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705-805 (Hard)|   Geometry|      
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alex90
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The Quadrilateral shown above is a square. Four circles are Updated on: 09 Sep 2019, 08:08
Originally posted by alex90 on 26 Apr 2013, 12:12.
Last edited by Gladiator59 on 09 Sep 2019, 08:08, edited 1 time in total.
Question stem and options fixed.
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75% (hard)

Question Stats:

58% (02:58) correct 42% (02:47) wrong based on 293 sessions

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The Quadrilateral shown above is a square. Four circles are tangent to the sides of the square and the small circle in the centre is tangent to each of the four circles. What is the ratio of the radius of the small circle to the side of the square?
a \(\frac{1}{2}\)
b \(\frac{1}{4}(\sqrt{2} -1)\)
c \(\frac{1}{2}(\sqrt{2}-1)\)
d \(\sqrt{2} -1\)
e \(2 (\sqrt{2}-2)\)
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B

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Zarrolou
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The Quadrilateral shown above is a square. Four circles are 26 Apr 2013, 12:39
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alex90
The Quadrilateral shown above is a square. Four circles are tangent to the sides of the square and the small circle in the centre is tangent to each of the four circles. What is the ratio of the small circle to the side of the square?
a 1/2
b 1/4(\sqrt{2} -1)
c 1/2(\sqrt{2}-1)
d \sqrt{2} -1
e 2 (\sqrt{2}-2)
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Lets assume that the radius of the big circle is 1.
Create a square connecting all the centers (look picture). Its side is 2, it diagonal is \(2\sqrt{2}\), this if formed by 2 radius and the diameter of the smaller circle
\(2\sqrt{2}-2(=2*radiusBig)=diameterSmall\) \(2(\sqrt{2}-1)=diamSmall\)
so radius small = \(\sqrt{2}-1\)

The ratio radius to side square (= 4) is \((\sqrt{2}-1)/4\)
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hitman5532
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The Quadrilateral shown above is a square. Four circles are 26 Apr 2013, 15:45
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alex90
The Quadrilateral shown above is a square. Four circles are tangent to the sides of the square and the small circle in the centre is tangent to each of the four circles. What is the ratio of the small circle to the side of the square?
a 1/2
b 1/4(\sqrt{2} -1)
c 1/2(\sqrt{2}-1)
d \sqrt{2} -1
e 2 (\sqrt{2}-2)
Show more


Is this question copied from the OG verbatim?

The section I highlighted in red seems to fail to aknowledge what ratio they asking (diameter of small circle vs side of square? Circumference of circle vs side of square? etc)
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The Quadrilateral shown above is a square. Four circles are 29 Apr 2013, 02:45
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I considered it as ratio of radius of small circle to the side of the square (yes, the question lacks clarity as to what ratio asked for)

Considering 2 as side of the square

(sqrt2 - 1)/2 = Radius of small circle
2 = Side of Square;Solving it further, we get ans = 1/(4.sqrt2-1)
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The Quadrilateral shown above is a square. Four circles are 01 May 2015, 03:11
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the wording of this problem is bad. it is not from og.

ok, the concept tested is simple.
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The Quadrilateral shown above is a square. Four circles are 08 Sep 2019, 11:22
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hitman5532
alex90
The Quadrilateral shown above is a square. Four circles are tangent to the sides of the square and the small circle in the centre is tangent to each of the four circles. What is the ratio of the small circle to the side of the square?
a 1/2
b 1/4(\sqrt{2} -1)
c 1/2(\sqrt{2}-1)
d \sqrt{2} -1
e 2 (\sqrt{2}-2)


Is this question copied from the OG verbatim?

The section I highlighted in red seems to fail to aknowledge what ratio they asking (diameter of small circle vs side of square? Circumference of circle vs side of square? etc)
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Mods: Gladiator59 generis

Could we please correct the wording of the Q and the formatting of answer choices? Thanks!

ratio of the small circle to the side of the square => "ratio of the radius of small circle to the side of the square"
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The Quadrilateral shown above is a square. Four circles are 09 Sep 2019, 08:09
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dabaobao thanks for pointing out. Fixed it.
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The Quadrilateral shown above is a square. Four circles are 08 Nov 2019, 12:00
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alex90
The Quadrilateral shown above is a square. Four circles are tangent to the sides of the square and the small circle in the centre is tangent to each of the four circles. What is the ratio of the radius of the small circle to the side of the square?
a \(\frac{1}{2}\)
b \(\frac{1}{4}(\sqrt{2} -1)\)
c \(\frac{1}{2}(\sqrt{2}-1)\)
d \(\sqrt{2} -1\)
e \(2 (\sqrt{2}-2)\)
Show more

Here's an alternative way to solve this question with very little math, but you need to draw the diagram as accurate as possible, and know that the \(\sqrt{2}\) = 1.41 ish

Right from the start, you can eliminate Answer Choice E because it gives you a negative value
Visually, you can see that about 10 radii from the smaller circle can fit into one side of the square.

The only ratio that gives us close to .1 is \(\frac{1}{4}(\sqrt{2} -1)\)

B is the Answer!
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The Quadrilateral shown above is a square. Four circles are 24 Dec 2020, 15:23
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