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 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

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)
= 42.25π

Answer: B

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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 = 5/2 = 2/5 and XB = 12/2 = 6
Triangle OXB is a right angles triangle.
OB = radius = \(\sqrt{6^2+2.5^2} = \sqrt{42.5}\)

Area = \(π*r^2 = 42.5π\)

B is the answer.

diagonal of figure ; 13
so radis - 6.5
area = 6.5^2 * pi = 42.25 pi
IMO B

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 = (π)(R^2) = (π)(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 36π and 49π. There's only one answer that fits...

Final Answer:

GMAT assassins aren't born, they're made,
Rich
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Solution

Given:

• ABCD is a rectangle.

To Find:

• Area of circle.

Approach and Working:

• Since ABCD is a rectangle, \(AC^2\) = \(AB^2\) + \(BC^2\)

o \(AC^2\) = \(12^2\) + \(5^2\) = 144 +25 = 169

 AC = 13

• ∆ ABC is a right triangle and by applying the property that angle in a semi-circle is 90 degrees.

o We can conclude that AC is the diameter of the triangle.
o Hence, radius of circle = \(\frac{13}{2}\)

• Therefore, area of circle = π * \((\frac{13}{2})^2\) = (\(\frac{169}{4}\)) π =42.25 π

Hence, option B is the correct answer.

Correct Answer: B
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We see that diagonal AC (of the rectangle) is also the diameter of the circle. When we make diagonal AC, we see that we create a right triangle, and more specifically a 5-12-13 right triangle, so the length of AC is 13.

Since the diameter is 13, the radius is 6.5, and the area of the circle is (6.5)^2 x π = 42.25π.

Answer: B
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A good solution of the above question:

Diagonal of ABCD = 13 (Using Pythagoras theorem)

Hence, Diameter of circle = Diagonal = 13 | Radius = 6.5

Area of circle = pi * r^2 = pi * (13/2) * (13/2) = 42.25 pi

can u plz clear this ?
the angle formed by diameter at circum is 90 degrees

but here we know angle is 90 degree so we are saying thtt chor di diamerter
so my qsn is ,is collary also true?

Hi vanam52923,

Yes - the corollary is ALSO true. If you have a right triangle inscribed in a circle, then the hypotenuse of the triangle is the diameter of the circle.

GMAT assassins aren't born, they're made,
Rich
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The diagonal of the rectangle is the diameter of the circle because the half triangle is the right angle.
So, diameter = 13
radius = 13/2
area of circle = pi r*r = 169/4 pi = 42.25pi

The Pythagorean triplets formula \(5:12:13\) is applied here. \(So, \ the \ Hypotenuse \ or \ Diameter \ is \ 13.\)

Radius\( = \frac{13}{2}=6.5\)

The area of the circular region\(=πr^2=π(6.5)^2=42.25π\)

The answer is \(B. \)
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Answer: Option B

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