Question: The diameter of circle S is equal in length to a side of a certain square. The diameter of circle T is equal in length to a diagonal of the same square. The area of circle T is how many times the area of circle S ?
- √2
- √2+1
- 2
- π
- √2π
“The diameter of circle S is equal in length to a side of a certain square.”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.
Solution and Explanation:
Approach Solution 1:
It is asked that how many times the area of circle S is The area of circle T
Let the radius of the circle S and the side of the square a be, respectively, s and t.
The side of a particular square is the same length as the diameter of circle S.
2s = a
s =\(\frac{a}{2}\)
A diagonal of the same square has the same length as the diameter of circle T.
2t = √2a
t = a/√2
How many times the area of circle S is the area of circle T?
\(\pi*t^2=x*\pi*s^2\)
\(a^{\frac{2}{2}}=x*a^{\frac{2}{4}}\)
x = 2
The answer is 2
Correct Answer: C
Approach Solution 2:
There is another approach to this question which is fairly simple
Let's answer the question by giving the dimensions some convenient values.
Say the square's sides are each two lengths long.
It follows that 2 = the diameter of circle S.
The diagonal has length \(2\sqrt2\) if 2 is equal to the length of each side of the square X.
Therefore, the diameter of circle T is equal to \(2\sqrt2\)
How many times the area of circle S is there in the area of circle T?
If 2 = the diameter of circle S,
then its radius = 1
Area = πr² = π(1²) = π
If 2√2 = the diameter of circle T,
then its radius = √2
Area = πr² = π(√2)² = 2π
The area of circle T is 2 times the area of circle S
The answer is 2
Correct Answer: C
Approach Solution 3:
How many times does the area of circle S equal the area of circle T?
Let s and t represent, respectively, the radius of the circle S and the side of the square a.
A particular square has sides that are the same length as the diameter of circle S.
2s = a
s =\(\frac{a}{2}\)
The same square's diagonal is the same length as circle T's diameter.
2t = √2a
t = a/√2
The area of circle T is how many times that of circle S?
\(\pi*t^2=x*\pi*s^2\)
\(a^{\frac{2}{2}}=x*a^{\frac{2}{4}}\)
x = 2
The answer is 2
Correct Answer: C
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