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GMAT Geometry-DS

by sanjay12590 » Sat Aug 16, 2014 5:44 am
Guys need help with this one!!

A circle C is drawn around a square S such that the sides of the square become the four chords of the circle. What is the area of square S?

(1) Had a circle been drawn such that the four sides of square S were tangents to the circle, the area of the circle would be 30 square centimetres less than the area of circle C

(2) Had a circle been drawn with the diagonal of square S as its radius, the area of the circle have been 180 square centimetres more than the area of circle C


A)Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
B)Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient
C)Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient
D)Each statement ALONE is sufficient
E)Statements (1) and (2) TOGETHER are NOT sufficient

Thanks!!
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Source: — Data Sufficiency |
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by GMATGuruNY » Sat Aug 16, 2014 7:29 am
sanjay12590 wrote:Guys need help with this one!!

A circle C is drawn around a square S such that the sides of the square become the four chords of the circle. What is the area of square S?

(1) Had a circle been drawn such that the four sides of square S were tangents to the circle, the area of the circle would be 30 square centimetres less than the area of circle C

(2) Had a circle been drawn with the diagonal of square S as its radius, the area of the circle have been 180 square centimetres more than the area of circle C
Statement 1:
Image
In the figure above, r = the radius of Circle C.

∆OAE is a 45-45-90 triangle.
In a 45-45-90 triangle, the sides are in the following ratio:
x : x : x√2.
In ∆OAE, x√2 = r.
Thus, OE -- the radius of the smaller circle -- is equal to r/√2.

Since the difference between the area of Circle C and the smaller circle is 30, we get:
πr² - π(r/√2)² = 30
πr² - πr²/2 = 30
2πr² - πr² = 60
πr² = 60.

Since the value of r can be determined, so can the length of AE, which is equal to 1/2 of the length of each side of square ABCD.
Thus, the area of the square ABCD can be determined.
SUFFICIENT.

Statement 2:
Check whether the figure in Statement 1 is the only case that will satisfy statement 2.

Let Circle X = the circle whose radius is equal to diagonal AC.
In statement 1, the area of Circle C = 60.
Diagonal AC = 2r.
If the radius of Circle C doubles to 2r, its area will QUADRUPLE.
Thus, the area of Circle X = 4*60 = 240.
Resulting difference:
Circle X - Circle C = 240-60 = 180.
Success!

If the area of Circle C decreases or increases, the difference between Circle X and Circle C will also decrease or increase.
Implication:
The figure in Statement 1 is the ONLY CASE that will satisfy Statement 2.
Since Statement 2 implies the same information as Statement 1 -- and Statement 1 is sufficient -- Statement 2 is also SUFFICIENT.

The correct answer is D.
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by GMATinsight » Sat Aug 16, 2014 9:26 am
sanjay12590 wrote:Guys need help with this one!!

A circle C is drawn around a square S such that the sides of the square become the four chords of the circle. What is the area of square S?

(1) Had a circle been drawn such that the four sides of square S were tangents to the circle, the area of the circle would be 30 square centimetres less than the area of circle C

(2) Had a circle been drawn with the diagonal of square S as its radius, the area of the circle have been 180 square centimetres more than the area of circle C
I would only say that this question requires the judgement to be made to save time which is quite easy even if the entire calculations are not done completely.

Question : What is the area of square ABCD (as per the figure drawn here)?

To calculate the area we require side of ABCD
Diameter of Circle = Diagonal of Square ABCD = √2 x The side of Square ABCD [Let's say 'a']
i.e. Diameter of Circle, C = a√2

Statement 1) Had a circle been drawn such that the four sides of square S were tangents to the circle, the area of the circle would be 30 square centimetres less than the area of circle C

Diameter of Circle inside Square ABCD = Side of the Square ABCD = a
(Pi/4)(a√2)^2 - (Pi/4)(a)^2 = 30
'a' can be calculated and therefore Area of Square too is determinable

SUFFICIENT

Statement 2) Had a circle been drawn with the diagonal of square S as its radius, the area of the circle have been 180 square centimetres more than the area of circle C
Area of circle with the diagonal of square S as its radius = (Pi)(a√2)^2 - (Pi/4)(a√2)^2 = 180
'a' can be calculated and therefore Area of Square too is determinable

SUFFICIENT

Answer: Option D
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