Round hole in a ??? peg

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kstv: Legendary Member; Posts: 610; Joined: Fri Jan 15, 2010 12:33 amFollowed by: 2 members; Thanked: 47 times

by kstv » Sun Apr 18, 2010 5:53 pm

A circle with a radius of x cm is inscribed inside a quadrilateral so that it touches all internal faces of the quadrilateral.

What is the area of the circle?
1. The diagonals of the quadrilateral bisect each other.
2. The perimeter of the quadrilateral is 40 cm.

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Source: Beat The GMAT — Data Sufficiency | Sun Apr 18, 2010 5:53 pm

liferocks: Legendary Member; Posts: 576; Joined: Sat Mar 13, 2010 8:31 pmFollowed by: 1 member; Thanked: 97 times

by liferocks » Sun Apr 18, 2010 6:03 pm

Basically the question is to determine the value of x

from statement 1 --> the quadrilateral is a parallelogram and the only parallelogram which can have a circle inscribed inside will be square as the distance between each pair of parallel sides is equal to the diameter of the circle. but we does not have information on value of x-- insuff

from statement 2-- cannot conclude any thing ---insuff

combining 1 and 2 we get side of the square=2x=10 -- sufficient

Ans C
can you please confirm the OA.

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thephoenix: Legendary Member; Posts: 1560; Joined: Tue Nov 17, 2009 2:38 amFollowed by: 5 members; Thanked: 137 times

by thephoenix » Sun Apr 18, 2010 6:14 pm

liferocks wrote:Basically the question is to determine the value of x
combining 1 and 2 we get side of the square=2x=10 -- sufficient
can you please confirm the OA.

IMO we can make it out from the question itself that the quadrilateral is either a square or a rhombus????isn't it??
the only possible case where a circle can touch all the 4 interfaces of a quadrilateral is square or rhombus
if i am correct then ans sud be B
pls comment

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liferocks: Legendary Member; Posts: 576; Joined: Sat Mar 13, 2010 8:31 pmFollowed by: 1 member; Thanked: 97 times

by liferocks » Sun Apr 18, 2010 6:21 pm

thephoenix wrote:
liferocks wrote:Basically the question is to determine the value of x
combining 1 and 2 we get side of the square=2x=10 -- sufficient
can you please confirm the OA.
IMO we can make it out from the question itself that the quadrilateral is either a square or a rhombus????isn't it??
the only possible case where a circle can touch all the 4 interfaces of a quadrilateral is square or rhombus
if i am correct then ans sud be B
pls comment

it can be kite as well. see this link for diagram https://en.wikipedia.org/wiki/Kite_(geometry)

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akhpad: Legendary Member; Posts: 809; Joined: Wed Mar 24, 2010 10:10 pmFollowed by: 4 members; Thanked: 50 times

by akhpad » Sun Apr 18, 2010 8:11 pm

Given in Q, diagonals are just bisector but they have not mentioned perpendicular bisector.

Square, rhombus and kite have diagonals perpendicular bisector. Diagonals of parallelogram are bisect each other.

A circle can be inscribed even if all sides of quadrilateral are different.

I believe that this is a broad level of Q.

I would go for E. Not sure.

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liferocks: Legendary Member; Posts: 576; Joined: Sat Mar 13, 2010 8:31 pmFollowed by: 1 member; Thanked: 97 times

by liferocks » Sun Apr 18, 2010 8:20 pm

akhp77 wrote:Given in Q, diagonals are just bisector but they have not mentioned perpendicular bisector.

Square, rhombus and kite have diagonals perpendicular bisector. Diagonals of parallelogram are bisect each other.

A circle can be inscribed even if all sides of quadrilateral are different.

I believe that this is a broad level of Q.

I would go for E. Not sure.

I think only parallelograms has diagonals which bisects each other and the only parallelogram with a circle inscribed within it can be a square or rhombus. Please correct me if wrong.

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Testluv: GMAT Instructor; Posts: 1302; Joined: Mon Oct 19, 2009 2:13 pmFollowed by: 164 members; Location: Toronto; Thanked: 539 times; GMAT Score:800

by Testluv » Mon Apr 19, 2010 1:36 am

What is the source of this question? I ask because instead of saying "faces" (which belong to 3-D figures), the question should have said "sides".
_______
The question stem tells us that "a circle with a radius of x cm is inscribed inside a quadrilateral so that it touches all internal faces of the quadrilateral."

The kinds of quadrilaterals in which a circle can be inscribed (such that all four sides of the quadrilateral touch the circle) include: rhombi, kites, and trapezoids. (Squares are a type of rhombus. A non-square rhombus is like a "pushed over" square).

(1) gives us no quantitative information, no numbers. And we don't get any quantitative informaton in the question stem either. Thus, (1) is clearly insufficient.

From (2), we get the perimeter of the quadrilateral. The area of a quadrilateral--and, therefore, the area of a circle inscribed in a quadrilateral--is maximized when its sides are equalized AND when they meet at right angles.

Thus, if their perimeters are the same, then the area of a square is bigger than the area of a "pushed over" square. This is because as we push the square over, its height decreases (accordingly, the diameter of the circle decreases). Insufficient.

(1) + (2):

(1) tells us that the diagonals bisect each other. To bisect is to cut in half. The diagonals of squares and other rhombi cut each other in half; the diagonals of kites and trapezoids do not.

So, now we know that the shape can be a square or a "pushed over" square, and that the sides are all equal. But, again, because the area of a quadrilateral is maximized when the sides meet at right angles, the circle will be bigger if it were inscribed in a square (than if it were inscribed in a "pushed over" square).

Choose E.

Kaplan Teacher in Toronto

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Mon Apr 19, 2010 1:52 am

kstv wrote:A circle with a radius of x cm is 1 inscribed inside a quadrilateral 2 so that it touches all internal faces of the quadrilateral.

What is the area of the circle?
1. The diagonals of the quadrilateral bisect each other.
2. The perimeter of the quadrilateral is 40 cm.

Please note: The word "Inscribed" in Geometry means what is there in the 2 above, so writing 2 after 1 is luxury here.

Area of a circle with radius x is known when x is known.

(1) There are only two types of quadrilaterals in which the diagonals bisect each other, a square, or a rhombus. So, if it's a square, x is unique, but if it's a rhombus, x would depend on few more factors than one. Nevertheless, be x unique or not, we don't have a number to play it further. Insufficient

(2) There are countless quadrilaterals for a given perimeter. Insufficient

Combining (1) and (2)

Contest remains

a square, or a rhombus?

[spoiler]E[/spoiler]

The mind is everything. What you think you become. -Lord Buddha

Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001

www.manyagroup.com

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Testluv: GMAT Instructor; Posts: 1302; Joined: Mon Oct 19, 2009 2:13 pmFollowed by: 164 members; Location: Toronto; Thanked: 539 times; GMAT Score:800

by Testluv » Mon Apr 19, 2010 1:56 am

This:

Please note: The word "Inscribed" in Geometry means what is there in the 2 above, so writing 2 after 1 is luxury here.

is 100% correct.

The question should have been worded:

A circle is inscribed in a quadrilateral. What is the area of the circle?

Kaplan Teacher in Toronto

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Testluv: GMAT Instructor; Posts: 1302; Joined: Mon Oct 19, 2009 2:13 pmFollowed by: 164 members; Location: Toronto; Thanked: 539 times; GMAT Score:800

by Testluv » Mon Apr 19, 2010 9:46 am

This:

A circle can be inscribed even if all sides of quadrilateral are different.

is correct. We can insrcribe a circle in a trapezoid all of whose sides are inequal.

Kaplan Teacher in Toronto

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