Summer is Coming! Join the Game of Timers Competition to Win Epic Prizes. Registration is Open. Game starts Mon July 1st.

 It is currently 20 Jul 2019, 15:12 ### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # What is the circumference of the circle above with center O?

Author Message
TAGS:

### Hide Tags

Manager  Joined: 02 Dec 2012
Posts: 174
What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

12 00:00

Difficulty:   5% (low)

Question Stats: 81% (01:15) correct 19% (01:33) wrong based on 948 sessions

### HideShow timer Statistics Attachment: Circle.png [ 10.62 KiB | Viewed 18828 times ]
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 $$5\pi$$.
Math Expert V
Joined: 02 Sep 2009
Posts: 56304
Re: What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

1
4 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.

_________________
##### General Discussion
Director  Joined: 29 Nov 2012
Posts: 731
Re: What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

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?
Math Expert V
Joined: 02 Sep 2009
Posts: 56304
Re: What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

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)
_________________
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 14590
Location: United States (CA)
GMAT 1: 800 Q51 V49 GRE 1: Q170 V170 Re: What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

2
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.

GMAT assassins aren't born, they're made,
Rich
_________________
760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****

# Rich Cohen

Co-Founder & GMAT Assassin Follow
Special Offer: Save \$75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/
Manager  S
Joined: 22 Nov 2016
Posts: 205
Location: United States
GPA: 3.4
Re: What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

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

_________________
Kudosity killed the cat but your kudos can save it.
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7612
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

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.
_________________
Intern  B
Joined: 19 Oct 2015
Posts: 26
Re: What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

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
Non-Human User Joined: 09 Sep 2013
Posts: 11720
Re: What is the circumference of the circle above with center O?  [#permalink]

### Show Tags

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________ Re: What is the circumference of the circle above with center O?   [#permalink] 18 Jul 2018, 08:36
Display posts from previous: Sort by

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