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

Question

https://gmatclub.com/forum/download/file.php?id=47579 If rectangle ABCD is inscribed in the circle above, what is the area of the circular region? A. 36.00π B. 42.25π C. 64.00π D. 84.50π E. 169.00π PS77502.01 OG2020 NEW QUESTION 2019-04-26_1231.png

BrentGMATPrepNow · Accepted Answer

Draw a line connecting points A and C. 
 https://i.imgur.com/fQaxM5q.png 
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 
 https://i.imgur.com/zXIBPKP.png 
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( 6.5 ²)
= 42.25π

Answer: B

RELATED VIDEO
 https://www.youtube.com/watch?v=xiyv2MacIUk

EMPOWERgmatRichC · Answer

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:  B

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

Chethan92 · Answer

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.

Archit3110 · Answer

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

EgmatQuantExpert · Answer

Solution

https://e-gmat.com/blogs/wp-content/uploads/2019/05/3.png

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

ScottTargetTestPrep · Answer

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

Ashank07 · Answer

A good solution of the above question:

https://www.youtube.com/watch?v=WCUcLIhuhX0

DipG · Answer

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

vanam52923 · Answer

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?

EMPOWERgmatRichC · Answer

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

PS91 · Answer

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

MHIKER · Answer

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.

GMATinsight · Answer

Answer: Option B

Video solution by  GMATinsight

https://www.youtube.com/watch?v=m8g_B7ipg-o

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=r2ao5maFNb4

bumpbot · Answer

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

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards