If rectangle ABCD is inscribed in the circle above, what is
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A
B
C
D
E
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 Brent@GMATPrepNow
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Draw a line connecting points A and C.
An important circle property (see video below for more info) tells us that, if we have a 90degree inscribed angle, then that angle must be containing ("holding") the DIAMETER of the circle.
So, we know that AC = the diameter of the circle.
To find the hypotenuse of the red triangle, we can EITHER apply the Pythagorean Theorem OR recognize that 5 and 12 are part of the Pythagorean triplet 51213
With either approach, we learn that AC = 13
IMPORTANT: if the diameter (AC) is 13, then the radius = 13/2 = 6.5
What is the area of the circular region?
Area of circle = Ï€rÂ²
So, area = Ï€(6.5Â²)
= 42.25Ï€
Answer: B
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Brent
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The rectangle is equally placed on the center of the circle.
Let \(O\) be the center of the circle.
And \(X\) be the midpoint of \(AB\).
Then \(OX = \frac{5}{2} \) and \(XB = \frac{12}{2} = 6\)
Triangle \(OXB\) is a right angles triangle.
\(OB = radius = \sqrt{6^2+2.5^2}=\sqrt{42.5}\)
Area \(= \pi \cdot r^2 = 42.5\pi\)
Hence, __B__
Let \(O\) be the center of the circle.
And \(X\) be the midpoint of \(AB\).
Then \(OX = \frac{5}{2} \) and \(XB = \frac{12}{2} = 6\)
Triangle \(OXB\) is a right angles triangle.
\(OB = radius = \sqrt{6^2+2.5^2}=\sqrt{42.5}\)
Area \(= \pi \cdot r^2 = 42.5\pi\)
Hence, __B__
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 Rich.C@EMPOWERgmat.com
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Hi All,
We're told that a rectangle with sides of 5 and 12 is inscribed in the circle. We're asked for the area of the circular region. This question is based on a number of math patterns that can help you to save time answering the question.
First, any square or rectangle that is inscribed in a circle will have a diagonal that IS the diameter of the circle. Second, when splitting a rectangle or square across its diagonal, you'll form two right triangles. With the given side lengths of the rectangle (5 and 12), we have a 5/12/13 right triangle, so we know the diameter of the circle is 13. The radius is half the diameter... radius = 6.5
Area of a circle = (Ï€)(R^2) = (Ï€)(6.5^2)
At this point, you don't actually have to calculate the value. Since 6^2 = 36 and 7^2 = 49, we know the correct answer has to be between 36Ï€ and 49Ï€. There's only one answer that fits...
Final Answer: B
GMAT assassins aren't born, they're made,
Rich
We're told that a rectangle with sides of 5 and 12 is inscribed in the circle. We're asked for the area of the circular region. This question is based on a number of math patterns that can help you to save time answering the question.
First, any square or rectangle that is inscribed in a circle will have a diagonal that IS the diameter of the circle. Second, when splitting a rectangle or square across its diagonal, you'll form two right triangles. With the given side lengths of the rectangle (5 and 12), we have a 5/12/13 right triangle, so we know the diameter of the circle is 13. The radius is half the diameter... radius = 6.5
Area of a circle = (Ï€)(R^2) = (Ï€)(6.5^2)
At this point, you don't actually have to calculate the value. Since 6^2 = 36 and 7^2 = 49, we know the correct answer has to be between 36Ï€ and 49Ï€. There's only one answer that fits...
Final Answer: B
GMAT assassins aren't born, they're made,
Rich