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# A square is inscribed in a big circle and a small circle is inscribed

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Math Expert
Joined: 02 Sep 2009
Posts: 64951
A square is inscribed in a big circle and a small circle is inscribed  [#permalink]

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02 Jun 2020, 04:28
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Difficulty:

45% (medium)

Question Stats:

74% (01:53) correct 26% (01:09) wrong based on 26 sessions

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A square is inscribed in a big circle and a small circle is inscribed in the square as shown in the figure above. (All of them are concentric and their center is the point O). If the radius of the big circle is √2 cm, what is the perimeter of the small circle?

A. 1
B. 2
C. 3
D. π
E. 2π

Attachment:

1.png [ 9.72 KiB | Viewed 319 times ]

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Re: A square is inscribed in a big circle and a small circle is inscribed  [#permalink]

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02 Jun 2020, 04:39
Bunuel wrote:

A square is inscribed in a big circle and a small circle is inscribed in the square as shown in the figure above. (All of them are concentric and their center is the point O). If the radius of the big circle is √2 cm, what is the perimeter of the small circle?

A. 1
B. 2
C. 3
D. π
E. 2π

Attachment:
1.png

IMO E.

Sol
Radius of outer circle=root2 that means diameter=2root2
the diagonal of the square=2root2 so the side must be 2
the side of the square is equalt to the diameter of teh inner circle so 1
and the permeter=2PiR=2Pi.
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Joined: 21 May 2020
Posts: 9
Re: A square is inscribed in a big circle and a small circle is inscribed  [#permalink]

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02 Jun 2020, 04:49
Diagonal of square = diameter of bigger circle = 2(root 2)
Side of square = 2 [if diagonal is 2(root 2)]
Side of square is diameter of smaller circle = 2
Perimeter of smaller circle is pi(diameter) = 2pi

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Re: A square is inscribed in a big circle and a small circle is inscribed  [#permalink]

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02 Jun 2020, 04:53
Bunuel wrote:

A square is inscribed in a big circle and a small circle is inscribed in the square as shown in the figure above. (All of them are concentric and their center is the point O). If the radius of the big circle is √2 cm, what is the perimeter of the small circle?

A. 1
B. 2
C. 3
D. π
E. 2π

Attachment:
1.png

Radius fo Small Circle, r = Side of square = a
Radius fo Big Circle, R = Diagonal of square = a√2

i.e. $$\frac{r}{R} = \frac{a}{a√2}$$

i.e. $$r = (\frac{1}{√2})*√2 = 1$$

$$Circumference = 2πr = 2π$$

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Re: A square is inscribed in a big circle and a small circle is inscribed  [#permalink]

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02 Jun 2020, 05:22
Bunuel wrote:

A square is inscribed in a big circle and a small circle is inscribed in the square as shown in the figure above. (All of them are concentric and their center is the point O). If the radius of the big circle is √2 cm, what is the perimeter of the small circle?

A. 1
B. 2
C. 3
D. π
E. 2π

Attachment:
1.png

Solution

• Let us assume that the side of the square is a.
• Now, Diameter of the big circle = diagonal of the square

o $$⟹2* \sqrt{2} = a*\sqrt{2} ⟹ a = 2$$
• Also, side of the square = diameter of the small circle
o Diameter of the small circle $$= a = 2$$
• Thus, the circumference of the small circle $$= \pi* diameter = 2*\pi$$
Thus, the correct answer is Option E.
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Re: A square is inscribed in a big circle and a small circle is inscribed  [#permalink]

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02 Jun 2020, 07:21
IMO E.

Diameter of the big circle = Diagonal of square

and Side of square = \sqrt{2} Side of square

Using above we can solve this.
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WhatsApp Image 2020-06-02 at 5.18.51 PM.jpeg [ 167.9 KiB | Viewed 170 times ]

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Re: A square is inscribed in a big circle and a small circle is inscribed   [#permalink] 02 Jun 2020, 07:21