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Math Expert V
Joined: 02 Sep 2009
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If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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16 00:00

Difficulty:   25% (medium)

Question Stats: 73% (01:29) correct 27% (01:28) wrong based on 1007 sessions

### HideShow timer Statistics 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

Attachment: 2019-04-26_1231.png [ 12.21 KiB | Viewed 11271 times ]

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Joined: 11 Sep 2015
Posts: 4886
GMAT 1: 770 Q49 V46
Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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Top Contributor
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Bunuel wrote: 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

Attachment:
2019-04-26_1231.png

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

RELATED VIDEO FROM MY COURSE

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##### General Discussion
NUS School Moderator V
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Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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2
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 = $$π*r^2 = 42.5π$$

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Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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diagonal of figure ; 13
area = 6.5^2 * pi = 42.25 pi
IMO B
Bunuel wrote: 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

Attachment:
2019-04-26_1231.png
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 16768
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GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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1
1
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...

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

Originally posted by EMPOWERgmatRichC on 27 Apr 2019, 15:57.
Last edited by EMPOWERgmatRichC on 22 May 2019, 08:51, edited 1 time in total.
e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3394
Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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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.

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Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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Bunuel wrote: 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

Attachment:
2019-04-26_1231.png

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.

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Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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Bunuel wrote: 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

Attachment:
2019-04-26_1231.png

A good solution of the above question:

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Joined: 19 Sep 2018
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Concentration: Strategy, Technology
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Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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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
Manager  S
Joined: 17 Jul 2017
Posts: 244
Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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GMATPrepNow wrote:
Bunuel wrote: 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

Attachment:
2019-04-26_1231.png

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

RELATED VIDEO FROM MY COURSE

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?
EMPOWERgmat Instructor V
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
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Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: If rectangle ABCD is inscribed in the circle above, what is the area  [#permalink]

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1
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
_________________ Re: If rectangle ABCD is inscribed in the circle above, what is the area   [#permalink] 04 Apr 2020, 14:02

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