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If rectangle ABCD is inscribed in the circle above, what is

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If rectangle ABCD is inscribed in the circle above, what is the area of the circular region?

A) 36.00Ï€
B) 42.25Ï€
C) 64.00Ï€
D) 84.50Ï€
E) 169.00Ï€

B

Source: Official Guide 2020

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by Brent@GMATPrepNow » Sat May 04, 2019 6:01 am
AbeNeedsAnswers wrote:Image

If rectangle ABCD is inscribed in the circle above, what is the area of the circular region?

A) 36.00Ï€
B) 42.25Ï€
C) 64.00Ï€
D) 84.50Ï€
E) 169.00Ï€

B

Source: Official Guide 2020
Draw a line connecting points A and C.
Image
An important circle property (see video below for more info) tells us that, if we have a 90-degree 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 5-12-13
Image
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 = πr²
So, area = π(6.5²)
= 42.25Ï€

Answer: B

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by swerve » Sat May 04, 2019 10:36 am
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__

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by Rich.C@EMPOWERgmat.com » Wed May 22, 2019 8:46 am
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
Contact Rich at Rich.C@empowergmat.com
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