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GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta

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Bunuel
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GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta [#permalink] GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta   21 Jul 2022, 08:00
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A
B
C
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55% (hard)

Question Stats:

59% (02:03) correct 41% (02:04) wrong based on 87 sessions
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Two circles are inscribed in a rectangle, as shown above. What is the distance between the centers of these circles?

A. 2
B. \(\sqrt{5}\)
C. 3
D. \(\sqrt{10}\)
E. \(2\sqrt{5}\)

 


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B
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Re: GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta [#permalink] GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta   21 Jul 2022, 08:05
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Bunuel wrote:

Two circles are inscribed in a rectangle, as shown above. What is the distance between the centers of these circles?

A. 2
B. \(\sqrt{5}\)
C. 3
D. \(\sqrt{10}\)
E. \(2\sqrt{5}\)

 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

Compete, Get Better, Win prizes and more

 



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Untitled.png


The diameter of each circle is 2. If the two circles were side-by-side and tangential, their centers would be separated by 2r = d= 2. But they are offset a bit vertically, so their centers are farther apart than 2. A is out. Do we want something a little more than 2 or a lot more than 2? Just a little. Certainly not more than 1.5x. C, D, and E are out.

Answer choice B.
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Re: GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta [#permalink] GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta   21 Jul 2022, 08:17
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the value of digonal is 3^2+4^2 ; √25 ; 5
area of the half of the rectangle 3*4/2 ; 6
let xy be the distance of two center circle
xy = √( 3-2r)^2 + ( 4-2r)^2
we can say three ∆ area
3r/2 + 4r/2 + 5r/2
12r/2 ; 6r
6r=6
r=1
substitute r in √( 3-2r)^2 + ( 4-2r)^2
xy = √(1+4) ; √5
OPTION B \(\sqrt{5}\)


Bunuel wrote:

Two circles are inscribed in a rectangle, as shown above. What is the distance between the centers of these circles?

A. 2
B. \(\sqrt{5}\)
C. 3
D. \(\sqrt{10}\)
E. \(2\sqrt{5}\)

 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

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Re: GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta [#permalink] GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta   21 Jul 2022, 08:25
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Hello everyone,

The correct answer is B.
The detailed explanation of the solution is in the pic. attached
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1.jpg
1.jpg [ 98.94 KiB | Viewed 2570 times ]

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Re: GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta [#permalink] GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta   21 Jul 2022, 08:33
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Bunuel wrote:

Two circles are inscribed in a rectangle, as shown above. What is the distance between the centers of these circles?

A. 2
B. \(\sqrt{5}\)
C. 3
D. \(\sqrt{10}\)
E. \(2\sqrt{5}\)

 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

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Attachment:
The attachment Untitled.png is no longer available


Refering to the picture, 3R (in red) = 3; R=1

Also, 4-2R=2.

Distance between the center will be 1ˆ2+2ˆ2= Dˆ2,

So D = \sqrt{5}

Option B is the correct answer
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WhatsApp Image 2022-07-21 at 8.59.06 PM.jpeg
WhatsApp Image 2022-07-21 at 8.59.06 PM.jpeg [ 69.7 KiB | Viewed 2526 times ]

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Re: GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta [#permalink] GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta   21 Jul 2022, 09:29
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In the figure below we extend the circle's radius to meet the sides and construct a right angled triangle as shown.

Attachment:
Screenshot 2022-07-21 214307.png
Screenshot 2022-07-21 214307.png [ 93.97 KiB | Viewed 2395 times ]


In \(\triangle ABC\) = \(\sqrt{(3-2r)^2 + (4-2r)^2}\) ----------- (1)

Now lets construct three triangles as shown below -

Attachment:
Screenshot 2022-07-21 214530.png
Screenshot 2022-07-21 214530.png [ 80.69 KiB | Viewed 2391 times ]


PM = 5 (3 - 4 -5 triplet)

Area of \(\triangle MPN\) = Area( \(\triangle MOP\) ) + Area( \(\triangle PON\) ) + Area( \(\triangle MON\) )

\(\frac{(4*3)}{2}\) = \(\frac{1}{2} * 4 * r +\frac{1}{2} * 3 * r + \frac{1}{2} * 5 * r\) = 6r

r = 1 -------------------- (2)

Substituting 2 in 1

AB =\( \sqrt{1^2 + 2^2}\) = \(\sqrt{5}\)

IMO B
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Re: GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta [#permalink] GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta   21 Jul 2022, 22:14
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Bunuel wrote:

Two circles are inscribed in a rectangle, as shown above. What is the distance between the centers of these circles?

A. 2
B. \(\sqrt{5}\)
C. 3
D. \(\sqrt{10}\)
E. \(2\sqrt{5}\)

 


This question was provided by GMAT Club
for the GMAT Club World Cup Competition

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Attachment:
The attachment Untitled.png is no longer available


Please refer to the attachment for the diagram and solution
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Circle.jpg
Circle.jpg [ 173.6 KiB | Viewed 705 times ]

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Re: GMAT Club World Cup 2022 (DAY 9): Two circles are inscribed in a recta [#permalink]
21 Jul 2022, 22:14
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Bunuel
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