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Circomference of circle O

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Source: — Data Sufficiency |
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by Patrick_GMATFix » Wed Feb 19, 2014 7:57 am
To find the circumference of a circle we need its radius. (1) allows us to find the radius since we know that the triangle's legs will follow the ratio of a 45-45-90 (isosceles right triangle). (2) allows us to find the circumference directly because the arc in question is exactly a quarter of the circle (central angle is 90 = 360/4).

The full solution below is taken from the GMATFix App.

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by [email protected] » Thu Feb 20, 2014 12:04 am
Hi LulaBrazilia,

Any time a "circle question" appears on the GMAT, you have to put your focus on the RADIUS. Just about any question you'll ever be asked that involves a circle will eventually come down to the radius.

Here, we have to be suspicious about the picture, since pictures in DS questions are not always drawn to scale. The prompt does tell us the O is the center of the circle, so we can make some deductions about the triangle within the circle:

1) The triangle is made up of 2 radii (OX and OZ), so those two legs of the triangle are equal.
2) The triangle must be a 45/45/90 triangle, so we'll need to know THOSE special rules to answer the question.

The prompt asks us for the Circumference of the circle. We'll need to know the radius to answer the question.

Fact 1: The perimeter of the triangle is 20 + 10(root2)

Since a 45/45/90 triangle has sides of X/X/X(root2), we know that: X + X + X(root2) = 20 + 10(root2).

We can solve for X, so we know the radius and we CAN answer the question.
Fact 1 is SUFFICIENT.

Fact 2: Arc XYZ = 5pi

This tells us that 1/4 of the circle's circumference = 5pi. We know it's 1/4 because a circle has 360 degree and the inner angle is a 90 degree angle (90/360 = 1/4). Since 1/4 of the circle = 5pi, the full circle = 20pi. This answers the question.
Fact 2 is SUFFICIENT.

Final Answer: D

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by Brent@GMATPrepNow » Thu Feb 20, 2014 8:52 am
LulaBrazilia wrote:Image
What is the circumference of the circle above with center O?

1) The perimeter of triangle OXZ is 20 + 10 root(2)
2) The length of arc XYZ is 5*pi
IMPORTANT: For geometry DS questions, we are typically checking to see whether the statements "lock" a particular angle or length into having just one value. This concept is discussed in much greater detail in our free video: https://www.gmatprepnow.com/module/gmat- ... cy?id=1103

This technique can save a lot of time. In fact, we can actually solve this question without applying any geometry.

First notice that the SHAPE of the diagram is already "locked in." That is, we have 2 radii that form a 90-degree angle and points X and Z are at the ends of the radii. The only part of this diagram that isn't locked in is the SIZE.

That is, the diagram can have many different sizes:
Image

SUPER IMPORTANT: If a statement locks in the SIZE of the diagram, then that statement must be sufficient.

Target question: What is the circumference of the circle above with center O?

Statement 1: The perimeter of triangle OXZ is 20 + 10 root(2)
All we need to do here is recognize that we COULD draw the diagram (whatever size we wish) and then measure the perimeter of ∆OXZ. If the perimeter of ∆OXZ is less than 20 + 10 root(2), then draw a larger diagram. If the perimeter of ∆OXZ is greater than 20 + 10 root(2), then draw a smaller diagram.
Notice that we could keep doing this until we find the ONE DIAGRAM IN THE ENTIRE UNIVERSE in which the perimeter of ∆OXZ is EXACTLY 20 + 10 root(2).
At that point, we could measure the circle and determine its circumference
Since we COULD answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: The length of arc XYZ is 5*pi
Here, we can recognize that arc XYZ is 1/4 of the entire circumference. So, the circumference must have length 20(pi)
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

ALTERNATIVELY, we COULD draw the diagram (whatever size we wish) and then measure the length of arc XYZ. If it's length is less than 5(pi) then draw a larger diagram. If the arc XYZ is greater than 5(pi), then draw a smaller diagram. We could keep doing this until we find the ONE DIAGRAM IN THE ENTIRE UNIVERSE in which arc XYZ has length 5(pi), at which point, we could measure the circle and determine its circumference
Since we COULD answer the target question with certainty, statement 2 is SUFFICIENT

Answer = D

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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