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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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Bunuel

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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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 3386 times ]

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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 3238 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 3238 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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Bunuel

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

 



Attachment:
The attachment Untitled.png is no longer available

Please refer to the attachment for the diagram and solution
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