (re: What is the circumference of the circle?). Even though there's just one variable in this equation, the Absolute Value implies that there were would be 2 solutions to the equation (and there were). Here, Fact 1 would be INSUFFICIENT.

GMAT assassins aren't born, they're made,
Rich
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Re: What is the circumference of the circle above with center O? [#permalink]
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Re: What is the circumference of the circle above with center O? [#permalink]
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